diff --git a/app.py b/app.py index 4c31ea23e70d6a7c63935668845481f7e024c8a1..efb00a8bee2853b791a58acf877ebea4ae8562de 100644 --- a/app.py +++ b/app.py @@ -941,9 +941,13 @@ def visual_provenance(record: dict[str, Any], visual_record: dict[str, Any]) -> if visual_record.get("frontend_seed"): visual_metric = metric_value(visual_record) shown_metric = metric_value(record) + notes = str(visual_record.get("notes") or "").lower() suffix = "" if visual_metric != shown_metric: - suffix = f" Rendering metric: {esc(metric_symbol(visual_record))} = {esc(visual_metric)}; displayed record metric: {esc(metric_symbol(record))} = {esc(shown_metric)}." + suffix = f" Verified coordinate metric: {esc(metric_symbol(visual_record))} = {esc(visual_metric)}; Friedman reported metric: {esc(metric_symbol(record))} = {esc(shown_metric)}." + if "image-seeded" in notes or "image seeded" in notes: + status = "recovered within tolerance" if not needs_recovery(record, visual_record) else "still outside recovery tolerance" + return f"Image-seeded from the Friedman source image and verified ({status})." + suffix return "Generated feasible coordinate rendering, not recovered from the Friedman source image." + suffix return "Coordinate rendering attached to this case." diff --git a/data/records.jsonl b/data/records.jsonl index c9ff16dd0b5ba6e924819c595b694a597dc51778..8e5c1bc4170812ddc2b883a31402a008b150b2d6 100644 --- a/data/records.jsonl +++ b/data/records.jsonl @@ -7,7 +7,7 @@ {"case": "cirincir@7", "density": 0.77774147574625, "friedman_analytical_or_proved": true, "friedman_credit": "Graham", "friedman_date": "1968", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc7.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "3", "friedman_metric_symbol": "r", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@7", "credit": "Graham", "date": "1968", "family_updated": "6/27/05", "metric_expression": "3", "metric_symbol": "r", "metric_value": 3.0, "reference_text": "r = 3 | Proved by Graham in 1968.", "source_image": "ccc7.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc7.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 3 | Proved by Graham in 1968.", "friedman_source_image": "ccc7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "2b1985e4094e6893", "improvement": -7.001351167312819e-05, "max_boundary_excess": -6.2947164267690425e-06, "max_pair_overlap_depth": -6.231680049051391e-05, "metric_expression": "3.00007", "metric_symbol": "r", "metric_value": 3.000070013511673, "n": 7, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 3.0, "setup": "cirincir", "side": 6.000140027023346, "solution_path": "data/solutions/generated/cirincir_007_2b1985e4094e6893.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/2b1985e4094e6893.svg", "tolerance": 1e-08, "verified": true} {"case": "cirincir@8", "density": 0.7324681263291527, "friedman_analytical_or_proved": true, "friedman_credit": "Braaksma", "friedman_date": "1963", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc8.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "3.304+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.304, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@8", "credit": "Braaksma", "date": "1963", "family_updated": "6/27/05", "metric_expression": "3.304+", "metric_symbol": "r", "metric_value": 3.304, "reference_text": "r = 3.304+ | Proved by Braaksma in 1963.", "source_image": "ccc8.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc8.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 3.304+ | Proved by Braaksma in 1963.", "friedman_source_image": "ccc8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "0193661647fd3854", "improvement": -0.0008414425517750956, "max_boundary_excess": -7.96872891495326e-06, "max_pair_overlap_depth": -4.3191348294779885e-05, "metric_expression": "3.30484", "metric_symbol": "r", "metric_value": 3.304841442551775, "n": 8, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 3.304, "setup": "cirincir", "side": 6.60968288510355, "solution_path": "data/solutions/generated/cirincir_008_0193661647fd3854.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/0193661647fd3854.svg", "tolerance": 1e-08, "verified": true} {"case": "cirincir@9", "density": 0.6893878045655164, "friedman_analytical_or_proved": true, "friedman_credit": "Pirl", "friedman_date": "1969", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc9.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "3.613+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.613, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@9", "credit": "Pirl", "date": "1969", "family_updated": "6/27/05", "metric_expression": "3.613+", "metric_symbol": "r", "metric_value": 3.613, "reference_text": "r = 3.613+ | Proved by Pirl in 1969.", "source_image": "ccc9.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc9.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 3.613+ | Proved by Pirl in 1969.", "friedman_source_image": "ccc9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "b51df67181f6833b", "improvement": -0.00017882622267562098, "max_boundary_excess": -4.656082665643879e-06, "max_pair_overlap_depth": -2.396342999144707e-05, "metric_expression": "3.61318", "metric_symbol": "r", "metric_value": 3.6131788262226756, "n": 9, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 3.613, "setup": "cirincir", "side": 7.226357652445351, "solution_path": "data/solutions/generated/cirincir_009_b51df67181f6833b.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/b51df67181f6833b.svg", "tolerance": 1e-08, "verified": true} -{"case": "cirincir@10", "density": 0.6877646887506464, "friedman_analytical_or_proved": true, "friedman_credit": "Pirl", "friedman_date": "1969", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc10.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "3.813+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.813, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@10", "credit": "Pirl", "date": "1969", "family_updated": "6/27/05", "metric_expression": "3.813+", "metric_symbol": "r", "metric_value": 3.813, "reference_text": "r = 3.813+ | Proved by Pirl in 1969.", "source_image": "ccc10.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc10.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 3.813+ | Proved by Pirl in 1969.", "friedman_source_image": "ccc10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "fd8234924faa6cc7", "improvement": -0.00011639924422013337, "max_boundary_excess": -7.647318408299242e-06, "max_pair_overlap_depth": -4.6116421250630424e-05, "metric_expression": "3.81312", "metric_symbol": "r", "metric_value": 3.8131163992442203, "n": 10, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 3.813, "setup": "cirincir", "side": 7.626232798488441, "solution_path": "data/solutions/generated/cirincir_010_fd8234924faa6cc7.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/fd8234924faa6cc7.svg", "tolerance": 1e-08, "verified": true} +{"case": "cirincir@10", "density": 0.6877649990272218, "friedman_analytical_or_proved": true, "friedman_credit": "Pirl", "friedman_date": "1969", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc10.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "3.813+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.813, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@10", "credit": "Pirl", "date": "1969", "family_updated": "6/27/05", "metric_expression": "3.813+", "metric_symbol": "r", "metric_value": 3.813, "reference_text": "r = 3.813+ | Proved by Pirl in 1969.", "source_image": "ccc10.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc10.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 3.813+ | Proved by Pirl in 1969.", "friedman_source_image": "ccc10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "be14c55679a54b28", "improvement": -0.00011553912424266954, "max_boundary_excess": -4.081326946625552e-06, "max_pair_overlap_depth": -3.967607896537828e-05, "metric_expression": "3.81312", "metric_symbol": "r", "metric_value": 3.813115539124243, "n": 10, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 3.813, "setup": "cirincir", "side": 7.626231078248486, "solution_path": "data/solutions/generated/cirincir_10_be14c55679a54b28.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/be14c55679a54b28.svg", "tolerance": 1e-08, "verified": true} {"case": "cirincir@11", "density": 0.714418364520267, "friedman_analytical_or_proved": true, "friedman_credit": "Melissen", "friedman_date": "1994", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc11.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "3.923+", "friedman_metric_symbol": "r", "friedman_metric_value": 3.923, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@11", "credit": "Melissen", "date": "1994", "family_updated": "6/27/05", "metric_expression": "3.923+", "metric_symbol": "r", "metric_value": 3.923, "reference_text": "r = 3.923+ | Proved by Melissen in 1994.", "source_image": "ccc11.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc11.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 3.923+ | Proved by Melissen in 1994.", "friedman_source_image": "ccc11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "e624f391927921fb", "improvement": -0.0009190348380849755, "max_boundary_excess": -2.92009600810772e-06, "max_pair_overlap_depth": -4.346382940223492e-05, "metric_expression": "3.92392", "metric_symbol": "r", "metric_value": 3.923919034838085, "n": 11, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 3.923, "setup": "cirincir", "side": 7.84783806967617, "solution_path": "data/solutions/generated/cirincir_011_e624f391927921fb.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/e624f391927921fb.svg", "tolerance": 1e-08, "verified": true} {"case": "cirincir@12", "density": 0.7389828254275508, "friedman_analytical_or_proved": false, "friedman_credit": "Kravitz", "friedman_date": "1967", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc12.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "4.029+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.029, "friedman_reference": {"analytical_or_proved": false, "case": "cirincir@12", "credit": "Kravitz", "date": "1967", "family_updated": "6/27/05", "metric_expression": "4.029+", "metric_symbol": "r", "metric_value": 4.029, "reference_text": "r = 4.029+ | Found by Kravitz in 1967.", "source_image": "ccc12.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc12.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 4.029+ | Found by Kravitz in 1967.", "friedman_source_image": "ccc12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "0d7fc8a14b1c4e0e", "improvement": -0.0007068210362959348, "max_boundary_excess": -8.338373973515445e-06, "max_pair_overlap_depth": -5.019209633871924e-05, "metric_expression": "4.02971", "metric_symbol": "r", "metric_value": 4.029706821036296, "n": 12, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 4.029, "setup": "cirincir", "side": 8.059413642072592, "solution_path": "data/solutions/generated/cirincir_012_0d7fc8a14b1c4e0e.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/0d7fc8a14b1c4e0e.svg", "tolerance": 1e-08, "verified": true} {"case": "cirincir@13", "density": 0.7244255598642273, "friedman_analytical_or_proved": true, "friedman_credit": "F", "friedman_date": "2003", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc13.gif", "friedman_family_updated": "6/27/05", "friedman_metric_expression": "2 + \u221a5 = 4.236+", "friedman_metric_symbol": "r", "friedman_metric_value": 4.236, "friedman_reference": {"analytical_or_proved": true, "case": "cirincir@13", "credit": "F", "date": "2003", "family_updated": "6/27/05", "metric_expression": "2 + \u221a5 = 4.236+", "metric_symbol": "r", "metric_value": 4.236, "reference_text": "r = 2 + \u221a5 = 4.236+ | Proved by F. Fodor in 2003.", "source_image": "ccc13.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc13.gif", "source_page": "https://erich-friedman.github.io/packing/cirincir/", "title": "Circles in Circles"}, "friedman_reference_text": "r = 2 + \u221a5 = 4.236+ | Proved by F. Fodor in 2003.", "friedman_source_image": "ccc13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirincir/", "friedman_title": "Circles in Circles", "frontend_seed": true, "id": "a118f33872e0cb30", "improvement": -0.00018378579659206196, "max_boundary_excess": -4.324667153277062e-06, "max_pair_overlap_depth": -4.7467215169572086e-05, "metric_expression": "4.23618", "metric_symbol": "r", "metric_value": 4.236183785796592, "n": 13, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 4.236, "setup": "cirincir", "side": 8.472367571593184, "solution_path": "data/solutions/generated/cirincir_013_a118f33872e0cb30.json", "source_url": "https://erich-friedman.github.io/packing/cirincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/a118f33872e0cb30.svg", "tolerance": 1e-08, "verified": true} @@ -75,7 +75,7 @@ {"case": "cirinsqu@7", "density": 0.669268877528072, "friedman_analytical_or_proved": true, "friedman_credit": "Schaer", "friedman_date": "1964", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs7.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "4 + \u221a3 = 5.732+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.732, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@7", "credit": "Schaer", "date": "1964", "family_updated": "10/9/10", "metric_expression": "4 + \u221a3 = 5.732+", "metric_symbol": "s", "metric_value": 5.732, "reference_text": "s = 4 + \u221a3 = 5.732+ | Proved by Schaer in 1964.", "source_image": "cs7.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs7.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 4 + \u221a3 = 5.732+ | Proved by Schaer in 1964.", "friedman_source_image": "cs7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "7e4d3361af467e5d", "improvement": -0.0002304452542292168, "max_boundary_excess": -2.33238067148811e-06, "max_pair_overlap_depth": -6.633595249638802e-05, "metric_expression": "5.73223", "metric_symbol": "s", "metric_value": 5.732230445254229, "n": 7, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 5.732, "setup": "cirinsqu", "side": 5.732230445254229, "solution_path": "data/solutions/generated/cirinsqu_007_7e4d3361af467e5d.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/7e4d3361af467e5d.svg", "tolerance": 1e-08, "verified": true} {"case": "cirinsqu@8", "density": 0.7309263604724501, "friedman_analytical_or_proved": true, "friedman_credit": "Schaer/Meir", "friedman_date": "1964", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs8.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "2 + \u221a2 + \u221a6 = 5.863+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.863, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@8", "credit": "Schaer/Meir", "date": "1964", "family_updated": "10/9/10", "metric_expression": "2 + \u221a2 + \u221a6 = 5.863+", "metric_symbol": "s", "metric_value": 5.863, "reference_text": "s = 2 + \u221a2 + \u221a6 = 5.863+ | Proved by Schaer/Meir in 1964.", "source_image": "cs8.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs8.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 2 + \u221a2 + \u221a6 = 5.863+ | Proved by Schaer/Meir in 1964.", "friedman_source_image": "cs8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "67f6750a9af94d2c", "improvement": -0.0008535799033406732, "max_boundary_excess": -2.247320970871769e-06, "max_pair_overlap_depth": -5.202375940349668e-05, "metric_expression": "5.86385", "metric_symbol": "s", "metric_value": 5.863853579903341, "n": 8, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 5.863, "setup": "cirinsqu", "side": 5.863853579903341, "solution_path": "data/solutions/generated/cirinsqu_008_67f6750a9af94d2c.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/67f6750a9af94d2c.svg", "tolerance": 1e-08, "verified": true} {"case": "cirinsqu@9", "density": 0.7853720614609764, "friedman_analytical_or_proved": true, "friedman_credit": "Schaer", "friedman_date": "1964", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs9.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@9", "credit": "Schaer", "date": "1964", "family_updated": "10/9/10", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "s = 6 | Proved by Schaer in 1964.", "source_image": "cs9.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs9.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 6 | Proved by Schaer in 1964.", "friedman_source_image": "cs9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "a35561fa5a518c8b", "improvement": -9.970453832330861e-05, "max_boundary_excess": -6.26657331825875e-06, "max_pair_overlap_depth": -3.06706256441025e-05, "metric_expression": "6.00010", "metric_symbol": "s", "metric_value": 6.000099704538323, "n": 9, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 6.0, "setup": "cirinsqu", "side": 6.000099704538323, "solution_path": "data/solutions/generated/cirinsqu_009_a35561fa5a518c8b.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/a35561fa5a518c8b.svg", "tolerance": 1e-08, "verified": true} -{"case": "cirinsqu@10", "density": 0.6900071476974307, "friedman_analytical_or_proved": true, "friedman_credit": "De Groot", "friedman_date": "1990", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs10.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "6.747+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.747, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@10", "credit": "De Groot", "date": "1990", "family_updated": "10/9/10", "metric_expression": "6.747+", "metric_symbol": "s", "metric_value": 6.747, "reference_text": "s = 6.747+ | Proved by De Groot in 1990.", "source_image": "cs10.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs10.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 6.747+ | Proved by De Groot in 1990.", "friedman_source_image": "cs10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "f12d4a2caa2e5d63", "improvement": -0.0005815422964463579, "max_boundary_excess": -5.479144465470043e-06, "max_pair_overlap_depth": -3.9606260198166154e-05, "metric_expression": "6.74758", "metric_symbol": "s", "metric_value": 6.747581542296446, "n": 10, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 6.747, "setup": "cirinsqu", "side": 6.747581542296446, "solution_path": "data/solutions/generated/cirinsqu_010_f12d4a2caa2e5d63.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/f12d4a2caa2e5d63.svg", "tolerance": 1e-08, "verified": true} +{"case": "cirinsqu@10", "density": 0.6900075339157021, "friedman_analytical_or_proved": true, "friedman_credit": "De Groot", "friedman_date": "1990", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs10.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "6.747+", "friedman_metric_symbol": "s", "friedman_metric_value": 6.747, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@10", "credit": "De Groot", "date": "1990", "family_updated": "10/9/10", "metric_expression": "6.747+", "metric_symbol": "s", "metric_value": 6.747, "reference_text": "s = 6.747+ | Proved by De Groot in 1990.", "source_image": "cs10.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs10.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 6.747+ | Proved by De Groot in 1990.", "friedman_source_image": "cs10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "522948d614943781", "improvement": -0.0005796538825402209, "max_boundary_excess": -1.6772848518975536e-06, "max_pair_overlap_depth": -4.0928067794876455e-05, "metric_expression": "6.74758", "metric_symbol": "r", "metric_value": 6.74757965388254, "n": 10, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 6.747, "setup": "cirinsqu", "side": 6.74757965388254, "solution_path": "data/solutions/generated/cirinsqu_10_522948d614943781.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/522948d614943781.svg", "tolerance": 1e-08, "verified": true} {"case": "cirinsqu@11", "density": 0.7006956988559802, "friedman_analytical_or_proved": true, "friedman_credit": "Peikert", "friedman_date": "1991", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs11.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "7.022+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.022, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@11", "credit": "Peikert", "date": "1991", "family_updated": "10/9/10", "metric_expression": "7.022+", "metric_symbol": "s", "metric_value": 7.022, "reference_text": "s = 7.022+ | Proved by Peikert in 1991.", "source_image": "cs11.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs11.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 7.022+ | Proved by Peikert in 1991.", "friedman_source_image": "cs11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "b0ae38086de8cdf9", "improvement": -0.0007394033277021705, "max_boundary_excess": -6.667419736983149e-06, "max_pair_overlap_depth": -2.585792898823769e-05, "metric_expression": "7.02274", "metric_symbol": "s", "metric_value": 7.022739403327702, "n": 11, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 7.022, "setup": "cirinsqu", "side": 7.022739403327702, "solution_path": "data/solutions/generated/cirinsqu_011_b0ae38086de8cdf9.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/b0ae38086de8cdf9.svg", "tolerance": 1e-08, "verified": true} {"case": "cirinsqu@12", "density": 0.7384288108172983, "friedman_analytical_or_proved": true, "friedman_credit": "Peikert", "friedman_date": "1991", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs12.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "2 + 15\u221a(2/17) = 7.144+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.144, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@12", "credit": "Peikert", "date": "1991", "family_updated": "10/9/10", "metric_expression": "2 + 15\u221a(2/17) = 7.144+", "metric_symbol": "s", "metric_value": 7.144, "reference_text": "s = 2 + 15\u221a(2/17) = 7.144+ | Proved by Peikert in 1991.", "source_image": "cs12.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs12.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 2 + 15\u221a(2/17) = 7.144+ | Proved by Peikert in 1991.", "friedman_source_image": "cs12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "e58733a8646795ab", "improvement": -0.0011482300231175913, "max_boundary_excess": -8.939770909988098e-06, "max_pair_overlap_depth": -5.93949466169974e-05, "metric_expression": "7.14515", "metric_symbol": "s", "metric_value": 7.145148230023118, "n": 12, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 7.144, "setup": "cirinsqu", "side": 7.145148230023118, "solution_path": "data/solutions/generated/cirinsqu_012_e58733a8646795ab.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/e58733a8646795ab.svg", "tolerance": 1e-08, "verified": true} {"case": "cirinsqu@13", "density": 0.7332320892098975, "friedman_analytical_or_proved": true, "friedman_credit": "Peikert", "friedman_date": "1991", "friedman_external_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs13.gif", "friedman_family_updated": "10/9/10", "friedman_metric_expression": "7.463+", "friedman_metric_symbol": "s", "friedman_metric_value": 7.463, "friedman_reference": {"analytical_or_proved": true, "case": "cirinsqu@13", "credit": "Peikert", "date": "1991", "family_updated": "10/9/10", "metric_expression": "7.463+", "metric_symbol": "s", "metric_value": 7.463, "reference_text": "s = 7.463+ | Proved by Peikert in 1991.", "source_image": "cs13.gif", "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs13.gif", "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", "title": "Circles in Squares"}, "friedman_reference_text": "s = 7.463+ | Proved by Peikert in 1991.", "friedman_source_image": "cs13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "friedman_title": "Circles in Squares", "frontend_seed": true, "id": "60997371c312edb4", "improvement": -0.0002137620977080701, "max_boundary_excess": -5.323292483705444e-06, "max_pair_overlap_depth": -3.8715255959331785e-05, "metric_expression": "7.46321", "metric_symbol": "s", "metric_value": 7.463213762097708, "n": 13, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 7.463, "setup": "cirinsqu", "side": 7.463213762097708, "solution_path": "data/solutions/generated/cirinsqu_013_60997371c312edb4.json", "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/60997371c312edb4.svg", "tolerance": 1e-08, "verified": true} @@ -618,7 +618,7 @@ {"case": "triincir@13", "density": 0.7678670311765523, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc13.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "\u221a(7/3) = 1.527+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.527, "friedman_reference": {"analytical_or_proved": true, "case": "triincir@13", "credit": "Erich Friedman", "date": "1997", "family_updated": "8/31/19", "metric_expression": "\u221a(7/3) = 1.527+", "metric_symbol": "r", "metric_value": 1.527, "reference_text": "r = \u221a(7/3) = 1.527+ | Found by Erich Friedman | in 1997.", "source_image": "tc13.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc13.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = \u221a(7/3) = 1.527+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "tc13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "2b014eeead965100", "improvement": -0.0005802936296193106, "max_boundary_excess": -3.995535488687807e-06, "max_pair_overlap_depth": -6.278738689302621e-06, "metric_expression": "1.52758", "metric_symbol": "r", "metric_value": 1.5275802936296192, "n": 13, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 1.527, "setup": "triincir", "side": 3.0551605872592384, "solution_path": "data/solutions/generated/triincir_013_2b014eeead965100.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/2b014eeead965100.svg", "tolerance": 1e-08, "verified": true} {"case": "triincir@14", "density": 0.7492543391037492, "friedman_analytical_or_proved": false, "friedman_credit": "DC after MM", "friedman_date": "August 2012", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc14.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.609+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.609, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@14", "credit": "DC after MM", "date": "August 2012", "family_updated": "8/31/19", "metric_expression": "1.609+", "metric_symbol": "r", "metric_value": 1.609, "reference_text": "r = 1.609+ | Found by DC after MM | in August 2012.", "source_image": "tc14.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc14.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.609+ | Found by DC after MM | in August 2012.", "friedman_source_image": "tc14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "da4114b8f6b7e383", "improvement": 0.0041857717254343285, "max_boundary_excess": -1.4204418656849072e-06, "max_pair_overlap_depth": -6.9567849071972e-06, "metric_expression": "1.60481", "metric_symbol": "r", "metric_value": 1.6048142282745657, "n": 14, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 1.609, "setup": "triincir", "side": 3.2096284565491313, "solution_path": "data/solutions/generated/triincir_014_da4114b8f6b7e383.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/da4114b8f6b7e383.svg", "tolerance": 1e-08, "verified": true} {"case": "triincir@15", "density": 0.7708979393444947, "friedman_analytical_or_proved": false, "friedman_credit": "David W", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc15.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.636+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.636, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@15", "credit": "David W", "date": "July 2002", "family_updated": "8/31/19", "metric_expression": "1.636+", "metric_symbol": "r", "metric_value": 1.636, "reference_text": "r = 1.636+ | Found by David W. Cantrell | in July 2002.", "source_image": "tc15.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc15.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.636+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "tc15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "b0b095b0aec160c0", "improvement": -0.0016555780650902019, "max_boundary_excess": -3.6977389892278723e-06, "max_pair_overlap_depth": -3.28910924740633e-06, "metric_expression": "1.63766", "metric_symbol": "r", "metric_value": 1.63765557806509, "n": 15, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 1.636, "setup": "triincir", "side": 3.27531115613018, "solution_path": "data/solutions/generated/triincir_015_b0b095b0aec160c0.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/b0b095b0aec160c0.svg", "tolerance": 1e-08, "verified": true} -{"case": "triincir@16", "density": 0.7478320195844922, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc16.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.687+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.687, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@16", "credit": "Maurizio Morandi", "date": "April 2008", "family_updated": "8/31/19", "metric_expression": "1.687+", "metric_symbol": "r", "metric_value": 1.687, "reference_text": "r = 1.687+ | Found by Maurizio Morandi | in April 2008.", "source_image": "tc16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc16.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.687+ | Found by Maurizio Morandi | in April 2008.", "friedman_source_image": "tc16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "07537d81776ccefd", "improvement": -0.030249287647702072, "max_boundary_excess": -5.423746678001606e-06, "max_pair_overlap_depth": -4.8671677235212485e-06, "metric_expression": "1.71725", "metric_symbol": "r", "metric_value": 1.7172492876477021, "n": 16, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 1.687, "setup": "triincir", "side": 3.4344985752954043, "solution_path": "data/solutions/generated/triincir_016_07537d81776ccefd.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/07537d81776ccefd.svg", "tolerance": 1e-08, "verified": true} +{"case": "triincir@16", "density": 0.7739404139048575, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc16.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.687+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.687, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@16", "credit": "Maurizio Morandi", "date": "April 2008", "family_updated": "8/31/19", "metric_expression": "1.687+", "metric_symbol": "r", "metric_value": 1.687, "reference_text": "r = 1.687+ | Found by Maurizio Morandi | in April 2008.", "source_image": "tc16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc16.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.687+ | Found by Maurizio Morandi | in April 2008.", "friedman_source_image": "tc16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "8bb5531a7df45934", "improvement": -0.0010356342518775818, "max_boundary_excess": -9.068784430077415e-07, "max_pair_overlap_depth": -9.622687440202782e-07, "metric_expression": "1.68804", "metric_symbol": "r", "metric_value": 1.6880356342518776, "n": 16, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 1.687, "setup": "triincir", "side": 3.3760712685037553, "solution_path": "data/solutions/generated/triincir_16_8bb5531a7df45934.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/8bb5531a7df45934.svg", "tolerance": 1e-08, "verified": true} {"case": "triincir@17", "density": 0.7824875965148136, "friedman_analytical_or_proved": false, "friedman_credit": "David W", "friedman_date": "July 2005", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc17.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "1.730+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.73, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@17", "credit": "David W", "date": "July 2005", "family_updated": "8/31/19", "metric_expression": "1.730+", "metric_symbol": "r", "metric_value": 1.73, "reference_text": "r = 1.730+ | Found by David W. Cantrell | in July 2005.", "source_image": "tc17.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc17.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.730+ | Found by David W. Cantrell | in July 2005.", "friedman_source_image": "tc17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "47c836632f7377eb", "improvement": -0.00045819251380652126, "max_boundary_excess": -2.3405989713065622e-06, "max_pair_overlap_depth": -3.2207400054717894e-06, "metric_expression": "1.73046", "metric_symbol": "r", "metric_value": 1.7304581925138065, "n": 17, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 1.73, "setup": "triincir", "side": 3.460916385027613, "solution_path": "data/solutions/generated/triincir_017_47c836632f7377eb.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/47c836632f7377eb.svg", "tolerance": 1e-08, "verified": true} {"case": "triincir@18", "density": 0.763334042584198, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc18.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "\u221a13/2 = 1.802+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.802, "friedman_reference": {"analytical_or_proved": true, "case": "triincir@18", "credit": "Erich Friedman", "date": "1997", "family_updated": "8/31/19", "metric_expression": "\u221a13/2 = 1.802+", "metric_symbol": "r", "metric_value": 1.802, "reference_text": "r = \u221a13/2 = 1.802+ | Found by Erich Friedman | in 1997.", "source_image": "tc18.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc18.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = \u221a13/2 = 1.802+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "tc18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "8d495d33c78cce38", "improvement": -0.000828100052461922, "max_boundary_excess": -1.0688499290090192e-06, "max_pair_overlap_depth": -2.7547144944017887e-06, "metric_expression": "1.80283", "metric_symbol": "r", "metric_value": 1.802828100052462, "n": 18, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 1.802, "setup": "triincir", "side": 3.605656200104924, "solution_path": "data/solutions/generated/triincir_018_8d495d33c78cce38.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/8d495d33c78cce38.svg", "tolerance": 1e-08, "verified": true} {"case": "triincir@19", "density": 0.7677097187833641, "friedman_analytical_or_proved": true, "friedman_credit": "Haym Hirsh", "friedman_date": "August 2019", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc19.gif", "friedman_family_updated": "8/31/19", "friedman_metric_expression": "\u221a(10/3) = 1.825+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.825, "friedman_reference": {"analytical_or_proved": true, "case": "triincir@19", "credit": "Haym Hirsh", "date": "August 2019", "family_updated": "8/31/19", "metric_expression": "\u221a(10/3) = 1.825+", "metric_symbol": "r", "metric_value": 1.825, "reference_text": "r = \u221a(10/3) = 1.825+ | Found by Haym Hirsh | in August 2019.", "source_image": "tc19.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc19.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = \u221a(10/3) = 1.825+ | Found by Haym Hirsh | in August 2019.", "friedman_source_image": "tc19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "25e61803121deba5", "improvement": -0.021943724350603322, "max_boundary_excess": -6.158152259239102e-07, "max_pair_overlap_depth": -1.4118207399826943e-06, "metric_expression": "1.84694", "metric_symbol": "r", "metric_value": 1.8469437243506033, "n": 19, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 1.825, "setup": "triincir", "side": 3.6938874487012066, "solution_path": "data/solutions/generated/triincir_019_25e61803121deba5.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/25e61803121deba5.svg", "tolerance": 1e-08, "verified": true} @@ -626,9 +626,9 @@ {"case": "triincir@21", "density": 0.7807486191939167, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc21.gif", "friedman_metric_expression": "1.910+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.91, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@21", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "8/31/19", "metric_expression": "1.910+", "metric_symbol": "r", "metric_value": 1.91, "reference_text": "r = 1.910+ | Found by Maurizio Morandi | in May 2008.", "source_image": "tc21.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc21.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.910+ | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tc21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "26dd3deb629b6129", "improvement": -0.015437458779652191, "max_boundary_excess": -9.498317063982853e-07, "max_pair_overlap_depth": -2.835010137536287e-06, "metric_expression": "1.92544", "metric_symbol": "r", "metric_value": 1.9254374587796521, "n": 21, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 1.91, "setup": "triincir", "side": 3.8508749175593042, "solution_path": "data/solutions/generated/triincir_021_26dd3deb629b6129.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/26dd3deb629b6129.svg", "tolerance": 1e-08, "verified": true} {"case": "triincir@22", "density": 0.7874465082238514, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "June 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc22.gif", "friedman_metric_expression": "1.943+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.943, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@22", "credit": "Maurizio Morandi", "date": "June 2008", "family_updated": "8/31/19", "metric_expression": "1.943+", "metric_symbol": "r", "metric_value": 1.943, "reference_text": "r = 1.943+ | Found by Maurizio Morandi | in June 2008.", "source_image": "tc22.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc22.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.943+ | Found by Maurizio Morandi | in June 2008.", "friedman_source_image": "tc22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "0907be89df20a3ff", "improvement": -0.019348742244891026, "max_boundary_excess": -4.030552757772199e-06, "max_pair_overlap_depth": -1.5844548821419835e-05, "metric_expression": "1.96235", "metric_symbol": "r", "metric_value": 1.962348742244891, "n": 22, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 1.943, "setup": "triincir", "side": 3.924697484489782, "solution_path": "data/solutions/generated/triincir_022_0907be89df20a3ff.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/0907be89df20a3ff.svg", "tolerance": 1e-08, "verified": true} {"case": "triincir@23", "density": 0.7918747649774476, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "June 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc23.gif", "friedman_metric_expression": "1.988+", "friedman_metric_symbol": "r", "friedman_metric_value": 1.988, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@23", "credit": "Maurizio Morandi", "date": "June 2008", "family_updated": "8/31/19", "metric_expression": "1.988+", "metric_symbol": "r", "metric_value": 1.988, "reference_text": "r = 1.988+ | Found by Maurizio Morandi | in June 2008.", "source_image": "tc23.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc23.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 1.988+ | Found by Maurizio Morandi | in June 2008.", "friedman_source_image": "tc23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "973cadfcac9f0399", "improvement": -0.012833950693412532, "max_boundary_excess": -2.3604816288269603e-06, "max_pair_overlap_depth": -6.566732802859576e-06, "metric_expression": "2.00083", "metric_symbol": "r", "metric_value": 2.0008339506934125, "n": 23, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 1.988, "setup": "triincir", "side": 4.001667901386825, "solution_path": "data/solutions/generated/triincir_023_973cadfcac9f0399.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/973cadfcac9f0399.svg", "tolerance": 1e-08, "verified": true} -{"case": "triincir@24", "density": 0.77037708306075, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc24.gif", "friedman_metric_expression": "2", "friedman_metric_symbol": "r", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "triincir@24", "credit": "Trivial", "date": "", "family_updated": "8/31/19", "metric_expression": "2", "metric_symbol": "r", "metric_value": 2.0, "reference_text": "r = 2 | Trivial.", "source_image": "tc24.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc24.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 2 | Trivial.", "friedman_source_image": "tc24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "9ebf7fbae73f62a7", "improvement": -0.07218881407517408, "max_boundary_excess": -1.2456668625482337e-06, "max_pair_overlap_depth": -4.316070905119318e-07, "metric_expression": "2.07219", "metric_symbol": "r", "metric_value": 2.072188814075174, "n": 24, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.0, "setup": "triincir", "side": 4.144377628150348, "solution_path": "data/solutions/generated/triincir_024_9ebf7fbae73f62a7.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/9ebf7fbae73f62a7.svg", "tolerance": 1e-08, "verified": true} +{"case": "triincir@24", "density": 0.82672890671807, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc24.gif", "friedman_metric_expression": "2", "friedman_metric_symbol": "r", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": true, "case": "triincir@24", "credit": "Trivial", "date": "", "family_updated": "8/31/19", "metric_expression": "2", "metric_symbol": "r", "metric_value": 2.0, "reference_text": "r = 2 | Trivial.", "source_image": "tc24.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc24.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 2 | Trivial.", "friedman_source_image": "tc24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "e1de703df3a339d6", "improvement": -0.00031983310403260745, "max_boundary_excess": -7.886123831735148e-06, "max_pair_overlap_depth": -6.330279525013083e-07, "metric_expression": "2.00032", "metric_symbol": "r", "metric_value": 2.0003198331040326, "n": 24, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 2.0, "setup": "triincir", "side": 4.000639666208065, "solution_path": "data/solutions/generated/triincir_24_e1de703df3a339d6.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/e1de703df3a339d6.svg", "tolerance": 1e-08, "verified": true} {"case": "triincir@25", "density": 0.814798554480224, "friedman_analytical_or_proved": false, "friedman_credit": "David W", "friedman_date": "July 2005", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc25.gif", "friedman_metric_expression": "2.055+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.055, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@25", "credit": "David W", "date": "July 2005", "family_updated": "8/31/19", "metric_expression": "2.055+", "metric_symbol": "r", "metric_value": 2.055, "reference_text": "r = 2.055+ | Found by David W. Cantrell | in July 2005.", "source_image": "tc25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc25.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 2.055+ | Found by David W. Cantrell | in July 2005.", "friedman_source_image": "tc25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "460bfc63493e0a99", "improvement": -0.0014599742843675045, "max_boundary_excess": -5.499842621325968e-07, "max_pair_overlap_depth": -5.533591762807433e-06, "metric_expression": "2.05646", "metric_symbol": "r", "metric_value": 2.0564599742843677, "n": 25, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.055, "setup": "triincir", "side": 4.112919948568735, "solution_path": "data/solutions/generated/triincir_025_460bfc63493e0a99.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/460bfc63493e0a99.svg", "tolerance": 1e-08, "verified": true} -{"case": "triincir@26", "density": 0.763639473601239, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc26.gif", "friedman_metric_expression": "2.109+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.109, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@26", "credit": "Maurizio Morandi", "date": "April 2008", "family_updated": "8/31/19", "metric_expression": "2.109+", "metric_symbol": "r", "metric_value": 2.109, "reference_text": "r = 2.109+ | Found by Maurizio Morandi | in April 2008.", "source_image": "tc26.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc26.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 2.109+ | Found by Maurizio Morandi | in April 2008.", "friedman_source_image": "tc26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "297d48c1e4300be9", "improvement": -0.057296364393150334, "max_boundary_excess": -2.064436773085987e-06, "max_pair_overlap_depth": -1.3195031345336616e-07, "metric_expression": "2.16630", "metric_symbol": "r", "metric_value": 2.1662963643931503, "n": 26, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.109, "setup": "triincir", "side": 4.332592728786301, "solution_path": "data/solutions/generated/triincir_026_297d48c1e4300be9.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/297d48c1e4300be9.svg", "tolerance": 1e-08, "verified": true} +{"case": "triincir@26", "density": 0.8055666262709128, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "April 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc26.gif", "friedman_metric_expression": "2.109+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.109, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@26", "credit": "Maurizio Morandi", "date": "April 2008", "family_updated": "8/31/19", "metric_expression": "2.109+", "metric_symbol": "r", "metric_value": 2.109, "reference_text": "r = 2.109+ | Found by Maurizio Morandi | in April 2008.", "source_image": "tc26.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc26.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 2.109+ | Found by Maurizio Morandi | in April 2008.", "friedman_source_image": "tc26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "8245fa6b2a472124", "improvement": -0.00016872463985961872, "max_boundary_excess": -3.221408735321063e-06, "max_pair_overlap_depth": -3.3631357487795555e-07, "metric_expression": "2.10917", "metric_symbol": "r", "metric_value": 2.1091687246398596, "n": 26, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 2.109, "setup": "triincir", "side": 4.218337449279719, "solution_path": "data/solutions/generated/triincir_26_8245fa6b2a472124.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/8245fa6b2a472124.svg", "tolerance": 1e-08, "verified": true} {"case": "triincir@27", "density": 0.791482135514487, "friedman_analytical_or_proved": false, "friedman_credit": "David W", "friedman_date": "July 2005", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc27.gif", "friedman_metric_expression": "2.138+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.138, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@27", "credit": "David W", "date": "July 2005", "family_updated": "8/31/19", "metric_expression": "2.138+", "metric_symbol": "r", "metric_value": 2.138, "reference_text": "r = 2.138+ | Found by David W. Cantrell | in July 2005.", "source_image": "tc27.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc27.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 2.138+ | Found by David W. Cantrell | in July 2005.", "friedman_source_image": "tc27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "0b682f2ff7f493f4", "improvement": -0.03038655259456835, "max_boundary_excess": -8.100188688686671e-07, "max_pair_overlap_depth": -1.2480037501427788e-05, "metric_expression": "2.16839", "metric_symbol": "r", "metric_value": 2.1683865525945683, "n": 27, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.138, "setup": "triincir", "side": 4.3367731051891365, "solution_path": "data/solutions/generated/triincir_027_0b682f2ff7f493f4.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/0b682f2ff7f493f4.svg", "tolerance": 1e-08, "verified": true} {"case": "triincir@28", "density": 0.7833608666700476, "friedman_analytical_or_proved": false, "friedman_credit": "David W", "friedman_date": "July 2005", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc28.gif", "friedman_metric_expression": "2.191+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.191, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@28", "credit": "David W", "date": "July 2005", "family_updated": "8/31/19", "metric_expression": "2.191+", "metric_symbol": "r", "metric_value": 2.191, "reference_text": "r = 2.191+ | Found by David W. Cantrell | in July 2005.", "source_image": "tc28.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc28.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 2.191+ | Found by David W. Cantrell | in July 2005.", "friedman_source_image": "tc28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "f7e451634cc50a5c", "improvement": -0.028593586242025903, "max_boundary_excess": -3.334899561835414e-06, "max_pair_overlap_depth": -1.391899336167235e-05, "metric_expression": "2.21959", "metric_symbol": "r", "metric_value": 2.2195935862420257, "n": 28, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.191, "setup": "triincir", "side": 4.4391871724840515, "solution_path": "data/solutions/generated/triincir_028_f7e451634cc50a5c.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/f7e451634cc50a5c.svg", "tolerance": 1e-08, "verified": true} {"case": "triincir@29", "density": 0.8182348688742829, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "June 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triincir/tc29.gif", "friedman_metric_expression": "2.215+", "friedman_metric_symbol": "r", "friedman_metric_value": 2.215, "friedman_reference": {"analytical_or_proved": false, "case": "triincir@29", "credit": "Maurizio Morandi", "date": "June 2008", "family_updated": "8/31/19", "metric_expression": "2.215+", "metric_symbol": "r", "metric_value": 2.215, "reference_text": "r = 2.215+ | Found by Maurizio Morandi | in June 2008.", "source_image": "tc29.gif", "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc29.gif", "source_page": "https://erich-friedman.github.io/packing/triincir/", "title": "Triangles in Circles"}, "friedman_reference_text": "r = 2.215+ | Found by Maurizio Morandi | in June 2008.", "friedman_source_image": "tc29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triincir/", "friedman_title": "Triangles in Circles", "frontend_seed": true, "id": "73bcaf90b9fd2226", "improvement": 0.004780586805306797, "max_boundary_excess": -3.1271053040349273e-06, "max_pair_overlap_depth": -1.2019034158372577e-05, "metric_expression": "2.21022", "metric_symbol": "r", "metric_value": 2.210219413194693, "n": 29, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.215, "setup": "triincir", "side": 4.420438826389386, "solution_path": "data/solutions/generated/triincir_029_73bcaf90b9fd2226.json", "source_url": "https://erich-friedman.github.io/packing/triincir/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/73bcaf90b9fd2226.svg", "tolerance": 1e-08, "verified": true} @@ -681,18 +681,18 @@ {"case": "triinhex@25", "density": 0.8871759236838516, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/25.gif", "friedman_metric_expression": "13/6 = 2.166+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.166, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@25", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "13/6 = 2.166+", "metric_symbol": "s", "metric_value": 2.166, "reference_text": "s = 13/6 = 2.166+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/25.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 13/6 = 2.166+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "71a7801cfb4a8d31", "improvement": -0.0011526610601886311, "max_boundary_excess": -6.780494307380179e-07, "max_pair_overlap_depth": -1.3308976067882838e-06, "metric_expression": "2.16715", "metric_symbol": "s", "metric_value": 2.1671526610601886, "n": 25, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.166, "setup": "triinhex", "side": 2.1671526610601886, "solution_path": "data/solutions/generated/triinhex_025_71a7801cfb4a8d31.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/71a7801cfb4a8d31.svg", "tolerance": 1e-08, "verified": true} {"case": "triinhex@26", "density": 0.878430458154361, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/26.gif", "friedman_metric_expression": "2.21991+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.21991, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@26", "credit": "Ian Watson", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "2.21991+", "metric_symbol": "s", "metric_value": 2.21991, "reference_text": "s = 2.21991+ | Found by Ian Watson | in April 2026.", "source_image": "26.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/26.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.21991+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "658d7f93b604b663", "improvement": -0.0011349903788469007, "max_boundary_excess": -6.149641376243232e-07, "max_pair_overlap_depth": -5.41804289733161e-06, "metric_expression": "2.22104", "metric_symbol": "s", "metric_value": 2.221044990378847, "n": 26, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.21991, "setup": "triinhex", "side": 2.221044990378847, "solution_path": "data/solutions/generated/triinhex_026_658d7f93b604b663.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/658d7f93b604b663.svg", "tolerance": 1e-08, "verified": true} {"case": "triinhex@27", "density": 0.8328322571122685, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/27.gif", "friedman_metric_expression": "2.29541+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.29541, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@27", "credit": "Ian Watson", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "2.29541+", "metric_symbol": "s", "metric_value": 2.29541, "reference_text": "s = 2.29541+ | Found by Ian Watson | in April 2026.", "source_image": "27.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/27.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.29541+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "11ad91d5c6b8788e", "improvement": -0.029078960495372286, "max_boundary_excess": -8.325360180907825e-06, "max_pair_overlap_depth": -2.435631066832755e-06, "metric_expression": "2.32449", "metric_symbol": "s", "metric_value": 2.3244889604953722, "n": 27, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.29541, "setup": "triinhex", "side": 2.3244889604953722, "solution_path": "data/solutions/generated/triinhex_027_11ad91d5c6b8788e.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/11ad91d5c6b8788e.svg", "tolerance": 1e-08, "verified": true} -{"case": "triinhex@28", "density": 0.8335778928127813, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/28.gif", "friedman_metric_expression": "2.32289+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.32289, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@28", "credit": "Ian Watson", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "2.32289+", "metric_symbol": "s", "metric_value": 2.32289, "reference_text": "s = 2.32289+ | Found by Ian Watson | in April 2026.", "source_image": "28.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/28.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.32289+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "2b319174d63885b4", "improvement": -0.0431947496358025, "max_boundary_excess": -3.599330529801011e-06, "max_pair_overlap_depth": -6.859690785454564e-06, "metric_expression": "2.36608", "metric_symbol": "s", "metric_value": 2.3660847496358026, "n": 28, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.32289, "setup": "triinhex", "side": 2.3660847496358026, "solution_path": "data/solutions/generated/triinhex_028_2b319174d63885b4.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/2b319174d63885b4.svg", "tolerance": 1e-08, "verified": true} -{"case": "triinhex@29", "density": 0.837502602292776, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/29.gif", "friedman_metric_expression": "7/3 = 2.333+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.333, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@29", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "7/3 = 2.333+", "metric_symbol": "s", "metric_value": 2.333, "reference_text": "s = 7/3 = 2.333+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "29.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/29.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 7/3 = 2.333+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "64c6a497fdd4bc1d", "improvement": -0.06931687053876034, "max_boundary_excess": -5.555832025645621e-06, "max_pair_overlap_depth": -3.7939764052641323e-06, "metric_expression": "2.40232", "metric_symbol": "s", "metric_value": 2.4023168705387605, "n": 29, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.333, "setup": "triinhex", "side": 2.4023168705387605, "solution_path": "data/solutions/generated/triinhex_029_64c6a497fdd4bc1d.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/64c6a497fdd4bc1d.svg", "tolerance": 1e-08, "verified": true} -{"case": "triinhex@30", "density": 0.8237939342513431, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/30.gif", "friedman_metric_expression": "19/8 = 2.375", "friedman_metric_symbol": "s", "friedman_metric_value": 2.375, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@30", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "19/8 = 2.375", "metric_symbol": "s", "metric_value": 2.375, "reference_text": "s = 19/8 = 2.375 | Found by Ignacio Vallejo | in April 2026.", "source_image": "30.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/30.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 19/8 = 2.375 | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "11d17f9e12268b02", "improvement": -0.08863126687001355, "max_boundary_excess": -2.046588857940021e-06, "max_pair_overlap_depth": -2.8430880139040937e-06, "metric_expression": "2.46363", "metric_symbol": "s", "metric_value": 2.4636312668700135, "n": 30, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.375, "setup": "triinhex", "side": 2.4636312668700135, "solution_path": "data/solutions/generated/triinhex_030_11d17f9e12268b02.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/11d17f9e12268b02.svg", "tolerance": 1e-08, "verified": true} -{"case": "triinhex@31", "density": 0.8505895807478101, "friedman_analytical_or_proved": true, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/31.gif", "friedman_metric_expression": "12/5 = 2.400", "friedman_metric_symbol": "s", "friedman_metric_value": 2.4, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@31", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "12/5 = 2.400", "metric_symbol": "s", "metric_value": 2.4, "reference_text": "s = 12/5 = 2.400 | Found by Emerson Connelly | in April 2026.", "source_image": "31.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/31.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 12/5 = 2.400 | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "1ce8108c850d5941", "improvement": -0.06459289672266566, "max_boundary_excess": -7.67227682312255e-06, "max_pair_overlap_depth": -3.638067419098334e-06, "metric_expression": "2.46459", "metric_symbol": "s", "metric_value": 2.4645928967226656, "n": 31, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.4, "setup": "triinhex", "side": 2.4645928967226656, "solution_path": "data/solutions/generated/triinhex_031_1ce8108c850d5941.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/1ce8108c850d5941.svg", "tolerance": 1e-08, "verified": true} +{"case": "triinhex@28", "density": 0.8636324031387004, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/28.gif", "friedman_metric_expression": "2.32289+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.32289, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@28", "credit": "Ian Watson", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "2.32289+", "metric_symbol": "s", "metric_value": 2.32289, "reference_text": "s = 2.32289+ | Found by Ian Watson | in April 2026.", "source_image": "28.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/28.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.32289+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "3435500023858121", "improvement": -0.0016601826518787277, "max_boundary_excess": -2.60736131059532e-06, "max_pair_overlap_depth": -1.0038160656700157e-05, "metric_expression": "2.32455", "metric_symbol": "r", "metric_value": 2.324550182651879, "n": 28, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 2.32289, "setup": "triinhex", "side": 2.324550182651879, "solution_path": "data/solutions/generated/triinhex_28_3435500023858121.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/3435500023858121.svg", "tolerance": 1e-08, "verified": true} +{"case": "triinhex@29", "density": 0.8876189549602443, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/29.gif", "friedman_metric_expression": "7/3 = 2.333+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.333, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@29", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "7/3 = 2.333+", "metric_symbol": "s", "metric_value": 2.333, "reference_text": "s = 7/3 = 2.333+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "29.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/29.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 7/3 = 2.333+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "4339d1ebd3160f94", "improvement": -0.0005122751747741461, "max_boundary_excess": -2.371335525980428e-06, "max_pair_overlap_depth": -4.303492490143057e-07, "metric_expression": "2.33351", "metric_symbol": "r", "metric_value": 2.3335122751747743, "n": 29, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 2.333, "setup": "triinhex", "side": 2.3335122751747743, "solution_path": "data/solutions/generated/triinhex_29_4339d1ebd3160f94.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/4339d1ebd3160f94.svg", "tolerance": 1e-08, "verified": true} +{"case": "triinhex@30", "density": 0.8862493271562808, "friedman_analytical_or_proved": true, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/30.gif", "friedman_metric_expression": "19/8 = 2.375", "friedman_metric_symbol": "s", "friedman_metric_value": 2.375, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@30", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "19/8 = 2.375", "metric_symbol": "s", "metric_value": 2.375, "reference_text": "s = 19/8 = 2.375 | Found by Ignacio Vallejo | in April 2026.", "source_image": "30.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/30.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 19/8 = 2.375 | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "7274d354e73d8050", "improvement": -0.00023750926269094919, "max_boundary_excess": -9.946881910542413e-07, "max_pair_overlap_depth": -3.1129912760885503e-06, "metric_expression": "2.37524", "metric_symbol": "r", "metric_value": 2.375237509262691, "n": 30, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 2.375, "setup": "triinhex", "side": 2.375237509262691, "solution_path": "data/solutions/generated/triinhex_30_7274d354e73d8050.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/7274d354e73d8050.svg", "tolerance": 1e-08, "verified": true} +{"case": "triinhex@31", "density": 0.896119877899338, "friedman_analytical_or_proved": true, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/31.gif", "friedman_metric_expression": "12/5 = 2.400", "friedman_metric_symbol": "s", "friedman_metric_value": 2.4, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@31", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "12/5 = 2.400", "metric_symbol": "s", "metric_value": 2.4, "reference_text": "s = 12/5 = 2.400 | Found by Emerson Connelly | in April 2026.", "source_image": "31.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/31.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 12/5 = 2.400 | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "cae7087451040120", "improvement": -0.0011658949471424762, "max_boundary_excess": -1.9409739826535097e-07, "max_pair_overlap_depth": -2.8541100203227643e-06, "metric_expression": "2.40117", "metric_symbol": "r", "metric_value": 2.4011658949471424, "n": 31, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 2.4, "setup": "triinhex", "side": 2.4011658949471424, "solution_path": "data/solutions/generated/triinhex_31_cae7087451040120.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/cae7087451040120.svg", "tolerance": 1e-08, "verified": true} {"case": "triinhex@32", "density": 0.8767870241121234, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/32.gif", "friedman_metric_expression": "2.465+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.465, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@32", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "2.465+", "metric_symbol": "s", "metric_value": 2.465, "reference_text": "s = 2.465+ | Found by Emerson Connelly | in April 2026.", "source_image": "32.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/32.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.465+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "8674581436e5e74d", "improvement": -0.0013363682337788063, "max_boundary_excess": -6.925122041572899e-07, "max_pair_overlap_depth": -1.2807310418028184e-06, "metric_expression": "2.46634", "metric_symbol": "s", "metric_value": 2.4663363682337787, "n": 32, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.465, "setup": "triinhex", "side": 2.4663363682337787, "solution_path": "data/solutions/generated/triinhex_032_8674581436e5e74d.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/8674581436e5e74d.svg", "tolerance": 1e-08, "verified": true} {"case": "triinhex@33", "density": 0.8799443949567897, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/33.gif", "friedman_metric_expression": "2.498+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.498, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@33", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "2.498+", "metric_symbol": "s", "metric_value": 2.498, "reference_text": "s = 2.498+ | Found by Emerson Connelly | in April 2026.", "source_image": "33.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/33.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.498+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "33.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "5b836a27d2bb958c", "improvement": -0.00207898817969987, "max_boundary_excess": -2.8634496562140136e-06, "max_pair_overlap_depth": -5.829272115620565e-06, "metric_expression": "2.50008", "metric_symbol": "s", "metric_value": 2.5000789881797, "n": 33, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.498, "setup": "triinhex", "side": 2.5000789881797, "solution_path": "data/solutions/generated/triinhex_033_5b836a27d2bb958c.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/5b836a27d2bb958c.svg", "tolerance": 1e-08, "verified": true} {"case": "triinhex@34", "density": 0.9066102013779274, "friedman_analytical_or_proved": true, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/35.gif", "friedman_metric_expression": "5/2 = 2.500", "friedman_metric_symbol": "s", "friedman_metric_value": 2.5, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@34", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "5/2 = 2.500", "metric_symbol": "s", "metric_value": 2.5, "reference_text": "s = 5/2 = 2.500 | Found by Emerson Connelly | in April 2026.", "source_image": "35.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/35.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 5/2 = 2.500 | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "35.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "a712744ff3b5bcc9", "improvement": -7.7851001305973e-05, "max_boundary_excess": -1.2753913921730298e-06, "max_pair_overlap_depth": -7.960301433351802e-06, "metric_expression": "2.50008", "metric_symbol": "s", "metric_value": 2.500077851001306, "n": 34, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.5, "setup": "triinhex", "side": 2.500077851001306, "solution_path": "data/solutions/generated/triinhex_034_a712744ff3b5bcc9.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/a712744ff3b5bcc9.svg", "tolerance": 1e-08, "verified": true} -{"case": "triinhex@35", "density": 0.8784557475555014, "friedman_analytical_or_proved": true, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/35.gif", "friedman_metric_expression": "5/2 = 2.500", "friedman_metric_symbol": "s", "friedman_metric_value": 2.5, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@35", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "5/2 = 2.500", "metric_symbol": "s", "metric_value": 2.5, "reference_text": "s = 5/2 = 2.500 | Found by Emerson Connelly | in April 2026.", "source_image": "35.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/35.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 5/2 = 2.500 | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "35.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "6c433acba2e3b71c", "improvement": -0.07690526384601659, "max_boundary_excess": -2.8573318260367013e-06, "max_pair_overlap_depth": -2.7042986540104508e-06, "metric_expression": "2.57691", "metric_symbol": "s", "metric_value": 2.5769052638460166, "n": 35, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.5, "setup": "triinhex", "side": 2.5769052638460166, "solution_path": "data/solutions/generated/triinhex_035_6c433acba2e3b71c.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/6c433acba2e3b71c.svg", "tolerance": 1e-08, "verified": true} +{"case": "triinhex@35", "density": 0.93316535041203, "friedman_analytical_or_proved": true, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/35.gif", "friedman_metric_expression": "5/2 = 2.500", "friedman_metric_symbol": "s", "friedman_metric_value": 2.5, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@35", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "5/2 = 2.500", "metric_symbol": "s", "metric_value": 2.5, "reference_text": "s = 5/2 = 2.500 | Found by Emerson Connelly | in April 2026.", "source_image": "35.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/35.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 5/2 = 2.500 | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "35.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "ec49778cf450c37b", "improvement": -0.00022500750012577342, "max_boundary_excess": -2.6207066397040256e-06, "max_pair_overlap_depth": -1.5837731757728513e-06, "metric_expression": "2.50023", "metric_symbol": "r", "metric_value": 2.5002250075001258, "n": 35, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 2.5, "setup": "triinhex", "side": 2.5002250075001258, "solution_path": "data/solutions/generated/triinhex_35_ec49778cf450c37b.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/ec49778cf450c37b.svg", "tolerance": 1e-08, "verified": true} {"case": "triinhex@36", "density": 0.893589353430048, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/36.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.592+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.592, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@36", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "2.592+", "metric_symbol": "s", "metric_value": 2.592, "reference_text": "s = 2.592+ | Found by Emerson Connelly | in April 2026.", "source_image": "36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/36.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.592+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "id": "e4e0e01451d6ca30", "improvement": 0.0007660070914519146, "max_boundary_excess": -3.7601717225028787e-07, "max_pair_overlap_depth": -3.324194429499272e-06, "n": 36, "notes": "Seeded from the certified top-10 candidate bundle.", "reference_side": 2.592, "setup": "triinhex", "side": 2.591233992908548, "solution_path": "data/solutions/triinhex_36.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-04-30T23:19:18+00:00", "submitted_by": "Nathan Roll / polygon-packer-c", "svg_path": "data/svg/e4e0e01451d6ca30.svg", "tolerance": 1e-08, "verified": true} {"case": "triinhex@37", "density": 0.8987469977236169, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/37.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.655+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.655, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@37", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "2.655+", "metric_symbol": "s", "metric_value": 2.655, "reference_text": "s = 2.655+ | Found by Emerson Connelly | in April 2026.", "source_image": "37.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/37.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.655+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "37.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "id": "c1697d9445319eaa", "improvement": 0.035571728604908426, "max_boundary_excess": -1.2650118661675414e-07, "max_pair_overlap_depth": -4.4643045216119503e-07, "n": 37, "notes": "Seeded from the certified top-10 candidate bundle.", "reference_side": 2.655, "setup": "triinhex", "side": 2.6194282713950914, "solution_path": "data/solutions/triinhex_37.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-04-30T23:19:18+00:00", "submitted_by": "Nathan Roll / polygon-packer-c", "svg_path": "data/svg/c1697d9445319eaa.svg", "tolerance": 1e-08, "verified": true} {"case": "triinhex@38", "density": 0.8907392561961515, "friedman_analytical_or_proved": true, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/39.gif", "friedman_metric_expression": "8/3 = 2.666+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.666, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@38", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "8/3 = 2.666+", "metric_symbol": "s", "metric_value": 2.666, "reference_text": "s = 8/3 = 2.666+ | Found by Emerson Connelly | in April 2026.", "source_image": "39.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/39.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 8/3 = 2.666+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "39.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "76fd196e8c79a5c9", "improvement": -0.0004956329108489754, "max_boundary_excess": -3.7850003642140776e-06, "max_pair_overlap_depth": -3.863933217385096e-06, "metric_expression": "2.66650", "metric_symbol": "s", "metric_value": 2.666495632910849, "n": 38, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.666, "setup": "triinhex", "side": 2.666495632910849, "solution_path": "data/solutions/generated/triinhex_038_76fd196e8c79a5c9.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/76fd196e8c79a5c9.svg", "tolerance": 1e-08, "verified": true} -{"case": "triinhex@39", "density": 0.8833522738592235, "friedman_analytical_or_proved": true, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/39.gif", "friedman_metric_expression": "8/3 = 2.666+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.666, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@39", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "8/3 = 2.666+", "metric_symbol": "s", "metric_value": 2.666, "reference_text": "s = 8/3 = 2.666+ | Found by Emerson Connelly | in April 2026.", "source_image": "39.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/39.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 8/3 = 2.666+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "39.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "cf1c7ccbe8f975b2", "improvement": -0.04662470457951118, "max_boundary_excess": -2.2386896785242527e-06, "max_pair_overlap_depth": -9.826603503571008e-06, "metric_expression": "2.71262", "metric_symbol": "s", "metric_value": 2.712624704579511, "n": 39, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.666, "setup": "triinhex", "side": 2.712624704579511, "solution_path": "data/solutions/generated/triinhex_039_cf1c7ccbe8f975b2.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/cf1c7ccbe8f975b2.svg", "tolerance": 1e-08, "verified": true} +{"case": "triinhex@39", "density": 0.9136318214810123, "friedman_analytical_or_proved": true, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/39.gif", "friedman_metric_expression": "8/3 = 2.666+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.666, "friedman_reference": {"analytical_or_proved": true, "case": "triinhex@39", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "8/3 = 2.666+", "metric_symbol": "s", "metric_value": 2.666, "reference_text": "s = 8/3 = 2.666+ | Found by Emerson Connelly | in April 2026.", "source_image": "39.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/39.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 8/3 = 2.666+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "39.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "frontend_seed": true, "id": "a314b4bb8467e5e8", "improvement": -0.0012951149704507614, "max_boundary_excess": -6.891404577705932e-07, "max_pair_overlap_depth": -6.19203720853001e-07, "metric_expression": "2.66730", "metric_symbol": "r", "metric_value": 2.6672951149704507, "n": 39, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 2.666, "setup": "triinhex", "side": 2.6672951149704507, "solution_path": "data/solutions/generated/triinhex_39_a314b4bb8467e5e8.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/a314b4bb8467e5e8.svg", "tolerance": 1e-08, "verified": true} {"case": "triinhex@40", "density": 0.9036048260198691, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinhex/40.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.719+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.719, "friedman_reference": {"analytical_or_proved": false, "case": "triinhex@40", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "2.719+", "metric_symbol": "s", "metric_value": 2.719, "reference_text": "s = 2.719+ | Found by Emerson Connelly | in April 2026.", "source_image": "40.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinhex/40.gif", "source_page": "https://erich-friedman.github.io/packing/triinhex/", "title": "Triangles in Hexagons"}, "friedman_reference_text": "s = 2.719+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "40.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinhex/", "friedman_title": "Triangles in Hexagons", "id": "0ebde2f6b4b26d2b", "improvement": 0.002779018588611848, "max_boundary_excess": -5.817255795307e-07, "max_pair_overlap_depth": -1.0085947836913078e-06, "n": 40, "notes": "Seeded from the certified top-10 candidate bundle.", "reference_side": 2.719, "setup": "triinhex", "side": 2.716220981411388, "solution_path": "data/solutions/triinhex_40.json", "source_url": "https://erich-friedman.github.io/packing/triinhex/", "submitted_at": "2026-04-30T23:19:18+00:00", "submitted_by": "Nathan Roll / polygon-packer-c", "svg_path": "data/svg/0ebde2f6b4b26d2b.svg", "tolerance": 1e-08, "verified": true} {"case": "triinoct@1", "density": 0.4152122478191623, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/1.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": ".46470+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.4647, "friedman_reference": {"analytical_or_proved": true, "case": "triinoct@1", "credit": "Trivial", "date": "", "family_updated": "3/22/26", "metric_expression": ".46470+", "metric_symbol": "s", "metric_value": 0.4647, "reference_text": "s = .46470+ | Trivial. |", "source_image": "1.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/1.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = .46470+ | Trivial. |", "friedman_source_image": "1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "92c66f395ff03ab9", "improvement": -4.251046545072912e-05, "max_boundary_excess": -2.3282842257895898e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "0.46474", "metric_symbol": "s", "metric_value": 0.46474251046545073, "n": 1, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 0.4647, "setup": "triinoct", "side": 0.46474251046545073, "solution_path": "data/solutions/generated/triinoct_001_92c66f395ff03ab9.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/92c66f395ff03ab9.svg", "tolerance": 1e-08, "verified": true} {"case": "triinoct@2", "density": 0.40823825072318354, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/2.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "\u221a3/\u221a(4+2\u221a2) = .66282+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.66282, "friedman_reference": {"analytical_or_proved": true, "case": "triinoct@2", "credit": "Trivial", "date": "", "family_updated": "3/22/26", "metric_expression": "\u221a3/\u221a(4+2\u221a2) = .66282+", "metric_symbol": "s", "metric_value": 0.66282, "reference_text": "s = \u221a3/\u221a(4+2\u221a2) = .66282+ | Trivial. |", "source_image": "2.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/2.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = \u221a3/\u221a(4+2\u221a2) = .66282+ | Trivial. |", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "5706480ca6648c20", "improvement": -1.5298424264065247e-05, "max_boundary_excess": -4.197393779037103e-06, "max_pair_overlap_depth": -5.033792755404498e-07, "metric_expression": "0.66284", "metric_symbol": "s", "metric_value": 0.662835298424264, "n": 2, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 0.66282, "setup": "triinoct", "side": 0.662835298424264, "solution_path": "data/solutions/generated/triinoct_002_5706480ca6648c20.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/5706480ca6648c20.svg", "tolerance": 1e-08, "verified": true} @@ -709,7 +709,7 @@ {"case": "triinoct@13", "density": 0.7604201292225432, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/13.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.23759+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.23759, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@13", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "3/22/26", "metric_expression": "1.23759+", "metric_symbol": "s", "metric_value": 1.23759, "reference_text": "s = 1.23759+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "13.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/13.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.23759+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "ebb5825b80b24749", "improvement": -0.0006144239243295324, "max_boundary_excess": -4.763415979262496e-06, "max_pair_overlap_depth": -3.350137919300167e-06, "metric_expression": "1.23820", "metric_symbol": "s", "metric_value": 1.2382044239243295, "n": 13, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 1.23759, "setup": "triinoct", "side": 1.2382044239243295, "solution_path": "data/solutions/generated/triinoct_013_ebb5825b80b24749.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/ebb5825b80b24749.svg", "tolerance": 1e-08, "verified": true} {"case": "triinoct@14", "density": 0.7632138549104823, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/14.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.27677+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.27677, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@14", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "3/22/26", "metric_expression": "1.27677+", "metric_symbol": "s", "metric_value": 1.27677, "reference_text": "s = 1.27677+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "14.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/14.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.27677+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "d46f9390cd318528", "improvement": -0.005821544544329926, "max_boundary_excess": -4.128121247700188e-06, "max_pair_overlap_depth": -1.821323936102015e-05, "metric_expression": "1.28259", "metric_symbol": "s", "metric_value": 1.2825915445443299, "n": 14, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 1.27677, "setup": "triinoct", "side": 1.2825915445443299, "solution_path": "data/solutions/generated/triinoct_014_d46f9390cd318528.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/d46f9390cd318528.svg", "tolerance": 1e-08, "verified": true} {"case": "triinoct@15", "density": 0.7924237027261557, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/15.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.30275+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.30275, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@15", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "3/22/26", "metric_expression": "1.30275+", "metric_symbol": "s", "metric_value": 1.30275, "reference_text": "s = 1.30275+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "15.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/15.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.30275+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "e14eda962017d06d", "improvement": -0.0001598827367932376, "max_boundary_excess": -1.3032093708899356e-06, "max_pair_overlap_depth": -6.895788391969582e-06, "metric_expression": "1.30291", "metric_symbol": "s", "metric_value": 1.3029098827367933, "n": 15, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 1.30275, "setup": "triinoct", "side": 1.3029098827367933, "solution_path": "data/solutions/generated/triinoct_015_e14eda962017d06d.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/e14eda962017d06d.svg", "tolerance": 1e-08, "verified": true} -{"case": "triinoct@16", "density": 0.7647910514290241, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/16.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.34584+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.34584, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@16", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "3/22/26", "metric_expression": "1.34584+", "metric_symbol": "s", "metric_value": 1.34584, "reference_text": "s = 1.34584+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/16.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.34584+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "7d95c6b48a2068e3", "improvement": -0.023893476019485993, "max_boundary_excess": -2.4464974440901898e-06, "max_pair_overlap_depth": -1.1018645437399499e-05, "metric_expression": "1.36973", "metric_symbol": "s", "metric_value": 1.369733476019486, "n": 16, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 1.34584, "setup": "triinoct", "side": 1.369733476019486, "solution_path": "data/solutions/generated/triinoct_016_7d95c6b48a2068e3.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/7d95c6b48a2068e3.svg", "tolerance": 1e-08, "verified": true} +{"case": "triinoct@16", "density": 0.7920450821484247, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/16.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.34584+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.34584, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@16", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "3/22/26", "metric_expression": "1.34584+", "metric_symbol": "s", "metric_value": 1.34584, "reference_text": "s = 1.34584+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/16.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.34584+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "640352ab60961c58", "improvement": -0.00012112963758759143, "max_boundary_excess": -4.464975954309231e-06, "max_pair_overlap_depth": -7.38266966116008e-07, "metric_expression": "1.34596", "metric_symbol": "r", "metric_value": 1.3459611296375875, "n": 16, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 1.34584, "setup": "triinoct", "side": 1.3459611296375875, "solution_path": "data/solutions/generated/triinoct_16_640352ab60961c58.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/640352ab60961c58.svg", "tolerance": 1e-08, "verified": true} {"case": "triinoct@17", "density": 0.7803021824951497, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/17.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.38964+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.38964, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@17", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "3/22/26", "metric_expression": "1.38964+", "metric_symbol": "s", "metric_value": 1.38964, "reference_text": "s = 1.38964+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "17.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/17.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.38964+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "854cf664c481474a", "improvement": -0.008145488421500424, "max_boundary_excess": -6.214706009791588e-08, "max_pair_overlap_depth": -5.371473842497032e-06, "metric_expression": "1.39779", "metric_symbol": "s", "metric_value": 1.3977854884215004, "n": 17, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 1.38964, "setup": "triinoct", "side": 1.3977854884215004, "solution_path": "data/solutions/generated/triinoct_017_854cf664c481474a.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/854cf664c481474a.svg", "tolerance": 1e-08, "verified": true} {"case": "triinoct@18", "density": 0.7984879014669996, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/18.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.41836+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.41836, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@18", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "3/22/26", "metric_expression": "1.41836+", "metric_symbol": "s", "metric_value": 1.41836, "reference_text": "s = 1.41836+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/18.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.41836+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "8e956085d3bcc5e4", "improvement": -0.00347617639937825, "max_boundary_excess": -2.3654127749911342e-06, "max_pair_overlap_depth": -6.1952459529668324e-06, "metric_expression": "1.42184", "metric_symbol": "s", "metric_value": 1.4218361763993783, "n": 18, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 1.41836, "setup": "triinoct", "side": 1.4218361763993783, "solution_path": "data/solutions/generated/triinoct_018_8e956085d3bcc5e4.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/8e956085d3bcc5e4.svg", "tolerance": 1e-08, "verified": true} {"case": "triinoct@19", "density": 0.785495925696758, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/19.gif", "friedman_family_updated": "3/22/26", "friedman_metric_expression": "1.46749+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.46749, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@19", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "3/22/26", "metric_expression": "1.46749+", "metric_symbol": "s", "metric_value": 1.46749, "reference_text": "s = 1.46749+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/19.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.46749+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "b85e7f72b874a80d", "improvement": -0.005338941628411087, "max_boundary_excess": -1.1956920861702103e-06, "max_pair_overlap_depth": -5.584055034923985e-06, "metric_expression": "1.47283", "metric_symbol": "s", "metric_value": 1.472828941628411, "n": 19, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 1.46749, "setup": "triinoct", "side": 1.472828941628411, "solution_path": "data/solutions/generated/triinoct_019_b85e7f72b874a80d.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/b85e7f72b874a80d.svg", "tolerance": 1e-08, "verified": true} @@ -720,9 +720,9 @@ {"case": "triinoct@24", "density": 0.8373304168984339, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/24.gif", "friedman_metric_expression": "1.60261+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.60261, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@24", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "3/22/26", "metric_expression": "1.60261+", "metric_symbol": "s", "metric_value": 1.60261, "reference_text": "s = 1.60261+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/24.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.60261+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "e197b92ecac28558", "improvement": -0.0006526103809141315, "max_boundary_excess": -2.085338835265612e-06, "max_pair_overlap_depth": -9.98840296584369e-06, "metric_expression": "1.60326", "metric_symbol": "s", "metric_value": 1.6032626103809142, "n": 24, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 1.60261, "setup": "triinoct", "side": 1.6032626103809142, "solution_path": "data/solutions/generated/triinoct_024_e197b92ecac28558.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/e197b92ecac28558.svg", "tolerance": 1e-08, "verified": true} {"case": "triinoct@25", "density": 0.8206390570044901, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/25.gif", "friedman_metric_expression": "1.63797+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.63797, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@25", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "3/22/26", "metric_expression": "1.63797+", "metric_symbol": "s", "metric_value": 1.63797, "reference_text": "s = 1.63797+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/25.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.63797+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "4e1b861b138850ae", "improvement": -0.014910250491960486, "max_boundary_excess": -2.6877619511415554e-06, "max_pair_overlap_depth": -7.2555966788498405e-06, "metric_expression": "1.65288", "metric_symbol": "s", "metric_value": 1.6528802504919604, "n": 25, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 1.63797, "setup": "triinoct", "side": 1.6528802504919604, "solution_path": "data/solutions/generated/triinoct_025_4e1b861b138850ae.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/4e1b861b138850ae.svg", "tolerance": 1e-08, "verified": true} {"case": "triinoct@26", "density": 0.8149644565721081, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/26.gif", "friedman_metric_expression": "1.67666+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.67666, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@26", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "3/22/26", "metric_expression": "1.67666+", "metric_symbol": "s", "metric_value": 1.67666, "reference_text": "s = 1.67666+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "26.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/26.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.67666+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "cbcb6c755dc046de", "improvement": -0.014812017272631595, "max_boundary_excess": -3.687228372083238e-07, "max_pair_overlap_depth": -1.2896587116006941e-06, "metric_expression": "1.69147", "metric_symbol": "s", "metric_value": 1.6914720172726316, "n": 26, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 1.67666, "setup": "triinoct", "side": 1.6914720172726316, "solution_path": "data/solutions/generated/triinoct_026_cbcb6c755dc046de.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/cbcb6c755dc046de.svg", "tolerance": 1e-08, "verified": true} -{"case": "triinoct@27", "density": 0.8010680431479879, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/27.gif", "friedman_metric_expression": "1.70882+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.70882, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@27", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "3/22/26", "metric_expression": "1.70882+", "metric_symbol": "s", "metric_value": 1.70882, "reference_text": "s = 1.70882+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "27.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/27.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.70882+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "836b5a3b2c412e9b", "improvement": -0.029759907189525592, "max_boundary_excess": -9.248042691084391e-07, "max_pair_overlap_depth": -7.87865755902395e-06, "metric_expression": "1.73858", "metric_symbol": "s", "metric_value": 1.7385799071895256, "n": 27, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 1.70882, "setup": "triinoct", "side": 1.7385799071895256, "solution_path": "data/solutions/generated/triinoct_027_836b5a3b2c412e9b.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/836b5a3b2c412e9b.svg", "tolerance": 1e-08, "verified": true} +{"case": "triinoct@27", "density": 0.8297818493323376, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/27.gif", "friedman_metric_expression": "1.70882+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.70882, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@27", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "3/22/26", "metric_expression": "1.70882+", "metric_symbol": "s", "metric_value": 1.70882, "reference_text": "s = 1.70882+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "27.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/27.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.70882+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "c7748eb19b34ff2e", "improvement": 0.0005858720037632548, "max_boundary_excess": -9.055699391789948e-06, "max_pair_overlap_depth": -3.877030372301249e-07, "metric_expression": "1.70823", "metric_symbol": "r", "metric_value": 1.7082341279962368, "n": 27, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 1.70882, "setup": "triinoct", "side": 1.7082341279962368, "solution_path": "data/solutions/generated/triinoct_27_c7748eb19b34ff2e.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/c7748eb19b34ff2e.svg", "tolerance": 1e-08, "verified": true} {"case": "triinoct@28", "density": 0.8165576484730356, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/28.gif", "friedman_metric_expression": "1.73954+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.73954, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@28", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "3/22/26", "metric_expression": "1.73954+", "metric_symbol": "s", "metric_value": 1.73954, "reference_text": "s = 1.73954+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "28.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/28.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.73954+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "728bf65e8a483e5e", "improvement": -0.014070219094066916, "max_boundary_excess": -4.0057635728717315e-07, "max_pair_overlap_depth": -1.0036991463513445e-05, "metric_expression": "1.75361", "metric_symbol": "s", "metric_value": 1.753610219094067, "n": 28, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 1.73954, "setup": "triinoct", "side": 1.753610219094067, "solution_path": "data/solutions/generated/triinoct_028_728bf65e8a483e5e.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/728bf65e8a483e5e.svg", "tolerance": 1e-08, "verified": true} -{"case": "triinoct@29", "density": 0.8071992437047276, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/29.gif", "friedman_metric_expression": "1.75576+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.75576, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@29", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "3/22/26", "metric_expression": "1.75576+", "metric_symbol": "s", "metric_value": 1.75576, "reference_text": "s = 1.75576+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "29.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/29.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.75576+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "f6d5bdc33f1fd083", "improvement": -0.03920549021863984, "max_boundary_excess": -2.872640643928272e-06, "max_pair_overlap_depth": -2.128542299273395e-05, "metric_expression": "1.79497", "metric_symbol": "s", "metric_value": 1.7949654902186398, "n": 29, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 1.75576, "setup": "triinoct", "side": 1.7949654902186398, "solution_path": "data/solutions/generated/triinoct_029_f6d5bdc33f1fd083.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/f6d5bdc33f1fd083.svg", "tolerance": 1e-08, "verified": true} +{"case": "triinoct@29", "density": 0.8439313801960897, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/29.gif", "friedman_metric_expression": "1.75576+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.75576, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@29", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "3/22/26", "metric_expression": "1.75576+", "metric_symbol": "s", "metric_value": 1.75576, "reference_text": "s = 1.75576+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "29.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/29.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.75576+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "0575780ea85d68dd", "improvement": 0.00029203482208850495, "max_boundary_excess": -3.530531415212805e-06, "max_pair_overlap_depth": -1.1528245769332202e-05, "metric_expression": "1.75547", "metric_symbol": "r", "metric_value": 1.7554679651779115, "n": 29, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 1.75576, "setup": "triinoct", "side": 1.7554679651779115, "solution_path": "data/solutions/generated/triinoct_29_0575780ea85d68dd.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/0575780ea85d68dd.svg", "tolerance": 1e-08, "verified": true} {"case": "triinoct@30", "density": 0.8241852827166171, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinoct/30.gif", "friedman_metric_expression": "1.78101+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.78101, "friedman_reference": {"analytical_or_proved": false, "case": "triinoct@30", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "3/22/26", "metric_expression": "1.78101+", "metric_symbol": "s", "metric_value": 1.78101, "reference_text": "s = 1.78101+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "30.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinoct/30.gif", "source_page": "https://erich-friedman.github.io/packing/triinoct/", "title": "Triangles in Octagons"}, "friedman_reference_text": "s = 1.78101+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinoct/", "friedman_title": "Triangles in Octagons", "frontend_seed": true, "id": "55871e661d418192", "improvement": -0.02573007339812161, "max_boundary_excess": -9.550198166939913e-07, "max_pair_overlap_depth": -3.435684308383813e-06, "metric_expression": "1.80674", "metric_symbol": "s", "metric_value": 1.8067400733981216, "n": 30, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 1.78101, "setup": "triinoct", "side": 1.8067400733981216, "solution_path": "data/solutions/generated/triinoct_030_55871e661d418192.json", "source_url": "https://erich-friedman.github.io/packing/triinoct/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/55871e661d418192.svg", "tolerance": 1e-08, "verified": true} {"case": "triinpen@1", "density": 0.4121951299103008, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/1.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": ".781+", "friedman_metric_symbol": "s", "friedman_metric_value": 0.781, "friedman_reference": {"analytical_or_proved": true, "case": "triinpen@1", "credit": "Trivial", "date": "", "family_updated": "4/29/26", "metric_expression": ".781+", "metric_symbol": "s", "metric_value": 0.781, "reference_text": "s = cos(42\u00b0) / cos(18\u00b0) | s = .781+ | Trivial. |", "source_image": "1.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/1.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = cos(42\u00b0) / cos(18\u00b0) | s = .781+ | Trivial. |", "friedman_source_image": "1.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "6d3af7533c544064", "improvement": -0.0004017935524769367, "max_boundary_excess": -5.010333690326338e-06, "max_pair_overlap_depth": 0.0, "metric_expression": "0.78140", "metric_symbol": "s", "metric_value": 0.781401793552477, "n": 1, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 0.781, "setup": "triinpen", "side": 0.781401793552477, "solution_path": "data/solutions/generated/triinpen_001_6d3af7533c544064.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/6d3af7533c544064.svg", "tolerance": 1e-08, "verified": true} {"case": "triinpen@2", "density": 0.4392461428307515, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/2.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "1.070+", "friedman_metric_symbol": "s", "friedman_metric_value": 1.07, "friedman_reference": {"analytical_or_proved": true, "case": "triinpen@2", "credit": "Trivial", "date": "", "family_updated": "4/29/26", "metric_expression": "1.070+", "metric_symbol": "s", "metric_value": 1.07, "reference_text": "s = \u221a(6/(3+\u221a5)) | s = 1.070+ | Trivial. |", "source_image": "2.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/2.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = \u221a(6/(3+\u221a5)) | s = 1.070+ | Trivial. |", "friedman_source_image": "2.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "eebb278fed1959e5", "improvement": -0.0005004432401414238, "max_boundary_excess": -7.540244142797655e-06, "max_pair_overlap_depth": -4.677606255020384e-06, "metric_expression": "1.07050", "metric_symbol": "s", "metric_value": 1.0705004432401415, "n": 2, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 1.07, "setup": "triinpen", "side": 1.0705004432401415, "solution_path": "data/solutions/generated/triinpen_002_eebb278fed1959e5.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/eebb278fed1959e5.svg", "tolerance": 1e-08, "verified": true} @@ -742,18 +742,18 @@ {"case": "triinpen@16", "density": 0.8497807683719915, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/16.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.17598+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.17598, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@16", "credit": "Ian Watson", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "2.17598+", "metric_symbol": "s", "metric_value": 2.17598, "reference_text": "s = 2.17598+ | Found by Ian Watson | in April 2026.", "source_image": "16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/16.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.17598+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "852225bc33edeebe", "improvement": -0.0008893975034314217, "max_boundary_excess": -2.337598971955443e-06, "max_pair_overlap_depth": -3.396060024770525e-05, "metric_expression": "2.17687", "metric_symbol": "s", "metric_value": 2.1768693975034314, "n": 16, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.17598, "setup": "triinpen", "side": 2.1768693975034314, "solution_path": "data/solutions/generated/triinpen_016_852225bc33edeebe.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/852225bc33edeebe.svg", "tolerance": 1e-08, "verified": true} {"case": "triinpen@17", "density": 0.8523626090077759, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/17.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.23658+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.23658, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@17", "credit": "Ian Watson", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "2.23658+", "metric_symbol": "s", "metric_value": 2.23658, "reference_text": "s = 2.23658+ | Found by Ian Watson | in April 2026.", "source_image": "17.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/17.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.23658+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "17.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "7726e3530d961600", "improvement": -0.003884657795955704, "max_boundary_excess": -1.8305901749986508e-06, "max_pair_overlap_depth": -7.65308285666233e-06, "metric_expression": "2.24046", "metric_symbol": "s", "metric_value": 2.2404646577959557, "n": 17, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.23658, "setup": "triinpen", "side": 2.2404646577959557, "solution_path": "data/solutions/generated/triinpen_017_7726e3530d961600.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/7726e3530d961600.svg", "tolerance": 1e-08, "verified": true} {"case": "triinpen@18", "density": 0.8386266626517904, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/18.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.29382+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.29382, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@18", "credit": "Ian Watson", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "2.29382+", "metric_symbol": "s", "metric_value": 2.29382, "reference_text": "s = 2.29382+ | Found by Ian Watson | in April 2026.", "source_image": "18.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/18.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.29382+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "18.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "286d64d1342c9a9f", "improvement": -0.0304027693812583, "max_boundary_excess": -2.3683231734867957e-06, "max_pair_overlap_depth": -9.23929068963325e-06, "metric_expression": "2.32422", "metric_symbol": "s", "metric_value": 2.3242227693812585, "n": 18, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.29382, "setup": "triinpen", "side": 2.3242227693812585, "solution_path": "data/solutions/generated/triinpen_018_286d64d1342c9a9f.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/286d64d1342c9a9f.svg", "tolerance": 1e-08, "verified": true} -{"case": "triinpen@19", "density": 0.831205150275822, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/19.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.344+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.344, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@19", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "4/29/26", "metric_expression": "2.344+", "metric_symbol": "s", "metric_value": 2.344, "reference_text": "s = 2.344+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/19.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.344+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "5f0b9905f54e5f87", "improvement": -0.05454858293536624, "max_boundary_excess": -1.8779608876987526e-06, "max_pair_overlap_depth": -1.5527720347900675e-05, "metric_expression": "2.39855", "metric_symbol": "s", "metric_value": 2.398548582935366, "n": 19, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.344, "setup": "triinpen", "side": 2.398548582935366, "solution_path": "data/solutions/generated/triinpen_019_5f0b9905f54e5f87.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/5f0b9905f54e5f87.svg", "tolerance": 1e-08, "verified": true} +{"case": "triinpen@19", "density": 0.868021512388566, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/19.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.344+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.344, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@19", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "4/29/26", "metric_expression": "2.344+", "metric_symbol": "s", "metric_value": 2.344, "reference_text": "s = 2.344+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "19.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/19.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.344+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "19.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "eaa1430599d74148", "improvement": -0.003131316702724707, "max_boundary_excess": -2.9139572150349214e-06, "max_pair_overlap_depth": -1.0705368373109536e-05, "metric_expression": "2.34713", "metric_symbol": "r", "metric_value": 2.3471313167027246, "n": 19, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 2.344, "setup": "triinpen", "side": 2.3471313167027246, "solution_path": "data/solutions/generated/triinpen_19_eaa1430599d74148.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/eaa1430599d74148.svg", "tolerance": 1e-08, "verified": true} {"case": "triinpen@20", "density": 0.8487102106795418, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/20.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.43350+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.4335, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@20", "credit": "Ian Watson", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "2.43350+", "metric_symbol": "s", "metric_value": 2.4335, "reference_text": "s = 2.43350+ | Found by Ian Watson | in April 2026.", "source_image": "20.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/20.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.43350+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "20.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "760faebc4d0b131b", "improvement": -0.001848490502489586, "max_boundary_excess": -7.176183787084704e-07, "max_pair_overlap_depth": -8.807499014995557e-06, "metric_expression": "2.43535", "metric_symbol": "s", "metric_value": 2.4353484905024896, "n": 20, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.4335, "setup": "triinpen", "side": 2.4353484905024896, "solution_path": "data/solutions/generated/triinpen_020_760faebc4d0b131b.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/760faebc4d0b131b.svg", "tolerance": 1e-08, "verified": true} {"case": "triinpen@21", "density": 0.8589998748063701, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/21.gif", "friedman_metric_expression": "2.479+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.479, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@21", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "4/29/26", "metric_expression": "2.479+", "metric_symbol": "s", "metric_value": 2.479, "reference_text": "s = 2.479+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "21.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/21.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.479+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "21.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "115c20fad199fe9d", "improvement": -0.0014982727861765355, "max_boundary_excess": -2.811024693767028e-06, "max_pair_overlap_depth": -2.8209410938284663e-06, "metric_expression": "2.48050", "metric_symbol": "s", "metric_value": 2.4804982727861766, "n": 21, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.479, "setup": "triinpen", "side": 2.4804982727861766, "solution_path": "data/solutions/generated/triinpen_021_115c20fad199fe9d.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/115c20fad199fe9d.svg", "tolerance": 1e-08, "verified": true} {"case": "triinpen@22", "density": 0.8367105270898347, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/22.gif", "friedman_metric_expression": "2.56740+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.5674, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@22", "credit": "Ian Watson", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "2.56740+", "metric_symbol": "s", "metric_value": 2.5674, "reference_text": "s = 2.56740+ | Found by Ian Watson | in April 2026.", "source_image": "22.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/22.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.56740+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "22.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "b3ecf4aa9beda411", "improvement": -0.0050654803778220625, "max_boundary_excess": -3.63122781255133e-06, "max_pair_overlap_depth": -1.9313990747837906e-06, "metric_expression": "2.57247", "metric_symbol": "s", "metric_value": 2.572465480377822, "n": 22, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.5674, "setup": "triinpen", "side": 2.572465480377822, "solution_path": "data/solutions/generated/triinpen_022_b3ecf4aa9beda411.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/b3ecf4aa9beda411.svg", "tolerance": 1e-08, "verified": true} {"case": "triinpen@23", "density": 0.8358219621726138, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/23.gif", "friedman_metric_expression": "2.61831+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.61831, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@23", "credit": "Ian Watson", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "2.61831+", "metric_symbol": "s", "metric_value": 2.61831, "reference_text": "s = 2.61831+ | Found by Ian Watson | in April 2026.", "source_image": "23.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/23.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.61831+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "23.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "b2f9cfacd50b72e9", "improvement": -0.013368670719241305, "max_boundary_excess": -5.909725797067009e-06, "max_pair_overlap_depth": -5.3149326107448935e-06, "metric_expression": "2.63168", "metric_symbol": "s", "metric_value": 2.6316786707192414, "n": 23, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.61831, "setup": "triinpen", "side": 2.6316786707192414, "solution_path": "data/solutions/generated/triinpen_023_b2f9cfacd50b72e9.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/b2f9cfacd50b72e9.svg", "tolerance": 1e-08, "verified": true} -{"case": "triinpen@24", "density": 0.8203286521496933, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/24.gif", "friedman_metric_expression": "2.66641+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.66641, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@24", "credit": "Ian Watson", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "2.66641+", "metric_symbol": "s", "metric_value": 2.66641, "reference_text": "s = 2.66641+ | Found by Ian Watson | in April 2026.", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/24.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.66641+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "76211a17aba30b36", "improvement": -0.04713802590265814, "max_boundary_excess": -7.766120852076588e-07, "max_pair_overlap_depth": -5.665654174702439e-07, "metric_expression": "2.71355", "metric_symbol": "s", "metric_value": 2.713548025902658, "n": 24, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.66641, "setup": "triinpen", "side": 2.713548025902658, "solution_path": "data/solutions/generated/triinpen_024_76211a17aba30b36.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/76211a17aba30b36.svg", "tolerance": 1e-08, "verified": true} -{"case": "triinpen@25", "density": 0.8170666213083482, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/25.gif", "friedman_metric_expression": "2.720+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.72, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@25", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "4/29/26", "metric_expression": "2.720+", "metric_symbol": "s", "metric_value": 2.72, "reference_text": "s = 2.720+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/25.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.720+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "cc904cb82492c4b0", "improvement": -0.055026288404456114, "max_boundary_excess": -3.831253149892822e-06, "max_pair_overlap_depth": -1.0171981972284527e-05, "metric_expression": "2.77503", "metric_symbol": "s", "metric_value": 2.7750262884044563, "n": 25, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.72, "setup": "triinpen", "side": 2.7750262884044563, "solution_path": "data/solutions/generated/triinpen_025_cc904cb82492c4b0.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/cc904cb82492c4b0.svg", "tolerance": 1e-08, "verified": true} +{"case": "triinpen@24", "density": 0.8488309418600493, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/24.gif", "friedman_metric_expression": "2.66641+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.66641, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@24", "credit": "Ian Watson", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "2.66641+", "metric_symbol": "s", "metric_value": 2.66641, "reference_text": "s = 2.66641+ | Found by Ian Watson | in April 2026.", "source_image": "24.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/24.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.66641+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "24.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "7bc3466b163ec849", "improvement": -0.0011908775198370414, "max_boundary_excess": -2.421732660736982e-06, "max_pair_overlap_depth": -3.3708625798434255e-05, "metric_expression": "2.66760", "metric_symbol": "r", "metric_value": 2.667600877519837, "n": 24, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 2.66641, "setup": "triinpen", "side": 2.667600877519837, "solution_path": "data/solutions/generated/triinpen_24_7bc3466b163ec849.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/7bc3466b163ec849.svg", "tolerance": 1e-08, "verified": true} +{"case": "triinpen@25", "density": 0.8484730741355839, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/25.gif", "friedman_metric_expression": "2.720+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.72, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@25", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "4/29/26", "metric_expression": "2.720+", "metric_symbol": "s", "metric_value": 2.72, "reference_text": "s = 2.720+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "25.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/25.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.720+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "25.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "d7a41b31b48a9cd4", "improvement": -0.0031828531428343965, "max_boundary_excess": -2.9946489466199756e-06, "max_pair_overlap_depth": -1.485388314126368e-06, "metric_expression": "2.72318", "metric_symbol": "r", "metric_value": 2.7231828531428346, "n": 25, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 2.72, "setup": "triinpen", "side": 2.7231828531428346, "solution_path": "data/solutions/generated/triinpen_25_d7a41b31b48a9cd4.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/d7a41b31b48a9cd4.svg", "tolerance": 1e-08, "verified": true} {"case": "triinpen@26", "density": 0.8287869991439658, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/26.gif", "friedman_metric_expression": "2.771+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.771, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@26", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "4/29/26", "metric_expression": "2.771+", "metric_symbol": "s", "metric_value": 2.771, "reference_text": "s = 2.771+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "26.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/26.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.771+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "810173856676a6fb", "improvement": -0.03890114272185885, "max_boundary_excess": -2.899135639086088e-06, "max_pair_overlap_depth": -1.3139310877985366e-05, "metric_expression": "2.80990", "metric_symbol": "s", "metric_value": 2.8099011427218588, "n": 26, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.771, "setup": "triinpen", "side": 2.8099011427218588, "solution_path": "data/solutions/generated/triinpen_026_810173856676a6fb.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/810173856676a6fb.svg", "tolerance": 1e-08, "verified": true} {"case": "triinpen@27", "density": 0.8321868307827671, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/27.gif", "friedman_metric_expression": "2.81761+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.81761, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@27", "credit": "Ian Watson", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "2.81761+", "metric_symbol": "s", "metric_value": 2.81761, "reference_text": "s = 2.81761+ | Found by Ian Watson | in April 2026.", "source_image": "27.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/27.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.81761+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "27.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "8b8ba91284e47068", "improvement": -0.03996274171783831, "max_boundary_excess": -3.050950539540987e-07, "max_pair_overlap_depth": -5.479231620197922e-06, "metric_expression": "2.85757", "metric_symbol": "s", "metric_value": 2.8575727417178385, "n": 27, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.81761, "setup": "triinpen", "side": 2.8575727417178385, "solution_path": "data/solutions/generated/triinpen_027_8b8ba91284e47068.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/8b8ba91284e47068.svg", "tolerance": 1e-08, "verified": true} {"case": "triinpen@28", "density": 0.8534363238461394, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "March 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/28.gif", "friedman_metric_expression": "2.868+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.868, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@28", "credit": "Ignacio Vallejo", "date": "March 2026", "family_updated": "4/29/26", "metric_expression": "2.868+", "metric_symbol": "s", "metric_value": 2.868, "reference_text": "s = 2.868+ | Found by Ignacio Vallejo | in March 2026.", "source_image": "28.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/28.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.868+ | Found by Ignacio Vallejo | in March 2026.", "friedman_source_image": "28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "6ad421ad9865e8bf", "improvement": -0.005553490865734201, "max_boundary_excess": -2.0498365378060392e-06, "max_pair_overlap_depth": -1.2746651528505737e-05, "metric_expression": "2.87355", "metric_symbol": "s", "metric_value": 2.873553490865734, "n": 28, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.868, "setup": "triinpen", "side": 2.873553490865734, "solution_path": "data/solutions/generated/triinpen_028_6ad421ad9865e8bf.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/6ad421ad9865e8bf.svg", "tolerance": 1e-08, "verified": true} {"case": "triinpen@29", "density": 0.8464177208582354, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/29.gif", "friedman_metric_expression": "2.903+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.903, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@29", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "2.903+", "metric_symbol": "s", "metric_value": 2.903, "reference_text": "s = 2.903+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "29.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/29.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.903+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "0d9d40fe56c0f6d4", "improvement": -0.03351657923241014, "max_boundary_excess": -2.7025059194940577e-06, "max_pair_overlap_depth": -2.3448291092131868e-06, "metric_expression": "2.93652", "metric_symbol": "s", "metric_value": 2.93651657923241, "n": 29, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.903, "setup": "triinpen", "side": 2.93651657923241, "solution_path": "data/solutions/generated/triinpen_029_0d9d40fe56c0f6d4.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/0d9d40fe56c0f6d4.svg", "tolerance": 1e-08, "verified": true} -{"case": "triinpen@30", "density": 0.8397676649377761, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/30.gif", "friedman_metric_expression": "2.952+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.952, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@30", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "2.952+", "metric_symbol": "s", "metric_value": 2.952, "reference_text": "s = 2.952+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "30.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/30.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.952+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "2b832e7a9fa0a7c5", "improvement": -0.04651955266880892, "max_boundary_excess": -7.656844312442343e-06, "max_pair_overlap_depth": -3.4133589315032964e-06, "metric_expression": "2.99852", "metric_symbol": "s", "metric_value": 2.998519552668809, "n": 30, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.952, "setup": "triinpen", "side": 2.998519552668809, "solution_path": "data/solutions/generated/triinpen_030_2b832e7a9fa0a7c5.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/2b832e7a9fa0a7c5.svg", "tolerance": 1e-08, "verified": true} +{"case": "triinpen@30", "density": 0.8662355090431414, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/30.gif", "friedman_metric_expression": "2.952+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.952, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@30", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "2.952+", "metric_symbol": "s", "metric_value": 2.952, "reference_text": "s = 2.952+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "30.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/30.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.952+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "29e4e0e684891d8a", "improvement": -0.00035425771249730786, "max_boundary_excess": -1.990812967166742e-06, "max_pair_overlap_depth": -2.800943013125945e-05, "metric_expression": "2.95235", "metric_symbol": "r", "metric_value": 2.9523542577124973, "n": 30, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 2.952, "setup": "triinpen", "side": 2.9523542577124973, "solution_path": "data/solutions/generated/triinpen_30_29e4e0e684891d8a.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/29e4e0e684891d8a.svg", "tolerance": 1e-08, "verified": true} {"case": "triinpen@31", "density": 0.8566123952654375, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/31.gif", "friedman_metric_expression": "2.992+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.992, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@31", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "2.992+", "metric_symbol": "s", "metric_value": 2.992, "reference_text": "s = 2.992+ | Found by Emerson Connelly | in April 2026.", "source_image": "31.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/31.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 2.992+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "579de3c893a5a64c", "improvement": -0.025967103693977922, "max_boundary_excess": -1.099100289551188e-06, "max_pair_overlap_depth": -2.29204214511336e-05, "metric_expression": "3.01797", "metric_symbol": "s", "metric_value": 3.017967103693978, "n": 31, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.992, "setup": "triinpen", "side": 3.017967103693978, "solution_path": "data/solutions/generated/triinpen_031_579de3c893a5a64c.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/579de3c893a5a64c.svg", "tolerance": 1e-08, "verified": true} {"case": "triinpen@32", "density": 0.8514556204542924, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/32.gif", "friedman_metric_expression": "3.029+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.029, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@32", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "3.029+", "metric_symbol": "s", "metric_value": 3.029, "reference_text": "s = 3.029+ | Found by Emerson Connelly | in April 2026.", "source_image": "32.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/32.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 3.029+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "70ed9a5e30a88f24", "improvement": -0.04652890272981658, "max_boundary_excess": -2.6611165642007356e-06, "max_pair_overlap_depth": -3.1395889643626163e-06, "metric_expression": "3.07553", "metric_symbol": "s", "metric_value": 3.0755289027298165, "n": 32, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 3.029, "setup": "triinpen", "side": 3.0755289027298165, "solution_path": "data/solutions/generated/triinpen_032_70ed9a5e30a88f24.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/70ed9a5e30a88f24.svg", "tolerance": 1e-08, "verified": true} {"case": "triinpen@33", "density": 0.872718269558225, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinpen/33.gif", "friedman_metric_expression": "3.078+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.078, "friedman_reference": {"analytical_or_proved": false, "case": "triinpen@33", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "3.078+", "metric_symbol": "s", "metric_value": 3.078, "reference_text": "s = 3.078+ | Found by Emerson Connelly | in April 2026.", "source_image": "33.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinpen/33.gif", "source_page": "https://erich-friedman.github.io/packing/triinpen/", "title": "Triangles in Pentagons"}, "friedman_reference_text": "s = 3.078+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "33.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinpen/", "friedman_title": "Triangles in Pentagons", "frontend_seed": true, "id": "13357e078d1002ad", "improvement": -0.006933225403836563, "max_boundary_excess": -1.6488057343371516e-06, "max_pair_overlap_depth": -2.3777449531969808e-05, "metric_expression": "3.08493", "metric_symbol": "s", "metric_value": 3.0849332254038364, "n": 33, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 3.078, "setup": "triinpen", "side": 3.0849332254038364, "solution_path": "data/solutions/generated/triinpen_033_13357e078d1002ad.json", "source_url": "https://erich-friedman.github.io/packing/triinpen/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/13357e078d1002ad.svg", "tolerance": 1e-08, "verified": true} @@ -773,10 +773,10 @@ {"case": "triinsqu@7", "density": 0.7576992844833931, "friedman_analytical_or_proved": false, "friedman_credit": "Erich Friedman", "friedman_date": "1996", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t7.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2", "friedman_metric_symbol": "s", "friedman_metric_value": 2.0, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@7", "credit": "Erich Friedman", "date": "1996", "family_updated": "4/29/26", "metric_expression": "2", "metric_symbol": "s", "metric_value": 2.0, "reference_text": "s = 2 | Found by Erich Friedman | in 1996.", "source_image": "t7.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t7.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2 | Found by Erich Friedman | in 1996.", "friedman_source_image": "t7.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "d4eecc11473f7bf7", "improvement": -9.626783869354227e-05, "max_boundary_excess": -7.867700488247924e-07, "max_pair_overlap_depth": -1.9950288099523666e-05, "metric_expression": "2.00010", "metric_symbol": "s", "metric_value": 2.0000962678386935, "n": 7, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.0, "setup": "triinsqu", "side": 2.0000962678386935, "solution_path": "data/solutions/generated/triinsqu_007_d4eecc11473f7bf7.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/d4eecc11473f7bf7.svg", "tolerance": 1e-08, "verified": true} {"case": "triinsqu@8", "density": 0.7868902975296777, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1996", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t8.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "3\u221a3/2\u20131/2 = 2.098+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.098, "friedman_reference": {"analytical_or_proved": true, "case": "triinsqu@8", "credit": "Erich Friedman", "date": "1996", "family_updated": "4/29/26", "metric_expression": "3\u221a3/2\u20131/2 = 2.098+", "metric_symbol": "s", "metric_value": 2.098, "reference_text": "s = 3\u221a3/2\u20131/2 = 2.098+ | Found by Erich Friedman | in 1996.", "source_image": "t8.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t8.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 3\u221a3/2\u20131/2 = 2.098+ | Found by Erich Friedman | in 1996.", "friedman_source_image": "t8.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "bd28b88a6ba9b99d", "improvement": -0.00015813059780489525, "max_boundary_excess": -1.0169734543243664e-06, "max_pair_overlap_depth": -2.636538231393093e-05, "metric_expression": "2.09816", "metric_symbol": "s", "metric_value": 2.0981581305978048, "n": 8, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.098, "setup": "triinsqu", "side": 2.0981581305978048, "solution_path": "data/solutions/generated/triinsqu_008_bd28b88a6ba9b99d.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/bd28b88a6ba9b99d.svg", "tolerance": 1e-08, "verified": true} {"case": "triinsqu@9", "density": 0.7446998211864622, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "July 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t9.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.287+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.287, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@9", "credit": "Maurizio Morandi", "date": "July 2008", "family_updated": "4/29/26", "metric_expression": "2.287+", "metric_symbol": "s", "metric_value": 2.287, "reference_text": "s = 2.287+ | Found by Maurizio Morandi | in July 2008.", "source_image": "t9.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t9.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.287+ | Found by Maurizio Morandi | in July 2008.", "friedman_source_image": "t9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "458e49fec5e5573c", "improvement": -0.0006045303821253789, "max_boundary_excess": -4.745466993473377e-07, "max_pair_overlap_depth": -5.124657140087585e-06, "metric_expression": "2.28760", "metric_symbol": "s", "metric_value": 2.2876045303821253, "n": 9, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.287, "setup": "triinsqu", "side": 2.2876045303821253, "solution_path": "data/solutions/generated/triinsqu_009_458e49fec5e5573c.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/458e49fec5e5573c.svg", "tolerance": 1e-08, "verified": true} -{"case": "triinsqu@10", "density": 0.7662113488997031, "friedman_analytical_or_proved": false, "friedman_credit": "David W", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t10.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.377+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.377, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@10", "credit": "David W", "date": "July 2002", "family_updated": "4/29/26", "metric_expression": "2.377+", "metric_symbol": "s", "metric_value": 2.377, "reference_text": "s = 2.377+ | Found by David W. Cantrell | in July 2002.", "source_image": "t10.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t10.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.377+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "t10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "5addeb4a05e6dd16", "improvement": -0.000256414102594249, "max_boundary_excess": -2.8710097643173447e-06, "max_pair_overlap_depth": -1.526365666765228e-05, "metric_expression": "2.37726", "metric_symbol": "s", "metric_value": 2.377256414102594, "n": 10, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.377, "setup": "triinsqu", "side": 2.377256414102594, "solution_path": "data/solutions/generated/triinsqu_010_5addeb4a05e6dd16.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/5addeb4a05e6dd16.svg", "tolerance": 1e-08, "verified": true} +{"case": "triinsqu@10", "density": 0.7662234065167978, "friedman_analytical_or_proved": false, "friedman_credit": "David W", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t10.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.377+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.377, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@10", "credit": "David W", "date": "July 2002", "family_updated": "4/29/26", "metric_expression": "2.377+", "metric_symbol": "s", "metric_value": 2.377, "reference_text": "s = 2.377+ | Found by David W. Cantrell | in July 2002.", "source_image": "t10.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t10.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.377+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "t10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "5b0eaf84f54fe48c", "improvement": -0.00023770927049060475, "max_boundary_excess": -6.324926068512563e-07, "max_pair_overlap_depth": -1.1896205976635876e-05, "metric_expression": "2.37724", "metric_symbol": "r", "metric_value": 2.3772377092704904, "n": 10, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 2.377, "setup": "triinsqu", "side": 2.3772377092704904, "solution_path": "data/solutions/generated/triinsqu_10_5b0eaf84f54fe48c.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/5b0eaf84f54fe48c.svg", "tolerance": 1e-08, "verified": true} {"case": "triinsqu@11", "density": 0.7671858882860172, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "June 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t11.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.490+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.49, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@11", "credit": "Maurizio Morandi", "date": "June 2008", "family_updated": "4/29/26", "metric_expression": "2.490+", "metric_symbol": "s", "metric_value": 2.49, "reference_text": "s = 2.490+ | Found by Maurizio Morandi | in June 2008.", "source_image": "t11.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t11.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.490+ | Found by Maurizio Morandi | in June 2008.", "friedman_source_image": "t11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "b95fbc7e7009bb63", "improvement": -0.0017034740442234408, "max_boundary_excess": -9.171896607131735e-07, "max_pair_overlap_depth": -3.917476745979309e-05, "metric_expression": "2.49170", "metric_symbol": "s", "metric_value": 2.4917034740442237, "n": 11, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.49, "setup": "triinsqu", "side": 2.4917034740442237, "solution_path": "data/solutions/generated/triinsqu_011_b95fbc7e7009bb63.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/b95fbc7e7009bb63.svg", "tolerance": 1e-08, "verified": true} {"case": "triinsqu@12", "density": 0.792908624011719, "friedman_analytical_or_proved": false, "friedman_credit": "Dominik Kamp", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t12.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.558+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.558, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@12", "credit": "Dominik Kamp", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "2.558+", "metric_symbol": "s", "metric_value": 2.558, "reference_text": "s = 2.558+ | Found by Dominik Kamp | in April 2026.", "source_image": "t12.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t12.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.558+ | Found by Dominik Kamp | in April 2026.", "friedman_source_image": "t12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "297283972e34795d", "improvement": -0.0019375473066665094, "max_boundary_excess": -2.5442542765183254e-06, "max_pair_overlap_depth": -1.2119755173123359e-05, "metric_expression": "2.55994", "metric_symbol": "s", "metric_value": 2.5599375473066663, "n": 12, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.558, "setup": "triinsqu", "side": 2.5599375473066663, "solution_path": "data/solutions/generated/triinsqu_012_297283972e34795d.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/297283972e34795d.svg", "tolerance": 1e-08, "verified": true} -{"case": "triinsqu@13", "density": 0.7984360278475032, "friedman_analytical_or_proved": false, "friedman_credit": "David W", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t13.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.595+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.595, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@13", "credit": "David W", "date": "July 2002", "family_updated": "4/29/26", "metric_expression": "2.595+", "metric_symbol": "s", "metric_value": 2.595, "reference_text": "s = 2.595+ | Found by David W. Cantrell | in July 2002.", "source_image": "t13.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t13.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.595+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "t13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "86b51c8e3c3e7262", "improvement": -0.060228686632984196, "max_boundary_excess": -1.2371100660487619e-06, "max_pair_overlap_depth": -2.3163827317262964e-05, "metric_expression": "2.65523", "metric_symbol": "s", "metric_value": 2.6552286866329844, "n": 13, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.595, "setup": "triinsqu", "side": 2.6552286866329844, "solution_path": "data/solutions/generated/triinsqu_013_86b51c8e3c3e7262.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/86b51c8e3c3e7262.svg", "tolerance": 1e-08, "verified": true} +{"case": "triinsqu@13", "density": 0.8355745862678572, "friedman_analytical_or_proved": false, "friedman_credit": "David W", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t13.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.595+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.595, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@13", "credit": "David W", "date": "July 2002", "family_updated": "4/29/26", "metric_expression": "2.595+", "metric_symbol": "s", "metric_value": 2.595, "reference_text": "s = 2.595+ | Found by David W. Cantrell | in July 2002.", "source_image": "t13.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t13.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.595+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "t13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "7628e283de25ee68", "improvement": -0.0005499010705918295, "max_boundary_excess": -1.8264727928819919e-06, "max_pair_overlap_depth": -1.755464108432747e-05, "metric_expression": "2.59555", "metric_symbol": "r", "metric_value": 2.595549901070592, "n": 13, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 2.595, "setup": "triinsqu", "side": 2.595549901070592, "solution_path": "data/solutions/generated/triinsqu_13_7628e283de25ee68.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/7628e283de25ee68.svg", "tolerance": 1e-08, "verified": true} {"case": "triinsqu@14", "density": 0.8018109132335658, "friedman_analytical_or_proved": false, "friedman_credit": "David W", "friedman_date": "July 2002", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t14.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.726+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.726, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@14", "credit": "David W", "date": "July 2002", "family_updated": "4/29/26", "metric_expression": "2.726+", "metric_symbol": "s", "metric_value": 2.726, "reference_text": "s = 2.726+ | Found by David W. Cantrell | in July 2002.", "source_image": "t14.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t14.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.726+ | Found by David W. Cantrell | in July 2002.", "friedman_source_image": "t14.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "76439832611f43c7", "improvement": -0.023655937979654595, "max_boundary_excess": -4.171338690111402e-06, "max_pair_overlap_depth": -4.970783313573701e-06, "metric_expression": "2.74966", "metric_symbol": "s", "metric_value": 2.7496559379796546, "n": 14, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.726, "setup": "triinsqu", "side": 2.7496559379796546, "solution_path": "data/solutions/generated/triinsqu_014_76439832611f43c7.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/76439832611f43c7.svg", "tolerance": 1e-08, "verified": true} {"case": "triinsqu@15", "density": 0.8033975182630823, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t15.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.8299+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.8299, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@15", "credit": "Ian Watson", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "2.8299+", "metric_symbol": "s", "metric_value": 2.8299, "reference_text": "s = 2.8299+ | Found by Ian Watson | in April 2026.", "source_image": "t15.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t15.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.8299+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "t15.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "52e52ea9a87dcceb", "improvement": -0.013452504217346561, "max_boundary_excess": -3.936575778440642e-07, "max_pair_overlap_depth": -1.4477048518035662e-05, "metric_expression": "2.84335", "metric_symbol": "s", "metric_value": 2.8433525042173464, "n": 15, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.8299, "setup": "triinsqu", "side": 2.8433525042173464, "solution_path": "data/solutions/generated/triinsqu_015_52e52ea9a87dcceb.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/52e52ea9a87dcceb.svg", "tolerance": 1e-08, "verified": true} {"case": "triinsqu@16", "density": 0.8226134516626242, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t16.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "2.9000+", "friedman_metric_symbol": "s", "friedman_metric_value": 2.9, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@16", "credit": "Ian Watson", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "2.9000+", "metric_symbol": "s", "metric_value": 2.9, "reference_text": "s = 2.9000+ | Found by Ian Watson | in April 2026.", "source_image": "t16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t16.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "s = 2.9000+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "t16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "2350476c07d7060c", "improvement": -0.002100218284964317, "max_boundary_excess": -2.0129376531587966e-06, "max_pair_overlap_depth": -5.176872598853383e-06, "metric_expression": "2.90210", "metric_symbol": "s", "metric_value": 2.9021002182849642, "n": 16, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 2.9, "setup": "triinsqu", "side": 2.9021002182849642, "solution_path": "data/solutions/generated/triinsqu_016_2350476c07d7060c.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/2350476c07d7060c.svg", "tolerance": 1e-08, "verified": true} @@ -795,9 +795,9 @@ {"case": "triinsqu@29", "density": 0.8511488797755807, "friedman_analytical_or_proved": false, "friedman_credit": "Ignacio Vallejo", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t29.gif", "friedman_metric_expression": "3.818+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.818, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@29", "credit": "Ignacio Vallejo", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "3.818+", "metric_symbol": "s", "metric_value": 3.818, "reference_text": "29. | | s = 3.818+ | Found by Ignacio Vallejo | in April 2026.", "source_image": "t29.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t29.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "29. | | s = 3.818+ | Found by Ignacio Vallejo | in April 2026.", "friedman_source_image": "t29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "580739cf26075b25", "improvement": -0.023019845203774647, "max_boundary_excess": -5.02170884848626e-08, "max_pair_overlap_depth": -7.455986605509057e-06, "metric_expression": "3.84102", "metric_symbol": "s", "metric_value": 3.8410198452037747, "n": 29, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 3.818, "setup": "triinsqu", "side": 3.8410198452037747, "solution_path": "data/solutions/generated/triinsqu_029_580739cf26075b25.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/580739cf26075b25.svg", "tolerance": 1e-08, "verified": true} {"case": "triinsqu@30", "density": 0.8453943070349464, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t30.gif", "friedman_metric_expression": "3.8662+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.8662, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@30", "credit": "Ian Watson", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "3.8662+", "metric_symbol": "s", "metric_value": 3.8662, "reference_text": "30. | | s = 3.8662+ | Found by Ian Watson | in April 2026.", "source_image": "t30.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t30.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "30. | | s = 3.8662+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "t30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "c1d8c0f9af9cd320", "improvement": -0.05375684533529945, "max_boundary_excess": -9.869941090398363e-07, "max_pair_overlap_depth": -3.480593055327219e-06, "metric_expression": "3.91996", "metric_symbol": "s", "metric_value": 3.9199568453352995, "n": 30, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 3.8662, "setup": "triinsqu", "side": 3.9199568453352995, "solution_path": "data/solutions/generated/triinsqu_030_c1d8c0f9af9cd320.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/c1d8c0f9af9cd320.svg", "tolerance": 1e-08, "verified": true} {"case": "triinsqu@31", "density": 0.8400189843781062, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t31.gif", "friedman_metric_expression": "3.936+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.936, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@31", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "3.936+", "metric_symbol": "s", "metric_value": 3.936, "reference_text": "31. | | s = 3.936+ | Found by Emerson Connelly | in April 2026.", "source_image": "t31.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t31.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "31. | | s = 3.936+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "272c8c45b9a6dd78", "improvement": -0.06148289712739752, "max_boundary_excess": -1.9869012023909734e-06, "max_pair_overlap_depth": -1.1060967226250717e-05, "metric_expression": "3.99748", "metric_symbol": "s", "metric_value": 3.9974828971273975, "n": 31, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 3.936, "setup": "triinsqu", "side": 3.9974828971273975, "solution_path": "data/solutions/generated/triinsqu_031_272c8c45b9a6dd78.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/272c8c45b9a6dd78.svg", "tolerance": 1e-08, "verified": true} -{"case": "triinsqu@32", "density": 0.833007660060693, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t32.gif", "friedman_metric_expression": "3.997+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.997, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@32", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "3.997+", "metric_symbol": "s", "metric_value": 3.997, "reference_text": "32. | | s = 3.997+ | Found by Emerson Connelly | in April 2026.", "source_image": "t32.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t32.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "32. | | s = 3.997+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "aa2ff1aceeba9dbd", "improvement": -0.0815032212073783, "max_boundary_excess": -5.66081673195531e-06, "max_pair_overlap_depth": -4.793291196181215e-06, "metric_expression": "4.07850", "metric_symbol": "s", "metric_value": 4.078503221207378, "n": 32, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 3.997, "setup": "triinsqu", "side": 4.078503221207378, "solution_path": "data/solutions/generated/triinsqu_032_aa2ff1aceeba9dbd.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/aa2ff1aceeba9dbd.svg", "tolerance": 1e-08, "verified": true} -{"case": "triinsqu@33", "density": 0.8366415016689572, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t33.gif", "friedman_metric_expression": "4.058+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.058, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@33", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "4.058+", "metric_symbol": "s", "metric_value": 4.058, "reference_text": "33. | | s = 4.058+ | Found by Emerson Connelly | in April 2026.", "source_image": "t33.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t33.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "33. | | s = 4.058+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t33.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "652d6c2cf21c385c", "improvement": -0.07473526355298876, "max_boundary_excess": -6.365030008288742e-08, "max_pair_overlap_depth": -1.4601549878645592e-06, "metric_expression": "4.13274", "metric_symbol": "s", "metric_value": 4.132735263552989, "n": 33, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 4.058, "setup": "triinsqu", "side": 4.132735263552989, "solution_path": "data/solutions/generated/triinsqu_033_652d6c2cf21c385c.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/652d6c2cf21c385c.svg", "tolerance": 1e-08, "verified": true} -{"case": "triinsqu@34", "density": 0.8455895674978956, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t34.gif", "friedman_metric_expression": "4.098+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.098, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@34", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "4.098+", "metric_symbol": "s", "metric_value": 4.098, "reference_text": "34. | | s = 4.098+ | Found by Emerson Connelly | in April 2026.", "source_image": "t34.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t34.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "34. | | s = 4.098+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t34.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "5f00ff525459125e", "improvement": -0.07463088632816373, "max_boundary_excess": -2.422656501976907e-07, "max_pair_overlap_depth": -3.1602008447872976e-06, "metric_expression": "4.17263", "metric_symbol": "s", "metric_value": 4.172630886328164, "n": 34, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 4.098, "setup": "triinsqu", "side": 4.172630886328164, "solution_path": "data/solutions/generated/triinsqu_034_5f00ff525459125e.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/5f00ff525459125e.svg", "tolerance": 1e-08, "verified": true} +{"case": "triinsqu@32", "density": 0.8675921732131934, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t32.gif", "friedman_metric_expression": "3.997+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.997, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@32", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "3.997+", "metric_symbol": "s", "metric_value": 3.997, "reference_text": "32. | | s = 3.997+ | Found by Emerson Connelly | in April 2026.", "source_image": "t32.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t32.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "32. | | s = 3.997+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "f42801b713b9238d", "improvement": 0.0006133968629469244, "max_boundary_excess": -2.208596980261035e-06, "max_pair_overlap_depth": -2.3341135757526388e-06, "metric_expression": "3.99639", "metric_symbol": "r", "metric_value": 3.996386603137053, "n": 32, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 3.997, "setup": "triinsqu", "side": 3.996386603137053, "solution_path": "data/solutions/generated/triinsqu_32_f42801b713b9238d.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/f42801b713b9238d.svg", "tolerance": 1e-08, "verified": true} +{"case": "triinsqu@33", "density": 0.8675949776849048, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t33.gif", "friedman_metric_expression": "4.058+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.058, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@33", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "4.058+", "metric_symbol": "s", "metric_value": 4.058, "reference_text": "33. | | s = 4.058+ | Found by Emerson Connelly | in April 2026.", "source_image": "t33.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t33.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "33. | | s = 4.058+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t33.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "a6f49f2f24228d1b", "improvement": -0.00034322121207797096, "max_boundary_excess": -1.0747182587902822e-06, "max_pair_overlap_depth": -2.8024820216354662e-05, "metric_expression": "4.05834", "metric_symbol": "r", "metric_value": 4.058343221212078, "n": 33, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 4.058, "setup": "triinsqu", "side": 4.058343221212078, "solution_path": "data/solutions/generated/triinsqu_33_a6f49f2f24228d1b.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/a6f49f2f24228d1b.svg", "tolerance": 1e-08, "verified": true} +{"case": "triinsqu@34", "density": 0.8767333914896571, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t34.gif", "friedman_metric_expression": "4.098+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.098, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@34", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "4.098+", "metric_symbol": "s", "metric_value": 4.098, "reference_text": "34. | | s = 4.098+ | Found by Emerson Connelly | in April 2026.", "source_image": "t34.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t34.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "34. | | s = 4.098+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t34.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "61b4d5d6bf22a206", "improvement": 0.00015051348449990343, "max_boundary_excess": -2.6262786795626285e-06, "max_pair_overlap_depth": -1.499292058504409e-05, "metric_expression": "4.09785", "metric_symbol": "r", "metric_value": 4.0978494865155, "n": 34, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 4.098, "setup": "triinsqu", "side": 4.0978494865155, "solution_path": "data/solutions/generated/triinsqu_34_61b4d5d6bf22a206.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/61b4d5d6bf22a206.svg", "tolerance": 1e-08, "verified": true} {"case": "triinsqu@35", "density": 0.8584202751980219, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t35.gif", "friedman_metric_expression": "9(2\u221a3\u20133) = 4.176+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.176, "friedman_reference": {"analytical_or_proved": true, "case": "triinsqu@35", "credit": "Erich Friedman", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "9(2\u221a3\u20133) = 4.176+", "metric_symbol": "s", "metric_value": 4.176, "reference_text": "35. | | s = 9(2\u221a3\u20133) = 4.176+ | Found by Erich Friedman | in April 2026.", "source_image": "t35.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t35.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "35. | | s = 9(2\u221a3\u20133) = 4.176+ | Found by Erich Friedman | in April 2026.", "friedman_source_image": "t35.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "0fd80a947c5904bb", "improvement": -0.025790132248165953, "max_boundary_excess": -2.0186743648470723e-06, "max_pair_overlap_depth": -1.0972661374369919e-05, "metric_expression": "4.20179", "metric_symbol": "s", "metric_value": 4.201790132248166, "n": 35, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 4.176, "setup": "triinsqu", "side": 4.201790132248166, "solution_path": "data/solutions/generated/triinsqu_035_0fd80a947c5904bb.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/0fd80a947c5904bb.svg", "tolerance": 1e-08, "verified": true} {"case": "triinsqu@36", "density": 0.8656950017914491, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t36.gif", "friedman_metric_expression": "4.214+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.214, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@36", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "4.214+", "metric_symbol": "s", "metric_value": 4.214, "reference_text": "36. | | s = 4.214+ | Found by Emerson Connelly | in April 2026.", "source_image": "t36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t36.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "36. | | s = 4.214+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "frontend_seed": true, "id": "73acd05c27fe131e", "improvement": -0.029450235060704877, "max_boundary_excess": -1.6988749159352698e-06, "max_pair_overlap_depth": -1.0359883173549633e-08, "metric_expression": "4.24345", "metric_symbol": "s", "metric_value": 4.243450235060705, "n": 36, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 4.214, "setup": "triinsqu", "side": 4.243450235060705, "solution_path": "data/solutions/generated/triinsqu_036_73acd05c27fe131e.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/73acd05c27fe131e.svg", "tolerance": 1e-08, "verified": true} {"case": "triinsqu@37", "density": 0.8777550041407036, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triinsqu/t37.gif", "friedman_family_updated": "4/29/26", "friedman_metric_expression": "4.276+", "friedman_metric_symbol": "s", "friedman_metric_value": 4.276, "friedman_reference": {"analytical_or_proved": false, "case": "triinsqu@37", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "4/29/26", "metric_expression": "4.276+", "metric_symbol": "s", "metric_value": 4.276, "reference_text": "37. | | s = 4.276+ | Found by Emerson Connelly | in April 2026.", "source_image": "t37.gif", "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t37.gif", "source_page": "https://erich-friedman.github.io/packing/triinsqu/", "title": "Triangles in Squares"}, "friedman_reference_text": "37. | | s = 4.276+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "t37.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triinsqu/", "friedman_title": "Triangles in Squares", "id": "8735d9f36ada5e93", "improvement": 0.0036726339396722096, "max_boundary_excess": -2.0547460484616992e-07, "max_pair_overlap_depth": -1.528656046745347e-06, "n": 37, "notes": "Seeded from the certified top-10 candidate bundle.", "reference_side": 4.276, "setup": "triinsqu", "side": 4.272327366060328, "solution_path": "data/solutions/triinsqu_37.json", "source_url": "https://erich-friedman.github.io/packing/triinsqu/", "submitted_at": "2026-04-30T23:19:18+00:00", "submitted_by": "Nathan Roll / polygon-packer-c", "svg_path": "data/svg/8735d9f36ada5e93.svg", "tolerance": 1e-08, "verified": true} @@ -814,7 +814,7 @@ {"case": "triintri@8", "density": 0.888817229166623, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@8", "credit": "Trivial", "date": "", "family_updated": "2/29/26", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "s = 3 | Trivial.", "source_image": "tt9.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 3 | Trivial.", "friedman_source_image": "tt9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "896797bdbe010b6f", "improvement": -0.00012093309333716107, "max_boundary_excess": -1.756783292246776e-06, "max_pair_overlap_depth": -1.7128373423530985e-06, "metric_expression": "3.00012", "metric_symbol": "s", "metric_value": 3.000120933093337, "n": 8, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 3.0, "setup": "triintri", "side": 3.000120933093337, "solution_path": "data/solutions/generated/triintri_008_896797bdbe010b6f.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/896797bdbe010b6f.svg", "tolerance": 1e-08, "verified": true} {"case": "triintri@9", "density": 0.9999350596545415, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "3", "friedman_metric_symbol": "s", "friedman_metric_value": 3.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@9", "credit": "Trivial", "date": "", "family_updated": "2/29/26", "metric_expression": "3", "metric_symbol": "s", "metric_value": 3.0, "reference_text": "s = 3 | Trivial.", "source_image": "tt9.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt9.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 3 | Trivial.", "friedman_source_image": "tt9.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "030d2c21a7344ace", "improvement": -9.741526284967605e-05, "max_boundary_excess": -1.8175499754633263e-06, "max_pair_overlap_depth": -5.983854353151415e-06, "metric_expression": "3.00010", "metric_symbol": "s", "metric_value": 3.0000974152628497, "n": 9, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 3.0, "setup": "triintri", "side": 3.0000974152628497, "solution_path": "data/solutions/generated/triintri_009_030d2c21a7344ace.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/030d2c21a7344ace.svg", "tolerance": 1e-08, "verified": true} {"case": "triintri@10", "density": 0.8162770652591925, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt10.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "7/2 = 3.500", "friedman_metric_symbol": "s", "friedman_metric_value": 3.5, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@10", "credit": "Erich Friedman", "date": "1997", "family_updated": "2/29/26", "metric_expression": "7/2 = 3.500", "metric_symbol": "s", "metric_value": 3.5, "reference_text": "s = 7/2 = 3.500 | Found by Erich Friedman | in 1997.", "source_image": "tt10.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt10.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 7/2 = 3.500 | Found by Erich Friedman | in 1997.", "friedman_source_image": "tt10.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "bcf7fa93627e2e22", "improvement": -0.00010604617003595607, "max_boundary_excess": -8.918936789115156e-08, "max_pair_overlap_depth": -1.2701382657187565e-05, "metric_expression": "3.50011", "metric_symbol": "s", "metric_value": 3.500106046170036, "n": 10, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 3.5, "setup": "triintri", "side": 3.500106046170036, "solution_path": "data/solutions/generated/triintri_010_bcf7fa93627e2e22.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/bcf7fa93627e2e22.svg", "tolerance": 1e-08, "verified": true} -{"case": "triintri@11", "density": 0.7606613339332848, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt11.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "9/4 + 9\u221a21/28 = 3.722+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.722, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@11", "credit": "Erich Friedman", "date": "1997", "family_updated": "2/29/26", "metric_expression": "9/4 + 9\u221a21/28 = 3.722+", "metric_symbol": "s", "metric_value": 3.722, "reference_text": "s = 9/4 + 9\u221a21/28 = 3.722+ | Found by Erich Friedman | in 1997.", "source_image": "tt11.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt11.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 9/4 + 9\u221a21/28 = 3.722+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "tt11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "2aa5afe3526231ee", "improvement": -0.08077536867961355, "max_boundary_excess": -3.529678376096273e-06, "max_pair_overlap_depth": -3.875027978134105e-06, "metric_expression": "3.80278", "metric_symbol": "s", "metric_value": 3.8027753686796135, "n": 11, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 3.722, "setup": "triintri", "side": 3.8027753686796135, "solution_path": "data/solutions/generated/triintri_011_2aa5afe3526231ee.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/2aa5afe3526231ee.svg", "tolerance": 1e-08, "verified": true} +{"case": "triintri@11", "density": 0.792568600317471, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt11.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "9/4 + 9\u221a21/28 = 3.722+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.722, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@11", "credit": "Erich Friedman", "date": "1997", "family_updated": "2/29/26", "metric_expression": "9/4 + 9\u221a21/28 = 3.722+", "metric_symbol": "s", "metric_value": 3.722, "reference_text": "s = 9/4 + 9\u221a21/28 = 3.722+ | Found by Erich Friedman | in 1997.", "source_image": "tt11.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt11.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 9/4 + 9\u221a21/28 = 3.722+ | Found by Erich Friedman | in 1997.", "friedman_source_image": "tt11.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "8c37202a929dc807", "improvement": -0.0034428993959463305, "max_boundary_excess": -2.7247990718051686e-06, "max_pair_overlap_depth": -2.4227159337045223e-05, "metric_expression": "3.72544", "metric_symbol": "r", "metric_value": 3.7254428993959463, "n": 11, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 3.722, "setup": "triintri", "side": 3.7254428993959463, "solution_path": "data/solutions/generated/triintri_11_8c37202a929dc807.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/8c37202a929dc807.svg", "tolerance": 1e-08, "verified": true} {"case": "triintri@12", "density": 0.785455836949608, "friedman_analytical_or_proved": true, "friedman_credit": "David W", "friedman_date": "July 2007", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt12.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "2 + 2cos(\u03c0/9) = 3.879+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.879, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@12", "credit": "David W", "date": "July 2007", "family_updated": "2/29/26", "metric_expression": "2 + 2cos(\u03c0/9) = 3.879+", "metric_symbol": "s", "metric_value": 3.879, "reference_text": "s = 2 + 2cos(\u03c0/9) = 3.879+ | Found by David W. Cantrell | in July 2007.", "source_image": "tt12.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt12.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 2 + 2cos(\u03c0/9) = 3.879+ | Found by David W. Cantrell | in July 2007.", "friedman_source_image": "tt12.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "2783dbd8fc940a50", "improvement": -0.029676586405010585, "max_boundary_excess": -2.5342122979254825e-07, "max_pair_overlap_depth": -1.2864706757542699e-05, "metric_expression": "3.90868", "metric_symbol": "s", "metric_value": 3.9086765864050106, "n": 12, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 3.879, "setup": "triintri", "side": 3.9086765864050106, "solution_path": "data/solutions/generated/triintri_012_2783dbd8fc940a50.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/2783dbd8fc940a50.svg", "tolerance": 1e-08, "verified": true} {"case": "triintri@13", "density": 0.8124441375462111, "friedman_analytical_or_proved": false, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt13.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "3.992+", "friedman_metric_symbol": "s", "friedman_metric_value": 3.992, "friedman_reference": {"analytical_or_proved": false, "case": "triintri@13", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "2/29/26", "metric_expression": "3.992+", "metric_symbol": "s", "metric_value": 3.992, "reference_text": "s = 3.992+ | Found by Maurizio Morandi | in May 2008.", "source_image": "tt13.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt13.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 3.992+ | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tt13.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "c0e9826b835df8c1", "improvement": -0.00813751466958923, "max_boundary_excess": -4.7058045362323485e-07, "max_pair_overlap_depth": -5.647083534121577e-07, "metric_expression": "4.00014", "metric_symbol": "s", "metric_value": 4.000137514669589, "n": 13, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 3.992, "setup": "triintri", "side": 4.000137514669589, "solution_path": "data/solutions/generated/triintri_013_c0e9826b835df8c1.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/c0e9826b835df8c1.svg", "tolerance": 1e-08, "verified": true} {"case": "triintri@14", "density": 0.8749380783267208, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "friedman_family_updated": "2/29/26", "friedman_metric_expression": "4", "friedman_metric_symbol": "s", "friedman_metric_value": 4.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@14", "credit": "Trivial", "date": "", "family_updated": "2/29/26", "metric_expression": "4", "metric_symbol": "s", "metric_value": 4.0, "reference_text": "s = 4 | Trivial.", "source_image": "tt16.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt16.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 4 | Trivial.", "friedman_source_image": "tt16.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "486f49c585ff26d0", "improvement": -0.0001415427657383006, "max_boundary_excess": -1.0358408447228967e-06, "max_pair_overlap_depth": -6.298432640994278e-07, "metric_expression": "4.00014", "metric_symbol": "s", "metric_value": 4.000141542765738, "n": 14, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 4.0, "setup": "triintri", "side": 4.000141542765738, "solution_path": "data/solutions/generated/triintri_014_486f49c585ff26d0.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:05:47+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/486f49c585ff26d0.svg", "tolerance": 1e-08, "verified": true} @@ -832,8 +832,8 @@ {"case": "triintri@26", "density": 0.8757579251966457, "friedman_analytical_or_proved": false, "friedman_credit": "Joah Frenzley", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt26.gif", "friedman_metric_expression": "5.406+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.406, "friedman_reference": {"analytical_or_proved": false, "case": "triintri@26", "credit": "Joah Frenzley", "date": "April 2026", "family_updated": "2/29/26", "metric_expression": "5.406+", "metric_symbol": "s", "metric_value": 5.406, "reference_text": "s = 5.406+ | Found by Joah Frenzley | in April 2026.", "source_image": "tt26.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt26.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 5.406+ | Found by Joah Frenzley | in April 2026.", "friedman_source_image": "tt26.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "8db318b73d49217d", "improvement": -0.04272181972430644, "max_boundary_excess": -1.2897298944239458e-06, "max_pair_overlap_depth": -1.0218222609881256e-05, "metric_expression": "5.44872", "metric_symbol": "s", "metric_value": 5.448721819724306, "n": 26, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 5.406, "setup": "triintri", "side": 5.448721819724306, "solution_path": "data/solutions/generated/triintri_026_8db318b73d49217d.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/8db318b73d49217d.svg", "tolerance": 1e-08, "verified": true} {"case": "triintri@27", "density": 0.8924996192164427, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt28.gif", "friedman_metric_expression": "11/2 = 5.500", "friedman_metric_symbol": "s", "friedman_metric_value": 5.5, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@27", "credit": "Erich Friedman", "date": "1997", "family_updated": "2/29/26", "metric_expression": "11/2 = 5.500", "metric_symbol": "s", "metric_value": 5.5, "reference_text": "s = 11/2 = 5.500 | Found by Erich Friedman | in 1997.", "source_image": "tt28.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt28.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 11/2 = 5.500 | Found by Erich Friedman | in 1997.", "friedman_source_image": "tt28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "f1c2be61e829f1e2", "improvement": -0.00019215549305418023, "max_boundary_excess": -1.8365186496628e-06, "max_pair_overlap_depth": -1.8064988390742442e-06, "metric_expression": "5.50019", "metric_symbol": "s", "metric_value": 5.500192155493054, "n": 27, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 5.5, "setup": "triintri", "side": 5.500192155493054, "solution_path": "data/solutions/generated/triintri_027_f1c2be61e829f1e2.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/f1c2be61e829f1e2.svg", "tolerance": 1e-08, "verified": true} {"case": "triintri@28", "density": 0.9255631973649003, "friedman_analytical_or_proved": true, "friedman_credit": "Erich Friedman", "friedman_date": "1997", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt28.gif", "friedman_metric_expression": "11/2 = 5.500", "friedman_metric_symbol": "s", "friedman_metric_value": 5.5, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@28", "credit": "Erich Friedman", "date": "1997", "family_updated": "2/29/26", "metric_expression": "11/2 = 5.500", "metric_symbol": "s", "metric_value": 5.5, "reference_text": "s = 11/2 = 5.500 | Found by Erich Friedman | in 1997.", "source_image": "tt28.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt28.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 11/2 = 5.500 | Found by Erich Friedman | in 1997.", "friedman_source_image": "tt28.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "9562d7d6e075a347", "improvement": -0.00016827626559567932, "max_boundary_excess": -2.296352348617567e-06, "max_pair_overlap_depth": -7.147352432124165e-08, "metric_expression": "5.50017", "metric_symbol": "s", "metric_value": 5.500168276265596, "n": 28, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 5.5, "setup": "triintri", "side": 5.500168276265596, "solution_path": "data/solutions/generated/triintri_028_9562d7d6e075a347.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/9562d7d6e075a347.svg", "tolerance": 1e-08, "verified": true} -{"case": "triintri@29", "density": 0.823477123057378, "friedman_analytical_or_proved": true, "friedman_credit": "David W", "friedman_date": "July 2007", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt29.gif", "friedman_metric_expression": "17/3 = 5.666+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.666, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@29", "credit": "David W", "date": "July 2007", "family_updated": "2/29/26", "metric_expression": "17/3 = 5.666+", "metric_symbol": "s", "metric_value": 5.666, "reference_text": "s = 17/3 = 5.666+ | Found by David W. Cantrell | in July 2007.", "source_image": "tt29.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt29.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 17/3 = 5.666+ | Found by David W. Cantrell | in July 2007.", "friedman_source_image": "tt29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "1e26d810ba567eb4", "improvement": -0.2683509950166121, "max_boundary_excess": -5.6694191623693234e-06, "max_pair_overlap_depth": -2.9353541475263256e-07, "metric_expression": "5.93435", "metric_symbol": "s", "metric_value": 5.9343509950166125, "n": 29, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 5.666, "setup": "triintri", "side": 5.9343509950166125, "solution_path": "data/solutions/generated/triintri_029_1e26d810ba567eb4.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/1e26d810ba567eb4.svg", "tolerance": 1e-08, "verified": true} -{"case": "triintri@30", "density": 0.8727149559817791, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt30.gif", "friedman_metric_expression": "23/4 = 5.75", "friedman_metric_symbol": "s", "friedman_metric_value": 5.75, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@30", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "2/29/26", "metric_expression": "23/4 = 5.75", "metric_symbol": "s", "metric_value": 5.75, "reference_text": "s = 23/4 = 5.75 | Found by Maurizio Morandi | in May 2008.", "source_image": "tt30.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt30.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 23/4 = 5.75 | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tt30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "1ace87e09c5f6fc8", "improvement": -0.11306107243262797, "max_boundary_excess": -1.1340847276256848e-05, "max_pair_overlap_depth": -5.548494655616132e-06, "metric_expression": "5.86306", "metric_symbol": "s", "metric_value": 5.863061072432628, "n": 30, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 5.75, "setup": "triintri", "side": 5.863061072432628, "solution_path": "data/solutions/generated/triintri_030_1ace87e09c5f6fc8.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/1ace87e09c5f6fc8.svg", "tolerance": 1e-08, "verified": true} +{"case": "triintri@29", "density": 0.9006896846506488, "friedman_analytical_or_proved": true, "friedman_credit": "David W", "friedman_date": "July 2007", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt29.gif", "friedman_metric_expression": "17/3 = 5.666+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.666, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@29", "credit": "David W", "date": "July 2007", "family_updated": "2/29/26", "metric_expression": "17/3 = 5.666+", "metric_symbol": "s", "metric_value": 5.666, "reference_text": "s = 17/3 = 5.666+ | Found by David W. Cantrell | in July 2007.", "source_image": "tt29.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt29.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 17/3 = 5.666+ | Found by David W. Cantrell | in July 2007.", "friedman_source_image": "tt29.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "f2204c752005e940", "improvement": -0.008288388557018855, "max_boundary_excess": -3.6782066596349484e-07, "max_pair_overlap_depth": -5.636043209800157e-06, "metric_expression": "5.67429", "metric_symbol": "r", "metric_value": 5.674288388557019, "n": 29, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 5.666, "setup": "triintri", "side": 5.674288388557019, "solution_path": "data/solutions/generated/triintri_29_f2204c752005e940.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/f2204c752005e940.svg", "tolerance": 1e-08, "verified": true} +{"case": "triintri@30", "density": 0.9060985977127352, "friedman_analytical_or_proved": true, "friedman_credit": "Maurizio Morandi", "friedman_date": "May 2008", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt30.gif", "friedman_metric_expression": "23/4 = 5.75", "friedman_metric_symbol": "s", "friedman_metric_value": 5.75, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@30", "credit": "Maurizio Morandi", "date": "May 2008", "family_updated": "2/29/26", "metric_expression": "23/4 = 5.75", "metric_symbol": "s", "metric_value": 5.75, "reference_text": "s = 23/4 = 5.75 | Found by Maurizio Morandi | in May 2008.", "source_image": "tt30.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt30.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 23/4 = 5.75 | Found by Maurizio Morandi | in May 2008.", "friedman_source_image": "tt30.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "149a91e0248df04e", "improvement": -0.004040286099913359, "max_boundary_excess": -1.2882428961650933e-06, "max_pair_overlap_depth": -6.414942533283785e-06, "metric_expression": "5.75404", "metric_symbol": "r", "metric_value": 5.754040286099913, "n": 30, "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", "record_type": "verified", "reference_side": 5.75, "setup": "triintri", "side": 5.754040286099913, "solution_path": "data/solutions/generated/triintri_30_149a91e0248df04e.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T02:08:58+00:00", "submitted_by": "polygon-packer-c image seed", "svg_path": "data/svg/149a91e0248df04e.svg", "tolerance": 1e-08, "verified": true} {"case": "triintri@31", "density": 0.8655205028371107, "friedman_analytical_or_proved": false, "friedman_credit": "Emerson Connelly", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt31.gif", "friedman_metric_expression": "5.912+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.912, "friedman_reference": {"analytical_or_proved": false, "case": "triintri@31", "credit": "Emerson Connelly", "date": "April 2026", "family_updated": "2/29/26", "metric_expression": "5.912+", "metric_symbol": "s", "metric_value": 5.912, "reference_text": "s = 5.912+ | Found by Emerson Connelly | in April 2026.", "source_image": "tt31.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt31.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 5.912+ | Found by Emerson Connelly | in April 2026.", "friedman_source_image": "tt31.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "e94aa7c2fc600373", "improvement": -0.07269699341354574, "max_boundary_excess": -1.1753168632427688e-06, "max_pair_overlap_depth": -1.577861272879927e-05, "metric_expression": "5.98470", "metric_symbol": "s", "metric_value": 5.984696993413546, "n": 31, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 5.912, "setup": "triintri", "side": 5.984696993413546, "solution_path": "data/solutions/generated/triintri_031_e94aa7c2fc600373.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/e94aa7c2fc600373.svg", "tolerance": 1e-08, "verified": true} {"case": "triintri@32", "density": 0.8888356604955802, "friedman_analytical_or_proved": false, "friedman_credit": "Ian Watson", "friedman_date": "April 2026", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt32.gif", "friedman_metric_expression": "5.951+", "friedman_metric_symbol": "s", "friedman_metric_value": 5.951, "friedman_reference": {"analytical_or_proved": false, "case": "triintri@32", "credit": "Ian Watson", "date": "April 2026", "family_updated": "2/29/26", "metric_expression": "5.951+", "metric_symbol": "s", "metric_value": 5.951, "reference_text": "s = 5.951+ | Found by Ian Watson | in April 2026.", "source_image": "tt32.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt32.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 5.951+ | Found by Ian Watson | in April 2026.", "friedman_source_image": "tt32.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "ea3eb3c434a6db88", "improvement": -0.049179653895976116, "max_boundary_excess": -2.19130606600491e-06, "max_pair_overlap_depth": -2.006545093746759e-06, "metric_expression": "6.00018", "metric_symbol": "s", "metric_value": 6.000179653895976, "n": 32, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 5.951, "setup": "triintri", "side": 6.000179653895976, "solution_path": "data/solutions/generated/triintri_032_ea3eb3c434a6db88.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/ea3eb3c434a6db88.svg", "tolerance": 1e-08, "verified": true} {"case": "triintri@33", "density": 0.9165918569883487, "friedman_analytical_or_proved": true, "friedman_credit": "Trivial", "friedman_date": "", "friedman_external_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "friedman_metric_expression": "6", "friedman_metric_symbol": "s", "friedman_metric_value": 6.0, "friedman_reference": {"analytical_or_proved": true, "case": "triintri@33", "credit": "Trivial", "date": "", "family_updated": "2/29/26", "metric_expression": "6", "metric_symbol": "s", "metric_value": 6.0, "reference_text": "s = 6 | Trivial. |", "source_image": "tt36.gif", "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt36.gif", "source_page": "https://erich-friedman.github.io/packing/triintri/", "title": "Triangles in Triangles"}, "friedman_reference_text": "s = 6 | Trivial. |", "friedman_source_image": "tt36.gif", "friedman_source_url": "https://erich-friedman.github.io/packing/triintri/", "friedman_title": "Triangles in Triangles", "frontend_seed": true, "id": "5e7c3bf3ddd6ef7e", "improvement": -0.0002448466611522093, "max_boundary_excess": -4.630617313298577e-07, "max_pair_overlap_depth": -7.583551231427066e-09, "metric_expression": "6.00024", "metric_symbol": "s", "metric_value": 6.000244846661152, "n": 33, "notes": "Generated seed layout for frontend figures with polygon-packer-c.", "record_type": "verified", "reference_side": 6.0, "setup": "triintri", "side": 6.000244846661152, "solution_path": "data/solutions/generated/triintri_033_5e7c3bf3ddd6ef7e.json", "source_url": "https://erich-friedman.github.io/packing/triintri/", "submitted_at": "2026-05-01T01:47:21+00:00", "submitted_by": "polygon-packer-c seed", "svg_path": "data/svg/5e7c3bf3ddd6ef7e.svg", "tolerance": 1e-08, "verified": true} diff --git a/data/solutions/generated/cirincir_10_be14c55679a54b28.json b/data/solutions/generated/cirincir_10_be14c55679a54b28.json new file mode 100644 index 0000000000000000000000000000000000000000..80f1267d098542377ef19e558d456b13a43905dd --- /dev/null +++ b/data/solutions/generated/cirincir_10_be14c55679a54b28.json @@ -0,0 +1,119 @@ +{ + "case": "cirincir@10", + "container": { + "radius": 3.813115539124243, + "type": "circle" + }, + "friedman_reference": { + "analytical_or_proved": true, + "case": "cirincir@10", + "credit": "Pirl", + "date": "1969", + "family_updated": "6/27/05", + "metric_expression": "3.813+", + "metric_symbol": "r", + "metric_value": 3.813, + "reference_text": "r = 3.813+ | Proved by Pirl in 1969.", + "source_image": "ccc10.gif", + "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc10.gif", + "source_page": "https://erich-friedman.github.io/packing/cirincir/", + "title": "Circles in Circles" + }, + "item": { + "radius": 1.0, + "type": "circle" + }, + "metadata": { + "certification_slack": 1.000040000600004, + "friedman_reference": { + "analytical_or_proved": true, + "case": "cirincir@10", + "credit": "Pirl", + "date": "1969", + "family_updated": "6/27/05", + "metric_expression": "3.813+", + "metric_symbol": "r", + "metric_value": 3.813, + "reference_text": "r = 3.813+ | Proved by Pirl in 1969.", + "source_image": "ccc10.gif", + "source_image_url": "https://erich-friedman.github.io/packing/cirincir/ccc10.gif", + "source_page": "https://erich-friedman.github.io/packing/cirincir/", + "title": "Circles in Circles" + }, + "generator": "friedman_image_seed", + "generator_profile": "best of cg/steepest/sa seed profiles", + "image_seed": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/site_images/cirincir_010.gif", + "image_seed_transform_error": 0.06260662900137194, + "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", + "refined_state": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/states/cirincir_010_image_refined.state", + "source_url": "https://erich-friedman.github.io/packing/cirincir/", + "submitted_at": "2026-05-01T02:08:58+00:00", + "submitted_by": "polygon-packer-c image seed" + }, + "placements": [ + { + "id": 1, + "rotation_radians": 4.250841583787497, + "x": -1.5495894413068694, + "y": 2.347809976552831 + }, + { + "id": 2, + "rotation_radians": 1.2049245946973615, + "x": 2.475977848974938, + "y": -1.3352782482689725 + }, + { + "id": 3, + "rotation_radians": 3.324535391146819, + "x": 0.9628155378992221, + "y": -2.6431837470626016 + }, + { + "id": 4, + "rotation_radians": 2.235924865008793, + "x": -2.7181525310229597, + "y": 0.724620859045127 + }, + { + "id": 5, + "rotation_radians": 5.015462004334633, + "x": 0.4022919298430151, + "y": 2.7841965332540686 + }, + { + "id": 6, + "rotation_radians": 0.28911623382725804, + "x": -1.0370560344683115, + "y": -2.614970847945628 + }, + { + "id": 7, + "rotation_radians": 0.652847074253164, + "x": 2.7375524795110038, + "y": 0.6476129213608892 + }, + { + "id": 8, + "rotation_radians": 2.753751059344826, + "x": 0.7444079617997427, + "y": 0.813627221395956 + }, + { + "id": 9, + "rotation_radians": 5.720342076194702, + "x": -0.6056692970283514, + "y": -0.6619913272422999 + }, + { + "id": 10, + "rotation_radians": 2.794472784545188, + "x": -2.512698883945597, + "y": -1.2648690591199467 + } + ], + "problem": "10 equal items for Circles in Circles", + "setup": "cirincir", + "source_image": "https://erich-friedman.github.io/packing/cirincir/ccc10.gif", + "source_page": "https://erich-friedman.github.io/packing/cirincir/" +} diff --git a/data/solutions/generated/cirinsqu_10_522948d614943781.json b/data/solutions/generated/cirinsqu_10_522948d614943781.json new file mode 100644 index 0000000000000000000000000000000000000000..b8f6c609632fc23206b2d20edc5b07695a38318e --- /dev/null +++ b/data/solutions/generated/cirinsqu_10_522948d614943781.json @@ -0,0 +1,122 @@ +{ + "case": "cirinsqu@10", + "container": { + "circumradius": 4.771259329856722, + "orientation_radians": 0.0, + "side_length": 6.74757965388254, + "sides": 4, + "type": "regular_polygon" + }, + "friedman_reference": { + "analytical_or_proved": true, + "case": "cirinsqu@10", + "credit": "De Groot", + "date": "1990", + "family_updated": "10/9/10", + "metric_expression": "6.747+", + "metric_symbol": "s", + "metric_value": 6.747, + "reference_text": "s = 6.747+ | Proved by De Groot in 1990.", + "source_image": "cs10.gif", + "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs10.gif", + "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", + "title": "Circles in Squares" + }, + "item": { + "radius": 1.0, + "type": "circle" + }, + "metadata": { + "certification_slack": 1.000040000600004, + "friedman_reference": { + "analytical_or_proved": true, + "case": "cirinsqu@10", + "credit": "De Groot", + "date": "1990", + "family_updated": "10/9/10", + "metric_expression": "6.747+", + "metric_symbol": "s", + "metric_value": 6.747, + "reference_text": "s = 6.747+ | Proved by De Groot in 1990.", + "source_image": "cs10.gif", + "source_image_url": "https://erich-friedman.github.io/packing/cirinsqu/cs10.gif", + "source_page": "https://erich-friedman.github.io/packing/cirinsqu/", + "title": "Circles in Squares" + }, + "generator": "friedman_image_seed", + "generator_profile": "best of cg/steepest/sa seed profiles", + "image_seed": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/site_images/cirinsqu_010.gif", + "image_seed_transform_error": 1.0052492910490174e-15, + "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", + "refined_state": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/states/cirinsqu_010_image_refined.state", + "source_url": "https://erich-friedman.github.io/packing/cirinsqu/", + "submitted_at": "2026-05-01T02:08:58+00:00", + "submitted_by": "polygon-packer-c image seed" + }, + "placements": [ + { + "id": 1, + "rotation_radians": 0.6631201483233983, + "x": -1.452583219963529, + "y": -1.9044085825632537 + }, + { + "id": 2, + "rotation_radians": 3.460415277605134, + "x": -0.038322427094046985, + "y": -3.3186863548500445 + }, + { + "id": 3, + "rotation_radians": 5.175384480811934, + "x": 1.4137529043718686, + "y": -1.9432788926909437 + }, + { + "id": 4, + "rotation_radians": 1.210268797097649, + "x": 2.828002842573363, + "y": -0.5289911772412871 + }, + { + "id": 5, + "rotation_radians": 1.2733469800666406, + "x": -3.357028277502991, + "y": -1.8343155996848248e-06 + }, + { + "id": 6, + "rotation_radians": 0.7406471839276207, + "x": -1.3591735215093275, + "y": 0.09348660293620727 + }, + { + "id": 7, + "rotation_radians": 5.70981309480042, + "x": 0.6312250856386635, + "y": -0.10267057200586589 + }, + { + "id": 8, + "rotation_radians": 0.7817877980872996, + "x": 2.045418000705354, + "y": 1.3116253947392875 + }, + { + "id": 9, + "rotation_radians": 3.1295383626226756, + "x": -1.2656324453185372, + "y": 2.091369376176176 + }, + { + "id": 10, + "rotation_radians": 1.2491990785030422, + "x": 0.6311280446094484, + "y": 2.7258703870584085 + } + ], + "problem": "10 equal items for Circles in Squares", + "setup": "cirinsqu", + "source_image": "https://erich-friedman.github.io/packing/cirinsqu/cs10.gif", + "source_page": "https://erich-friedman.github.io/packing/cirinsqu/" +} diff --git a/data/solutions/generated/triincir_16_8bb5531a7df45934.json b/data/solutions/generated/triincir_16_8bb5531a7df45934.json new file mode 100644 index 0000000000000000000000000000000000000000..d2a4a4ca51016b1f6a4723dca82e73e7f327eb28 --- /dev/null +++ b/data/solutions/generated/triincir_16_8bb5531a7df45934.json @@ -0,0 +1,157 @@ +{ + "case": "triincir@16", + "container": { + "radius": 1.6880356342518776, + "type": "circle" + }, + "friedman_reference": { + "analytical_or_proved": false, + "case": "triincir@16", + "credit": "Maurizio Morandi", + "date": "April 2008", + "family_updated": "8/31/19", + "metric_expression": "1.687+", + "metric_symbol": "r", + "metric_value": 1.687, + "reference_text": "r = 1.687+ | Found by Maurizio Morandi | in April 2008.", + "source_image": "tc16.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc16.gif", + "source_page": "https://erich-friedman.github.io/packing/triincir/", + "title": "Triangles in Circles" + }, + "item": { + "circumradius": 0.5773502691896258, + "side_length": 1.0, + "sides": 3, + "type": "regular_polygon" + }, + "metadata": { + "certification_slack": 1.0000500010000102, + "friedman_reference": { + "analytical_or_proved": false, + "case": "triincir@16", + "credit": "Maurizio Morandi", + "date": "April 2008", + "family_updated": "8/31/19", + "metric_expression": "1.687+", + "metric_symbol": "r", + "metric_value": 1.687, + "reference_text": "r = 1.687+ | Found by Maurizio Morandi | in April 2008.", + "source_image": "tc16.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc16.gif", + "source_page": "https://erich-friedman.github.io/packing/triincir/", + "title": "Triangles in Circles" + }, + "generator": "friedman_image_seed", + "generator_profile": "best of cg/steepest/sa seed profiles", + "image_seed": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/site_images/triincir_016.gif", + "image_seed_transform_error": 0.027220545981561973, + "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", + "refined_state": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/states/triincir_016_image_refined.state", + "source_url": "https://erich-friedman.github.io/packing/triincir/", + "submitted_at": "2026-05-01T02:08:58+00:00", + "submitted_by": "polygon-packer-c image seed" + }, + "placements": [ + { + "id": 1, + "rotation_radians": 3.664836111795197, + "x": 0.17784122734082114, + "y": -1.2571047992820488 + }, + { + "id": 2, + "rotation_radians": 4.714168249798268, + "x": 0.6790906403670012, + "y": -0.9651644644659932 + }, + { + "id": 3, + "rotation_radians": 4.714791780188679, + "x": -0.6756068095649385, + "y": -0.9701790597822169 + }, + { + "id": 4, + "rotation_radians": 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+ "metric_expression": "2", + "metric_symbol": "r", + "metric_value": 2.0, + "reference_text": "r = 2 | Trivial.", + "source_image": "tc24.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc24.gif", + "source_page": "https://erich-friedman.github.io/packing/triincir/", + "title": "Triangles in Circles" + }, + "generator": "friedman_image_seed", + "generator_profile": "best of cg/steepest/sa seed profiles", + "image_seed": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/site_images/triincir_024.gif", + "image_seed_transform_error": 0.007191923516456017, + "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", + "refined_state": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/states/triincir_024_image_refined.state", + "source_url": "https://erich-friedman.github.io/packing/triincir/", + "submitted_at": "2026-05-01T02:08:58+00:00", + "submitted_by": 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"https://erich-friedman.github.io/packing/triincir/tc24.gif", + "source_page": "https://erich-friedman.github.io/packing/triincir/" +} diff --git a/data/solutions/generated/triincir_26_8245fa6b2a472124.json b/data/solutions/generated/triincir_26_8245fa6b2a472124.json new file mode 100644 index 0000000000000000000000000000000000000000..a8d0f45b4d795bb45322cc215b5cb6a2c58dc48e --- /dev/null +++ b/data/solutions/generated/triincir_26_8245fa6b2a472124.json @@ -0,0 +1,217 @@ +{ + "case": "triincir@26", + "container": { + "radius": 2.1091687246398596, + "type": "circle" + }, + "friedman_reference": { + "analytical_or_proved": false, + "case": "triincir@26", + "credit": "Maurizio Morandi", + "date": "April 2008", + "family_updated": "8/31/19", + "metric_expression": "2.109+", + "metric_symbol": "r", + "metric_value": 2.109, + "reference_text": "r = 2.109+ | Found by Maurizio Morandi | in April 2008.", + "source_image": "tc26.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc26.gif", + "source_page": "https://erich-friedman.github.io/packing/triincir/", + "title": "Triangles in Circles" + }, + "item": { + "circumradius": 0.5773502691896258, + "side_length": 1.0, + "sides": 3, + "type": "regular_polygon" + }, + "metadata": { + "certification_slack": 1.000040000600004, + "friedman_reference": { + "analytical_or_proved": false, + "case": "triincir@26", + "credit": "Maurizio Morandi", + "date": "April 2008", + "family_updated": "8/31/19", + "metric_expression": "2.109+", + "metric_symbol": "r", + "metric_value": 2.109, + "reference_text": "r = 2.109+ | Found by Maurizio Morandi | in April 2008.", + "source_image": "tc26.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triincir/tc26.gif", + "source_page": "https://erich-friedman.github.io/packing/triincir/", + "title": "Triangles in Circles" + }, + "generator": "friedman_image_seed", + "generator_profile": "best of cg/steepest/sa seed 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b/data/solutions/generated/triinhex_28_3435500023858121.json new file mode 100644 index 0000000000000000000000000000000000000000..cb8a3e38fa7b17c7a952acd1f3b92e340788aae7 --- /dev/null +++ b/data/solutions/generated/triinhex_28_3435500023858121.json @@ -0,0 +1,232 @@ +{ + "case": "triinhex@28", + "container": { + "circumradius": 2.3245501826518793, + "orientation_radians": 0.0, + "side_length": 2.324550182651879, + "sides": 6, + "type": "regular_polygon" + }, + "friedman_reference": { + "analytical_or_proved": false, + "case": "triinhex@28", + "credit": "Ian Watson", + "date": "April 2026", + "family_updated": "4/29/26", + "metric_expression": "2.32289+", + "metric_symbol": "s", + "metric_value": 2.32289, + "reference_text": "s = 2.32289+ | Found by Ian Watson | in April 2026.", + "source_image": "28.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinhex/28.gif", + "source_page": "https://erich-friedman.github.io/packing/triinhex/", + "title": "Triangles in 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--git a/data/solutions/generated/triinhex_29_4339d1ebd3160f94.json b/data/solutions/generated/triinhex_29_4339d1ebd3160f94.json new file mode 100644 index 0000000000000000000000000000000000000000..cd3a65caa6c7463ed590cb8fb62a4e8542b0f734 --- /dev/null +++ b/data/solutions/generated/triinhex_29_4339d1ebd3160f94.json @@ -0,0 +1,238 @@ +{ + "case": "triinhex@29", + "container": { + "circumradius": 2.3335122751747748, + "orientation_radians": 0.0, + "side_length": 2.3335122751747743, + "sides": 6, + "type": "regular_polygon" + }, + "friedman_reference": { + "analytical_or_proved": true, + "case": "triinhex@29", + "credit": "Ignacio Vallejo", + "date": "April 2026", + "family_updated": "4/29/26", + "metric_expression": "7/3 = 2.333+", + "metric_symbol": "s", + "metric_value": 2.333, + "reference_text": "s = 7/3 = 2.333+ | Found by Ignacio Vallejo | in April 2026.", + "source_image": "29.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinhex/29.gif", + "source_page": "https://erich-friedman.github.io/packing/triinhex/", + "title": "Triangles in Hexagons" + }, + "item": { + "circumradius": 0.5773502691896258, + "side_length": 1.0, + "sides": 3, + "type": "regular_polygon" + }, + "metadata": { + "certification_slack": 1.0000500010000102, + "friedman_reference": { + "analytical_or_proved": true, + "case": "triinhex@29", + "credit": "Ignacio Vallejo", + "date": "April 2026", + "family_updated": "4/29/26", + "metric_expression": "7/3 = 2.333+", + "metric_symbol": "s", + "metric_value": 2.333, + "reference_text": "s = 7/3 = 2.333+ | Found by Ignacio Vallejo | in April 2026.", + "source_image": "29.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinhex/29.gif", + "source_page": "https://erich-friedman.github.io/packing/triinhex/", + "title": "Triangles in Hexagons" + }, + "generator": "friedman_image_seed", + "generator_profile": "best of cg/steepest/sa seed profiles", + "image_seed": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/site_images/triinhex_029.gif", + "image_seed_transform_error": 0.07906520028491007, + "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", + "refined_state": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/states/triinhex_029_image_refined.state", + "source_url": "https://erich-friedman.github.io/packing/triinhex/", + "submitted_at": "2026-05-01T02:08:58+00:00", + "submitted_by": "polygon-packer-c image seed" + }, + "placements": [ + { + "id": 1, + "rotation_radians": 3.665197390618569, + "x": 0.6667249012775238, + "y": -1.732195770703404 + }, + { + "id": 2, + "rotation_radians": 3.667253655144894, + "x": -0.7778413840254516, + "y": -1.5397734147449378 + }, + { + "id": 3, + "rotation_radians": 4.712394252117648, + "x": 1.1667445825097287, + "y": -1.4435043302803225 + }, + { + "id": 4, + "rotation_radians": 4.71239660655649, + "x": 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equal items for Triangles in Hexagons", + "setup": "triinhex", + "source_image": "https://erich-friedman.github.io/packing/triinhex/29.gif", + "source_page": "https://erich-friedman.github.io/packing/triinhex/" +} diff --git a/data/solutions/generated/triinhex_30_7274d354e73d8050.json b/data/solutions/generated/triinhex_30_7274d354e73d8050.json new file mode 100644 index 0000000000000000000000000000000000000000..d89ec61ef9a899baea51329d6bf98d4f63302c6e --- /dev/null +++ b/data/solutions/generated/triinhex_30_7274d354e73d8050.json @@ -0,0 +1,244 @@ +{ + "case": "triinhex@30", + "container": { + "circumradius": 2.3752375092626914, + "orientation_radians": 0.0, + "side_length": 2.375237509262691, + "sides": 6, + "type": "regular_polygon" + }, + "friedman_reference": { + "analytical_or_proved": true, + "case": "triinhex@30", + "credit": "Ignacio Vallejo", + "date": "April 2026", + "family_updated": "4/29/26", + "metric_expression": "19/8 = 2.375", + "metric_symbol": "s", + "metric_value": 2.375, + "reference_text": "s = 19/8 = 2.375 | Found by Ignacio Vallejo | in April 2026.", + "source_image": "30.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinhex/30.gif", + "source_page": "https://erich-friedman.github.io/packing/triinhex/", + "title": "Triangles in Hexagons" + }, + "item": { + "circumradius": 0.5773502691896258, + "side_length": 1.0, + "sides": 3, + "type": "regular_polygon" + }, + "metadata": { + "certification_slack": 1.0000600015000203, + "friedman_reference": { + "analytical_or_proved": true, + "case": "triinhex@30", + "credit": "Ignacio Vallejo", + "date": "April 2026", + "family_updated": "4/29/26", + "metric_expression": "19/8 = 2.375", + "metric_symbol": "s", + "metric_value": 2.375, + "reference_text": "s = 19/8 = 2.375 | Found by Ignacio Vallejo | in April 2026.", + "source_image": "30.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinhex/30.gif", + "source_page": "https://erich-friedman.github.io/packing/triinhex/", + "title": "Triangles in Hexagons" + }, + "generator": "friedman_image_seed", + "generator_profile": "best of cg/steepest/sa seed profiles", + "image_seed": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/site_images/triinhex_030.gif", + "image_seed_transform_error": 0.08866036628188438, + "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", + "refined_state": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/states/triinhex_030_image_refined.state", + "source_url": "https://erich-friedman.github.io/packing/triinhex/", + "submitted_at": "2026-05-01T02:08:58+00:00", + "submitted_by": "polygon-packer-c image seed" + }, + "placements": [ + { + "id": 1, + "rotation_radians": 0.5236024615256353, + "x": -0.16531613879444262, + "y": 1.7683251677172405 + }, + { + "id": 2, + "rotation_radians": 0.5235684875505077, + "x": 0.761172030870743, + 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"orientation_radians": 0.0, + "side_length": 2.4011658949471424, + "sides": 6, + "type": "regular_polygon" + }, + "friedman_reference": { + "analytical_or_proved": true, + "case": "triinhex@31", + "credit": "Emerson Connelly", + "date": "April 2026", + "family_updated": "4/29/26", + "metric_expression": "12/5 = 2.400", + "metric_symbol": "s", + "metric_value": 2.4, + "reference_text": "s = 12/5 = 2.400 | Found by Emerson Connelly | in April 2026.", + "source_image": "31.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinhex/31.gif", + "source_page": "https://erich-friedman.github.io/packing/triinhex/", + "title": "Triangles in Hexagons" + }, + "item": { + "circumradius": 0.5773502691896258, + "side_length": 1.0, + "sides": 3, + "type": "regular_polygon" + }, + "metadata": { + "certification_slack": 1.000040000600004, + "friedman_reference": { + "analytical_or_proved": true, + "case": "triinhex@31", + "credit": "Emerson Connelly", + "date": "April 2026", + "family_updated": "4/29/26", + "metric_expression": "12/5 = 2.400", + "metric_symbol": "s", + "metric_value": 2.4, + "reference_text": "s = 12/5 = 2.400 | Found by Emerson Connelly | in April 2026.", + "source_image": "31.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinhex/31.gif", + "source_page": "https://erich-friedman.github.io/packing/triinhex/", + "title": "Triangles in Hexagons" + }, + "generator": "friedman_image_seed", + "generator_profile": "best of cg/steepest/sa seed profiles", + "image_seed": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/site_images/triinhex_031.gif", + "image_seed_transform_error": 0.11402163884943653, + "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", + "refined_state": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/states/triinhex_031_image_refined.state", + "source_url": 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"https://erich-friedman.github.io/packing/triinhex/" +} diff --git a/data/solutions/generated/triinhex_35_ec49778cf450c37b.json b/data/solutions/generated/triinhex_35_ec49778cf450c37b.json new file mode 100644 index 0000000000000000000000000000000000000000..690a3f6645a7532ccc43fd146ed8c10f1aee3acf --- /dev/null +++ b/data/solutions/generated/triinhex_35_ec49778cf450c37b.json @@ -0,0 +1,274 @@ +{ + "case": "triinhex@35", + "container": { + "circumradius": 2.500225007500126, + "orientation_radians": 0.0, + "side_length": 2.5002250075001258, + "sides": 6, + "type": "regular_polygon" + }, + "friedman_reference": { + "analytical_or_proved": true, + "case": "triinhex@35", + "credit": "Emerson Connelly", + "date": "April 2026", + "family_updated": "4/29/26", + "metric_expression": "5/2 = 2.500", + "metric_symbol": "s", + "metric_value": 2.5, + "reference_text": "s = 5/2 = 2.500 | Found by Emerson Connelly | in April 2026.", + "source_image": "35.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinhex/35.gif", + "source_page": "https://erich-friedman.github.io/packing/triinhex/", + "title": "Triangles in Hexagons" + }, + "item": { + "circumradius": 0.5773502691896258, + "side_length": 1.0, + "sides": 3, + "type": "regular_polygon" + }, + "metadata": { + "certification_slack": 1.0000500010000102, + "friedman_reference": { + "analytical_or_proved": true, + "case": "triinhex@35", + "credit": "Emerson Connelly", + "date": "April 2026", + "family_updated": "4/29/26", + "metric_expression": "5/2 = 2.500", + "metric_symbol": "s", + "metric_value": 2.5, + "reference_text": "s = 5/2 = 2.500 | Found by Emerson Connelly | in April 2026.", + "source_image": "35.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinhex/35.gif", + "source_page": "https://erich-friedman.github.io/packing/triinhex/", + "title": "Triangles in Hexagons" + }, + "generator": "friedman_image_seed", + "generator_profile": "best of cg/steepest/sa seed 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a/data/solutions/generated/triinhex_39_a314b4bb8467e5e8.json b/data/solutions/generated/triinhex_39_a314b4bb8467e5e8.json new file mode 100644 index 0000000000000000000000000000000000000000..665c3470ad0a5376ba1200e6aab523eda3651b48 --- /dev/null +++ b/data/solutions/generated/triinhex_39_a314b4bb8467e5e8.json @@ -0,0 +1,298 @@ +{ + "case": "triinhex@39", + "container": { + "circumradius": 2.667295114970451, + "orientation_radians": 0.0, + "side_length": 2.6672951149704507, + "sides": 6, + "type": "regular_polygon" + }, + "friedman_reference": { + "analytical_or_proved": true, + "case": "triinhex@39", + "credit": "Emerson Connelly", + "date": "April 2026", + "family_updated": "4/29/26", + "metric_expression": "8/3 = 2.666+", + "metric_symbol": "s", + "metric_value": 2.666, + "reference_text": "s = 8/3 = 2.666+ | Found by Emerson Connelly | in April 2026.", + "source_image": "39.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinhex/39.gif", + "source_page": "https://erich-friedman.github.io/packing/triinhex/", + "title": "Triangles in Hexagons" + }, + "item": { + "circumradius": 0.5773502691896258, + "side_length": 1.0, + "sides": 3, + "type": "regular_polygon" + }, + "metadata": { + "certification_slack": 1.000040000600004, + "friedman_reference": { + "analytical_or_proved": true, + "case": "triinhex@39", + "credit": "Emerson Connelly", + "date": "April 2026", + "family_updated": "4/29/26", + "metric_expression": "8/3 = 2.666+", + "metric_symbol": "s", + "metric_value": 2.666, + "reference_text": "s = 8/3 = 2.666+ | Found by Emerson Connelly | in April 2026.", + "source_image": "39.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinhex/39.gif", + "source_page": "https://erich-friedman.github.io/packing/triinhex/", + "title": "Triangles in Hexagons" + }, + "generator": "friedman_image_seed", + "generator_profile": "best of cg/steepest/sa seed profiles", + "image_seed": 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"credit": "Ignacio Vallejo", + "date": "March 2026", + "family_updated": "3/22/26", + "metric_expression": "1.34584+", + "metric_symbol": "s", + "metric_value": 1.34584, + "reference_text": "s = 1.34584+ | Found by Ignacio Vallejo | in March 2026.", + "source_image": "16.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinoct/16.gif", + "source_page": "https://erich-friedman.github.io/packing/triinoct/", + "title": "Triangles in Octagons" + }, + "item": { + "circumradius": 0.5773502691896258, + "side_length": 1.0, + "sides": 3, + "type": "regular_polygon" + }, + "metadata": { + "certification_slack": 1.0000500010000102, + "friedman_reference": { + "analytical_or_proved": false, + "case": "triinoct@16", + "credit": "Ignacio Vallejo", + "date": "March 2026", + "family_updated": "3/22/26", + "metric_expression": "1.34584+", + "metric_symbol": "s", + "metric_value": 1.34584, + "reference_text": "s = 1.34584+ | Found by Ignacio Vallejo | in March 2026.", + "source_image": "16.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinoct/16.gif", + "source_page": "https://erich-friedman.github.io/packing/triinoct/", + "title": "Triangles in Octagons" + }, + "generator": "friedman_image_seed", + "generator_profile": "best of cg/steepest/sa seed profiles", + "image_seed": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/site_images/triinoct_016.gif", + "image_seed_transform_error": 0.03902795442976851, + "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", + "refined_state": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/states/triinoct_016_image_refined.state", + "source_url": "https://erich-friedman.github.io/packing/triinoct/", + "submitted_at": "2026-05-01T02:08:58+00:00", + "submitted_by": "polygon-packer-c image seed" + }, + "placements": [ + { + "id": 1, + "rotation_radians": 1.1794624492443122, + "x": 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"https://erich-friedman.github.io/packing/triinoct/" +} diff --git a/data/solutions/generated/triinoct_27_c7748eb19b34ff2e.json b/data/solutions/generated/triinoct_27_c7748eb19b34ff2e.json new file mode 100644 index 0000000000000000000000000000000000000000..7d64dbaebc1f999fc25273a9641653a7b9b5d183 --- /dev/null +++ b/data/solutions/generated/triinoct_27_c7748eb19b34ff2e.json @@ -0,0 +1,226 @@ +{ + "case": "triinoct@27", + "container": { + "circumradius": 2.2319154469777747, + "orientation_radians": 0.0, + "side_length": 1.7082341279962368, + "sides": 8, + "type": "regular_polygon" + }, + "friedman_reference": { + "analytical_or_proved": false, + "case": "triinoct@27", + "credit": "Ignacio Vallejo", + "date": "March 2026", + "family_updated": "3/22/26", + "metric_expression": "1.70882+", + "metric_symbol": "s", + "metric_value": 1.70882, + "reference_text": "s = 1.70882+ | Found by Ignacio Vallejo | in March 2026.", + "source_image": "27.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinoct/27.gif", + "source_page": "https://erich-friedman.github.io/packing/triinoct/", + "title": "Triangles in Octagons" + }, + "item": { + "circumradius": 0.5773502691896258, + "side_length": 1.0, + "sides": 3, + "type": "regular_polygon" + }, + "metadata": { + "certification_slack": 1.000040000600004, + "friedman_reference": { + "analytical_or_proved": false, + "case": "triinoct@27", + "credit": "Ignacio Vallejo", + "date": "March 2026", + "family_updated": "3/22/26", + "metric_expression": "1.70882+", + "metric_symbol": "s", + "metric_value": 1.70882, + "reference_text": "s = 1.70882+ | Found by Ignacio Vallejo | in March 2026.", + "source_image": "27.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinoct/27.gif", + "source_page": "https://erich-friedman.github.io/packing/triinoct/", + "title": "Triangles in Octagons" + }, + "generator": "friedman_image_seed", + "generator_profile": "best of cg/steepest/sa seed 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"https://erich-friedman.github.io/packing/triinoct/" +} diff --git a/data/solutions/generated/triinoct_29_0575780ea85d68dd.json b/data/solutions/generated/triinoct_29_0575780ea85d68dd.json new file mode 100644 index 0000000000000000000000000000000000000000..516689f2a6a333b56a3732ffc9cead23a6fa7794 --- /dev/null +++ b/data/solutions/generated/triinoct_29_0575780ea85d68dd.json @@ -0,0 +1,238 @@ +{ + "case": "triinoct@29", + "container": { + "circumradius": 2.2936294293283517, + "orientation_radians": 0.0, + "side_length": 1.7554679651779115, + "sides": 8, + "type": "regular_polygon" + }, + "friedman_reference": { + "analytical_or_proved": false, + "case": "triinoct@29", + "credit": "Ignacio Vallejo", + "date": "March 2026", + "family_updated": "3/22/26", + "metric_expression": "1.75576+", + "metric_symbol": "s", + "metric_value": 1.75576, + "reference_text": "s = 1.75576+ | Found by Ignacio Vallejo | in March 2026.", + "source_image": "29.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinoct/29.gif", + "source_page": "https://erich-friedman.github.io/packing/triinoct/", + "title": "Triangles in Octagons" + }, + "item": { + "circumradius": 0.5773502691896258, + "side_length": 1.0, + "sides": 3, + "type": "regular_polygon" + }, + "metadata": { + "certification_slack": 1.0000500010000102, + "friedman_reference": { + "analytical_or_proved": false, + "case": "triinoct@29", + "credit": "Ignacio Vallejo", + "date": "March 2026", + "family_updated": "3/22/26", + "metric_expression": "1.75576+", + "metric_symbol": "s", + "metric_value": 1.75576, + "reference_text": "s = 1.75576+ | Found by Ignacio Vallejo | in March 2026.", + "source_image": "29.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinoct/29.gif", + "source_page": "https://erich-friedman.github.io/packing/triinoct/", + "title": "Triangles in Octagons" + }, + "generator": "friedman_image_seed", + "generator_profile": "best of cg/steepest/sa seed 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+ "metric_symbol": "s", + "metric_value": 2.344, + "reference_text": "s = 2.344+ | Found by Ignacio Vallejo | in March 2026.", + "source_image": "19.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinpen/19.gif", + "source_page": "https://erich-friedman.github.io/packing/triinpen/", + "title": "Triangles in Pentagons" + }, + "item": { + "circumradius": 0.5773502691896258, + "side_length": 1.0, + "sides": 3, + "type": "regular_polygon" + }, + "metadata": { + "certification_slack": 1.0000600015000203, + "friedman_reference": { + "analytical_or_proved": false, + "case": "triinpen@19", + "credit": "Ignacio Vallejo", + "date": "March 2026", + "family_updated": "4/29/26", + "metric_expression": "2.344+", + "metric_symbol": "s", + "metric_value": 2.344, + "reference_text": "s = 2.344+ | Found by Ignacio Vallejo | in March 2026.", + "source_image": "19.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinpen/19.gif", + "source_page": "https://erich-friedman.github.io/packing/triinpen/", + "title": "Triangles in Pentagons" + }, + "generator": "friedman_image_seed", + "generator_profile": "best of cg/steepest/sa seed profiles", + "image_seed": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/site_images/triinpen_019.gif", + "image_seed_transform_error": 0.19191760684266418, + "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", + "refined_state": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/states/triinpen_019_image_refined.state", + "source_url": "https://erich-friedman.github.io/packing/triinpen/", + "submitted_at": "2026-05-01T02:08:58+00:00", + "submitted_by": "polygon-packer-c image seed" + }, + "placements": [ + { + "id": 1, + "rotation_radians": -0.8234793182253294, + "x": 0.05193324028631625, + "y": -1.4041134957506725 + }, + { + "id": 2, + "rotation_radians": -1.8659030736407047, + "x": 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"metric_expression": "2.66641+", + "metric_symbol": "s", + "metric_value": 2.66641, + "reference_text": "s = 2.66641+ | Found by Ian Watson | in April 2026.", + "source_image": "24.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinpen/24.gif", + "source_page": "https://erich-friedman.github.io/packing/triinpen/", + "title": "Triangles in Pentagons" + }, + "item": { + "circumradius": 0.5773502691896258, + "side_length": 1.0, + "sides": 3, + "type": "regular_polygon" + }, + "metadata": { + "certification_slack": 1.0000800028000563, + "friedman_reference": { + "analytical_or_proved": false, + "case": "triinpen@24", + "credit": "Ian Watson", + "date": "April 2026", + "family_updated": "4/29/26", + "metric_expression": "2.66641+", + "metric_symbol": "s", + "metric_value": 2.66641, + "reference_text": "s = 2.66641+ | Found by Ian Watson | in April 2026.", + "source_image": "24.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinpen/24.gif", + "source_page": "https://erich-friedman.github.io/packing/triinpen/", + "title": "Triangles in Pentagons" + }, + "generator": "friedman_image_seed", + "generator_profile": "best of cg/steepest/sa seed profiles", + "image_seed": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/site_images/triinpen_024.gif", + "image_seed_transform_error": 0.018608034921599067, + "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", + "refined_state": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/states/triinpen_024_image_refined.state", + "source_url": "https://erich-friedman.github.io/packing/triinpen/", + "submitted_at": "2026-05-01T02:08:58+00:00", + "submitted_by": "polygon-packer-c image seed" + }, + "placements": [ + { + "id": 1, + "rotation_radians": -1.3485999595384843, + "x": 0.5726425650344327, + "y": -1.5945156883075178 + }, + { + "id": 2, + "rotation_radians": -0.837774324027092, + "x": 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b/data/solutions/generated/triinpen_25_d7a41b31b48a9cd4.json new file mode 100644 index 0000000000000000000000000000000000000000..28fd86e366985c787fd99f57685a20d457e59abc --- /dev/null +++ b/data/solutions/generated/triinpen_25_d7a41b31b48a9cd4.json @@ -0,0 +1,214 @@ +{ + "case": "triinpen@25", + "container": { + "circumradius": 2.3164776953163666, + "orientation_radians": 0.0, + "side_length": 2.7231828531428346, + "sides": 5, + "type": "regular_polygon" + }, + "friedman_reference": { + "analytical_or_proved": false, + "case": "triinpen@25", + "credit": "Ignacio Vallejo", + "date": "March 2026", + "family_updated": "4/29/26", + "metric_expression": "2.720+", + "metric_symbol": "s", + "metric_value": 2.72, + "reference_text": "s = 2.720+ | Found by Ignacio Vallejo | in March 2026.", + "source_image": "25.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinpen/25.gif", + "source_page": "https://erich-friedman.github.io/packing/triinpen/", + "title": "Triangles in 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b/data/solutions/generated/triinpen_30_29e4e0e684891d8a.json @@ -0,0 +1,244 @@ +{ + "case": "triinpen@30", + "container": { + "circumradius": 2.5114225358647224, + "orientation_radians": 0.0, + "side_length": 2.9523542577124973, + "sides": 5, + "type": "regular_polygon" + }, + "friedman_reference": { + "analytical_or_proved": false, + "case": "triinpen@30", + "credit": "Ignacio Vallejo", + "date": "April 2026", + "family_updated": "4/29/26", + "metric_expression": "2.952+", + "metric_symbol": "s", + "metric_value": 2.952, + "reference_text": "s = 2.952+ | Found by Ignacio Vallejo | in April 2026.", + "source_image": "30.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinpen/30.gif", + "source_page": "https://erich-friedman.github.io/packing/triinpen/", + "title": "Triangles in Pentagons" + }, + "item": { + "circumradius": 0.5773502691896258, + "side_length": 1.0, + "sides": 3, + "type": "regular_polygon" + }, + "metadata": { + "certification_slack": 1.0000800028000563, + "friedman_reference": { + "analytical_or_proved": false, + "case": "triinpen@30", + "credit": "Ignacio Vallejo", + "date": "April 2026", + "family_updated": "4/29/26", + "metric_expression": "2.952+", + "metric_symbol": "s", + "metric_value": 2.952, + "reference_text": "s = 2.952+ | Found by Ignacio Vallejo | in April 2026.", + "source_image": "30.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinpen/30.gif", + "source_page": "https://erich-friedman.github.io/packing/triinpen/", + "title": "Triangles in Pentagons" + }, + "generator": "friedman_image_seed", + "generator_profile": "best of cg/steepest/sa seed profiles", + "image_seed": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/site_images/triinpen_030.gif", + "image_seed_transform_error": 0.02712819042244896, + "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", + "refined_state": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/states/triinpen_030_image_refined.state", + "source_url": "https://erich-friedman.github.io/packing/triinpen/", + "submitted_at": "2026-05-01T02:08:58+00:00", + "submitted_by": "polygon-packer-c image seed" + }, + "placements": [ + { + "id": 1, + "rotation_radians": 1.257791018520027, + "x": 0.5982822415639047, + "y": 1.8391629067344863 + }, + { + "id": 2, + "rotation_radians": 0.8419654445488478, + "x": -0.25516146063113654, + "y": 1.7476752872056927 + }, + { + "id": 3, + "rotation_radians": 0.861402581881756, + "x": -1.1956430685796726, + "y": 1.4319778653845623 + }, + { + "id": 4, + "rotation_radians": 1.9041670625105545, + "x": -0.6309162438150439, + "y": 1.299386107003686 + }, + { + "id": 5, + "rotation_radians": 0.2097516435986189, + "x": 0.4201157592822993, + "y": 1.2895037778099498 + }, + { + "id": 6, + "rotation_radians": 1.6683826585410848, + "x": 1.2342386803932806, + "y": 1.2606829934258466 + 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"source_page": "https://erich-friedman.github.io/packing/triinpen/" +} diff --git a/data/solutions/generated/triinsqu_10_5b0eaf84f54fe48c.json b/data/solutions/generated/triinsqu_10_5b0eaf84f54fe48c.json new file mode 100644 index 0000000000000000000000000000000000000000..214a225af9a2a11a82bef9c06b3dac3aff1c6231 --- /dev/null +++ b/data/solutions/generated/triinsqu_10_5b0eaf84f54fe48c.json @@ -0,0 +1,124 @@ +{ + "case": "triinsqu@10", + "container": { + "circumradius": 1.6809609047175382, + "orientation_radians": 0.0, + "side_length": 2.3772377092704904, + "sides": 4, + "type": "regular_polygon" + }, + "friedman_reference": { + "analytical_or_proved": false, + "case": "triinsqu@10", + "credit": "David W", + "date": "July 2002", + "family_updated": "4/29/26", + "metric_expression": "2.377+", + "metric_symbol": "s", + "metric_value": 2.377, + "reference_text": "s = 2.377+ | Found by David W. Cantrell | in July 2002.", + "source_image": "t10.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t10.gif", + "source_page": "https://erich-friedman.github.io/packing/triinsqu/", + "title": "Triangles in Squares" + }, + "item": { + "circumradius": 0.5773502691896258, + "side_length": 1.0, + "sides": 3, + "type": "regular_polygon" + }, + "metadata": { + "certification_slack": 1.0000600015000203, + "friedman_reference": { + "analytical_or_proved": false, + "case": "triinsqu@10", + "credit": "David W", + "date": "July 2002", + "family_updated": "4/29/26", + "metric_expression": "2.377+", + "metric_symbol": "s", + "metric_value": 2.377, + "reference_text": "s = 2.377+ | Found by David W. Cantrell | in July 2002.", + "source_image": "t10.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t10.gif", + "source_page": "https://erich-friedman.github.io/packing/triinsqu/", + "title": "Triangles in Squares" + }, + "generator": "friedman_image_seed", + "generator_profile": "best of cg/steepest/sa seed profiles", + "image_seed": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/site_images/triinsqu_010.gif", + "image_seed_transform_error": 2.707242505775539e-16, + "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", + "refined_state": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/states/triinsqu_010_image_refined.state", + "source_url": "https://erich-friedman.github.io/packing/triinsqu/", + "submitted_at": "2026-05-01T02:08:58+00:00", + "submitted_by": "polygon-packer-c image seed" + }, + "placements": [ + { + "id": 1, + "rotation_radians": 1.308985064767752, + "x": -0.27794719030982695, + "y": 0.9947539230027213 + }, + { + "id": 2, + "rotation_radians": 1.955178788048696, + "x": 0.4440646321481831, + "y": 0.7452481397177833 + }, + { + "id": 3, + "rotation_radians": 0.26180221150823135, + "x": -0.42740742300669543, + "y": 0.43700741893265777 + }, + { + "id": 4, + "rotation_radians": 1.3090244797838342, + "x": -0.9851214242224371, + "y": 0.2875516602455915 + }, + { + "id": 5, + "rotation_radians": 0.9080792876924499, + "x": 0.5373081909346326, + "y": 0.16010090682141648 + }, + { + "id": 6, + "rotation_radians": 1.955270520505862, + "x": 1.1091226237932865, + "y": 0.08001888675945196 + }, + { + "id": 7, + "rotation_radians": 0.677434884494812, + "x": -0.7579375678905592, + "y": -0.44093798695167374 + }, + { + "id": 8, + "rotation_radians": 1.7245709590750349, + "x": -0.1769968692759869, + "y": -0.5397764869409933 + }, + { + "id": 9, + "rotation_radians": 0.2618001659706557, + "x": 0.6314820977789997, + "y": -0.6412179375842701 + }, + { + "id": 10, + "rotation_radians": 0.6774638052346492, + "x": -0.0885312744105007, + "y": -1.110366621682777 + } + ], + "problem": "10 equal items for Triangles in Squares", + "setup": "triinsqu", + "source_image": "https://erich-friedman.github.io/packing/triinsqu/t10.gif", + "source_page": "https://erich-friedman.github.io/packing/triinsqu/" +} diff --git a/data/solutions/generated/triinsqu_13_7628e283de25ee68.json b/data/solutions/generated/triinsqu_13_7628e283de25ee68.json new file mode 100644 index 0000000000000000000000000000000000000000..c938e384d608d6469e95dd74564411e4efff9537 --- /dev/null +++ b/data/solutions/generated/triinsqu_13_7628e283de25ee68.json @@ -0,0 +1,142 @@ +{ + "case": "triinsqu@13", + "container": { + "circumradius": 1.8353309359550882, + "orientation_radians": 0.0, + "side_length": 2.595549901070592, + "sides": 4, + "type": "regular_polygon" + }, + "friedman_reference": { + "analytical_or_proved": false, + "case": "triinsqu@13", + "credit": "David W", + "date": "July 2002", + "family_updated": "4/29/26", + "metric_expression": "2.595+", + "metric_symbol": "s", + "metric_value": 2.595, + "reference_text": "s = 2.595+ | Found by David W. Cantrell | in July 2002.", + "source_image": "t13.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t13.gif", + "source_page": "https://erich-friedman.github.io/packing/triinsqu/", + "title": "Triangles in Squares" + }, + "item": { + "circumradius": 0.5773502691896258, + "side_length": 1.0, + "sides": 3, + "type": "regular_polygon" + }, + "metadata": { + "certification_slack": 1.0000600015000203, + "friedman_reference": { + "analytical_or_proved": false, + "case": "triinsqu@13", + "credit": "David W", + "date": "July 2002", + "family_updated": "4/29/26", + "metric_expression": "2.595+", + "metric_symbol": "s", + "metric_value": 2.595, + "reference_text": "s = 2.595+ | Found by David W. Cantrell | in July 2002.", + "source_image": "t13.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t13.gif", + "source_page": "https://erich-friedman.github.io/packing/triinsqu/", + "title": "Triangles in Squares" + }, + "generator": "friedman_image_seed", + "generator_profile": "best of cg/steepest/sa seed profiles", + "image_seed": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/site_images/triinsqu_013.gif", + "image_seed_transform_error": 2.327997847980024e-16, + "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", + "refined_state": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/states/triinsqu_013_image_refined.state", + "source_url": "https://erich-friedman.github.io/packing/triinsqu/", + "submitted_at": "2026-05-01T02:08:58+00:00", + "submitted_by": "polygon-packer-c image seed" + }, + "placements": [ + { + "id": 1, + "rotation_radians": 4.468528422254128, + "x": 0.13942942224715082, + "y": -1.2750079192475021 + }, + { + "id": 2, + "rotation_radians": 4.974187330836345, + "x": -0.5703272166278136, + "y": -0.8567308093432104 + }, + { + "id": 3, + "rotation_radians": 3.43377089748683, + "x": 0.3045021582492138, + "y": -0.7147063468804329 + }, + { + "id": 4, + "rotation_radians": 4.477445157233715, + "x": 0.8593936654699389, + "y": -0.5487919577641657 + }, + { + "id": 5, + "rotation_radians": 3.927000215311017, + "x": -0.7198190580220029, + "y": -0.29897064587529637 + }, + { + "id": 6, + "rotation_radians": 4.974205460919863, + "x": -1.2775728946526788, + "y": -0.14941107645864005 + }, + { + "id": 7, + "rotation_radians": 4.974191604185533, + "x": -0.14627248823585123, + "y": -0.05597003703164697 + }, + { + "id": 8, + "rotation_radians": 3.926468471931825, + "x": 1.2714089286429957, + "y": 0.15527837619255375 + }, + { + "id": 9, + "rotation_radians": 4.966658737067106, + "x": 0.710920367706504, + "y": 0.3079094268283681 + 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b/data/solutions/generated/triinsqu_32_f42801b713b9238d.json @@ -0,0 +1,256 @@ +{ + "case": "triinsqu@32", + "container": { + "circumradius": 2.8258720673212823, + "orientation_radians": 0.0, + "side_length": 3.996386603137053, + "sides": 4, + "type": "regular_polygon" + }, + "friedman_reference": { + "analytical_or_proved": false, + "case": "triinsqu@32", + "credit": "Emerson Connelly", + "date": "April 2026", + "family_updated": "4/29/26", + "metric_expression": "3.997+", + "metric_symbol": "s", + "metric_value": 3.997, + "reference_text": "32. | | s = 3.997+ | Found by Emerson Connelly | in April 2026.", + "source_image": "t32.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t32.gif", + "source_page": "https://erich-friedman.github.io/packing/triinsqu/", + "title": "Triangles in Squares" + }, + "item": { + "circumradius": 0.5773502691896258, + "side_length": 1.0, + "sides": 3, + "type": "regular_polygon" + }, + "metadata": { + "certification_slack": 1.0000800028000563, + "friedman_reference": { + "analytical_or_proved": false, + "case": "triinsqu@32", + "credit": "Emerson Connelly", + "date": "April 2026", + "family_updated": "4/29/26", + "metric_expression": "3.997+", + "metric_symbol": "s", + "metric_value": 3.997, + "reference_text": "32. | | s = 3.997+ | Found by Emerson Connelly | in April 2026.", + "source_image": "t32.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t32.gif", + "source_page": "https://erich-friedman.github.io/packing/triinsqu/", + "title": "Triangles in Squares" + }, + "generator": "friedman_image_seed", + "generator_profile": "best of cg/steepest/sa seed profiles", + "image_seed": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/site_images/triinsqu_032.gif", + "image_seed_transform_error": 4.34913614441221e-16, + "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", + "refined_state": 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"metric_expression": "4.058+", + "metric_symbol": "s", + "metric_value": 4.058, + "reference_text": "33. | | s = 4.058+ | Found by Emerson Connelly | in April 2026.", + "source_image": "t33.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t33.gif", + "source_page": "https://erich-friedman.github.io/packing/triinsqu/", + "title": "Triangles in Squares" + }, + "item": { + "circumradius": 0.5773502691896258, + "side_length": 1.0, + "sides": 3, + "type": "regular_polygon" + }, + "metadata": { + "certification_slack": 1.0000700021000353, + "friedman_reference": { + "analytical_or_proved": false, + "case": "triinsqu@33", + "credit": "Emerson Connelly", + "date": "April 2026", + "family_updated": "4/29/26", + "metric_expression": "4.058+", + "metric_symbol": "s", + "metric_value": 4.058, + "reference_text": "33. | | s = 4.058+ | Found by Emerson Connelly | in April 2026.", + "source_image": "t33.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t33.gif", + "source_page": "https://erich-friedman.github.io/packing/triinsqu/", + "title": "Triangles in Squares" + }, + "generator": "friedman_image_seed", + "generator_profile": "best of cg/steepest/sa seed profiles", + "image_seed": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/site_images/triinsqu_033.gif", + "image_seed_transform_error": 4.776843691548918e-16, + "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", + "refined_state": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/states/triinsqu_033_image_refined.state", + "source_url": "https://erich-friedman.github.io/packing/triinsqu/", + "submitted_at": "2026-05-01T02:08:58+00:00", + "submitted_by": "polygon-packer-c image seed" + }, + "placements": [ + { + "id": 1, + "rotation_radians": 4.9741864577723955, + "x": -0.1497887601653326, + "y": -2.3116210510448347 + }, + { + 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a/data/solutions/generated/triinsqu_34_61b4d5d6bf22a206.json b/data/solutions/generated/triinsqu_34_61b4d5d6bf22a206.json new file mode 100644 index 0000000000000000000000000000000000000000..95b28dec26d79420f8df49fd90f10277bf36cefe --- /dev/null +++ b/data/solutions/generated/triinsqu_34_61b4d5d6bf22a206.json @@ -0,0 +1,268 @@ +{ + "case": "triinsqu@34", + "container": { + "circumradius": 2.897617160196922, + "orientation_radians": 0.0, + "side_length": 4.0978494865155, + "sides": 4, + "type": "regular_polygon" + }, + "friedman_reference": { + "analytical_or_proved": false, + "case": "triinsqu@34", + "credit": "Emerson Connelly", + "date": "April 2026", + "family_updated": "4/29/26", + "metric_expression": "4.098+", + "metric_symbol": "s", + "metric_value": 4.098, + "reference_text": "34. | | s = 4.098+ | Found by Emerson Connelly | in April 2026.", + "source_image": "t34.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triinsqu/t34.gif", + "source_page": 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0000000000000000000000000000000000000000..b2d04a4a3550b3df9fa7353edf25cc6c1a661e37 --- /dev/null +++ b/data/solutions/generated/triintri_11_8c37202a929dc807.json @@ -0,0 +1,130 @@ +{ + "case": "triintri@11", + "container": { + "circumradius": 2.1508854608168297, + "orientation_radians": 0.0, + "side_length": 3.7254428993959463, + "sides": 3, + "type": "regular_polygon" + }, + "friedman_reference": { + "analytical_or_proved": true, + "case": "triintri@11", + "credit": "Erich Friedman", + "date": "1997", + "family_updated": "2/29/26", + "metric_expression": "9/4 + 9\u221a21/28 = 3.722+", + "metric_symbol": "s", + "metric_value": 3.722, + "reference_text": "s = 9/4 + 9\u221a21/28 = 3.722+ | Found by Erich Friedman | in 1997.", + "source_image": "tt11.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt11.gif", + "source_page": "https://erich-friedman.github.io/packing/triintri/", + "title": "Triangles in Triangles" + }, + "item": { + "circumradius": 0.5773502691896258, + "side_length": 1.0, + "sides": 3, + "type": "regular_polygon" + }, + "metadata": { + "certification_slack": 1.0000600015000203, + "friedman_reference": { + "analytical_or_proved": true, + "case": "triintri@11", + "credit": "Erich Friedman", + "date": "1997", + "family_updated": "2/29/26", + "metric_expression": "9/4 + 9\u221a21/28 = 3.722+", + "metric_symbol": "s", + "metric_value": 3.722, + "reference_text": "s = 9/4 + 9\u221a21/28 = 3.722+ | Found by Erich Friedman | in 1997.", + "source_image": "tt11.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt11.gif", + "source_page": "https://erich-friedman.github.io/packing/triintri/", + "title": "Triangles in Triangles" + }, + "generator": "friedman_image_seed", + "generator_profile": "best of cg/steepest/sa seed profiles", + "image_seed": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/site_images/triintri_011.gif", + "image_seed_transform_error": 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a/data/solutions/generated/triintri_29_f2204c752005e940.json b/data/solutions/generated/triintri_29_f2204c752005e940.json new file mode 100644 index 0000000000000000000000000000000000000000..1e31f4057bc0d2126e058810a2255eab8bd50c9c --- /dev/null +++ b/data/solutions/generated/triintri_29_f2204c752005e940.json @@ -0,0 +1,238 @@ +{ + "case": "triintri@29", + "container": { + "circumradius": 3.276051928592963, + "orientation_radians": 0.0, + "side_length": 5.674288388557019, + "sides": 3, + "type": "regular_polygon" + }, + "friedman_reference": { + "analytical_or_proved": true, + "case": "triintri@29", + "credit": "David W", + "date": "July 2007", + "family_updated": "2/29/26", + "metric_expression": "17/3 = 5.666+", + "metric_symbol": "s", + "metric_value": 5.666, + "reference_text": "s = 17/3 = 5.666+ | Found by David W. Cantrell | in July 2007.", + "source_image": "tt29.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt29.gif", + "source_page": "https://erich-friedman.github.io/packing/triintri/", + "title": "Triangles in Triangles" + }, + "item": { + "circumradius": 0.5773502691896258, + "side_length": 1.0, + "sides": 3, + "type": "regular_polygon" + }, + "metadata": { + "certification_slack": 1.0000500010000102, + "friedman_reference": { + "analytical_or_proved": true, + "case": "triintri@29", + "credit": "David W", + "date": "July 2007", + "family_updated": "2/29/26", + "metric_expression": "17/3 = 5.666+", + "metric_symbol": "s", + "metric_value": 5.666, + "reference_text": "s = 17/3 = 5.666+ | Found by David W. Cantrell | in July 2007.", + "source_image": "tt29.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt29.gif", + "source_page": "https://erich-friedman.github.io/packing/triintri/", + "title": "Triangles in Triangles" + }, + "generator": "friedman_image_seed", + "generator_profile": "best of cg/steepest/sa seed profiles", + "image_seed": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/site_images/triintri_029.gif", + "image_seed_transform_error": 0.08567121199249543, + "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", + "refined_state": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/states/triintri_029_image_refined.state", + "source_url": "https://erich-friedman.github.io/packing/triintri/", + "submitted_at": "2026-05-01T02:08:58+00:00", + "submitted_by": "polygon-packer-c image seed" + }, + "placements": [ + { + "id": 1, + "rotation_radians": 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"triintri@30", + "container": { + "circumradius": 3.3220967081077366, + "orientation_radians": 0.0, + "side_length": 5.754040286099913, + "sides": 3, + "type": "regular_polygon" + }, + "friedman_reference": { + "analytical_or_proved": true, + "case": "triintri@30", + "credit": "Maurizio Morandi", + "date": "May 2008", + "family_updated": "2/29/26", + "metric_expression": "23/4 = 5.75", + "metric_symbol": "s", + "metric_value": 5.75, + "reference_text": "s = 23/4 = 5.75 | Found by Maurizio Morandi | in May 2008.", + "source_image": "tt30.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt30.gif", + "source_page": "https://erich-friedman.github.io/packing/triintri/", + "title": "Triangles in Triangles" + }, + "item": { + "circumradius": 0.5773502691896258, + "side_length": 1.0, + "sides": 3, + "type": "regular_polygon" + }, + "metadata": { + "certification_slack": 1.0000600015000203, + "friedman_reference": { + "analytical_or_proved": true, + "case": "triintri@30", + "credit": "Maurizio Morandi", + "date": "May 2008", + "family_updated": "2/29/26", + "metric_expression": "23/4 = 5.75", + "metric_symbol": "s", + "metric_value": 5.75, + "reference_text": "s = 23/4 = 5.75 | Found by Maurizio Morandi | in May 2008.", + "source_image": "tt30.gif", + "source_image_url": "https://erich-friedman.github.io/packing/triintri/tt30.gif", + "source_page": "https://erich-friedman.github.io/packing/triintri/", + "title": "Triangles in Triangles" + }, + "generator": "friedman_image_seed", + "generator_profile": "best of cg/steepest/sa seed profiles", + "image_seed": "/Users/nathanroll/Documents/polygon-packer-c/bench_outputs/friedman_image_seeds/site_images/triintri_030.gif", + "image_seed_transform_error": 0.014633043011461478, + "notes": "Image-seeded from the Friedman source image and refined with polygon-packer-c.", + "refined_state": 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