{
  "schema_version": 1,
  "title": "Reproduction: Planar Symmetric Pattern Generation",
  "emoji": "🧩",
  "space_id": "ProCreations/repro-planar-symmetric-pattern-generation",
  "paper": {
    "arxiv_id": "2606.02073"
  },
  "tags": [
    "icml2026-repro",
    "paper-nbU2LNYdZN"
  ],
  "updated_at": "2026-07-27T07:10:00+00:00",
  "root": {
    "slug": "index",
    "title": "Reproduction: Planar Symmetric Pattern Generation",
    "file": "pages/index.md",
    "children": [
      {
        "slug": "executive-summary",
        "title": "Executive summary",
        "file": "pages/executive-summary/page.md",
        "children": []
      },
      {
        "slug": "claim-1-conjugacy",
        "title": "Claim 1: Theorem 3.2 proves that any planar symmetry group G is conjugate, via an invertible linear transformation, to a subgroup of an affine reflection group, allowing arbitrary planar symmetries to be embedded into higher-symmetry reflection groups (Theorem 3.2).",
        "file": "pages/claim-1-conjugacy/page.md",
        "children": []
      },
      {
        "slug": "claim-2-approximation",
        "title": "Claim 2: Theorem 3.3 shows that for any G-invariant continuous function and any epsilon > 0, there exist coefficients h_1,...,h_r in the proposed decomposition whose approximation error is below epsilon, establishing universal approximation of symmetric functions (Theorem 3.3).",
        "file": "pages/claim-2-approximation/page.md",
        "children": []
      },
      {
        "slug": "claim-3-representation",
        "title": "Claim 3: The symmetric representation combines Wa-invariant high-symmetry coefficient functions h_i with fixed low-symmetry G-invariant basis functions eta_i via f(x) = sum_i h_i(x) eta_i(x), avoiding the boundary discontinuities of naive asymmetric-unit extension for non-reflective transformations (Equation 3, Section 3.1, Figure 2).",
        "file": "pages/claim-3-representation/page.md",
        "children": []
      },
      {
        "slug": "claim-4-all-groups",
        "title": "Claim 4: Algorithm 1 generates symmetric patterns across all 17 wallpaper (planar) groups by mapping query points to the asymmetric unit and combining computed coefficients with fixed bases (Section 6.1, Algorithm 1).",
        "file": "pages/claim-4-all-groups/page.md",
        "children": []
      },
      {
        "slug": "claim-5-connectivity",
        "title": "Claim 5: A Virtual Temperature Method (VTM) loss enforces global connectivity for manufacturable paper-cutting patterns while preserving the required planar symmetry (Section 6.2, Equation 6).",
        "file": "pages/claim-5-connectivity/page.md",
        "children": []
      },
      {
        "slug": "claim-6-metamaterials",
        "title": "Claim 6: The framework extends to topology optimization using homogenization-based mechanical property objectives under symmetry constraints, and Theorem B.9 shows the theoretical construction generalizes to n=3 (space groups), with zero-shot extension to metamaterial design demonstrated in Figure 8 (Section 6.3, Equation 8, Theorem B.9, Figure 8).",
        "file": "pages/claim-6-metamaterials/page.md",
        "children": []
      },
      {
        "slug": "conclusion",
        "title": "Conclusion",
        "file": "pages/conclusion/page.md",
        "children": []
      }
    ]
  },
  "agent_view_tokens": 5000,
  "revision": "1785136200000000000"
}