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Browse files- .DS_Store +0 -0
- README.md +133 -71
- The_Sphere_of_Realization_LLM_Training_Dataset_rev6_files.jsonl +123 -0
- The_Sphere_of_Realization_documents.jsonl +0 -0
- coordinates/The_Sphere_of_Realization_coordinates.jsonl.gz +3 -0
- images/metadata.jsonl +41 -0
- vision_openai_chat/The_Sphere_of_Realization_vision_openai_chat_train.jsonl +0 -0
- vision_openai_chat/The_Sphere_of_Realization_vision_openai_chat_validation.jsonl +2 -2
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README.md
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---
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pretty_name:
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language:
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license: other
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license_name: author-controlled
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license_details:
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task_categories:
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tags:
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size_categories:
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configs:
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---
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-
# The Sphere of Realization (2nd Edition) — LLM Training Dataset (
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**Source work:** *The Sphere of Realization: The Mathematical Path of Harmonious Being*, 2nd Edition — volume two of the collected works.
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**Author:** Parker Emmerson (Yaohushuason), Independent Researcher — ORCID 0009-0007-1288-3292 — Parkeremmerson@icloud.com
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note = {Independent Researcher. ORCID 0009-0007-1288-3292}
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}
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```
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---
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pretty_name: The Sphere of Realization (2nd Edition) — LLM Training Dataset (rev6)
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language:
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- en
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license: other
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license_name: author-controlled
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license_details: Text of The Sphere of Realization, © Parker Emmerson (Yaohushuason). Terms of use for training are set by the author.
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task_categories:
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- text-generation
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- question-answering
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- image-to-text
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- visual-question-answering
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tags:
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- mathematics
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- mathematical-physics
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- wolfram-language
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- mathematica
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- 3d-plots
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- stl
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- fine-tuning
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- instruction-tuning
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- continued-pretraining
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size_categories:
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- 10K<n<100K
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configs:
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- config_name: sft_openai_chat
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data_files:
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- split: train
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path: sft_openai_chat/The_Sphere_of_Realization_sft_openai_chat_train.jsonl
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- split: validation
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path: sft_openai_chat/The_Sphere_of_Realization_sft_openai_chat_validation.jsonl
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- config_name: sft_sharegpt
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data_files:
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- split: train
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path: sft_sharegpt/The_Sphere_of_Realization_sft_sharegpt_train.jsonl
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- split: validation
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path: sft_sharegpt/The_Sphere_of_Realization_sft_sharegpt_validation.jsonl
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- config_name: sft_alpaca
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data_files:
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- split: train
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path: sft_alpaca/The_Sphere_of_Realization_sft_alpaca_train.json
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- split: validation
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path: sft_alpaca/The_Sphere_of_Realization_sft_alpaca_validation.json
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- config_name: sft_prompt_completion
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data_files:
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- split: train
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path: sft_prompt_completion/The_Sphere_of_Realization_sft_prompt_completion_train.jsonl
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- split: validation
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path: sft_prompt_completion/The_Sphere_of_Realization_sft_prompt_completion_validation.jsonl
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- config_name: vision_llava
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data_files:
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- split: train
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path: vision_llava/The_Sphere_of_Realization_vision_llava_train.json
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- split: validation
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path: vision_llava/The_Sphere_of_Realization_vision_llava_validation.json
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- config_name: plot_index
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data_files:
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- split: train
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path: The_Sphere_of_Realization_plot_index.jsonl
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- config_name: pretraining
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default: true
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data_files:
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- split: train
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path: pretraining/The_Sphere_of_Realization_pretraining_corpus_train.jsonl
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- split: validation
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path: pretraining/The_Sphere_of_Realization_pretraining_corpus_validation.jsonl
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- config_name: plot_data_sft_openai_chat
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data_files:
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- split: train
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path: plot_data/The_Sphere_of_Realization_plot_data_sft_openai_chat_train_*.jsonl.gz
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- split: validation
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path: plot_data/The_Sphere_of_Realization_plot_data_sft_openai_chat_validation_*.jsonl.gz
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- config_name: plot_data_pretraining
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data_files:
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- split: train
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path: plot_data/The_Sphere_of_Realization_plot_data_pretraining_train_*.jsonl.gz
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- split: validation
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path: plot_data/The_Sphere_of_Realization_plot_data_pretraining_validation_*.jsonl.gz
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- config_name: images
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data_files:
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- split: train
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path: images/**
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- config_name: coordinates
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data_files:
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- split: train
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path: coordinates/The_Sphere_of_Realization_coordinates.jsonl.gz
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- config_name: documents
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data_files:
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- split: train
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path: The_Sphere_of_Realization_documents.jsonl
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- config_name: vision_openai_chat
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data_files:
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- split: train
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path: vision_openai_chat/The_Sphere_of_Realization_vision_openai_chat_train.jsonl
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- split: validation
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path: vision_openai_chat/The_Sphere_of_Realization_vision_openai_chat_validation.jsonl
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- config_name: files
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data_files:
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- split: train
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path: The_Sphere_of_Realization_LLM_Training_Dataset_rev6_files.jsonl
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---
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# The Sphere of Realization (2nd Edition) — LLM Training Dataset (rev6)
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**Source work:** *The Sphere of Realization: The Mathematical Path of Harmonious Being*, 2nd Edition — volume two of the collected works.
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**Author:** Parker Emmerson (Yaohushuason), Independent Researcher — ORCID 0009-0007-1288-3292 — Parkeremmerson@icloud.com
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note = {Independent Researcher. ORCID 0009-0007-1288-3292}
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}
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```
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## Changes from The_Sphere_of_Realization_LLM_Training_Dataset_rev5: every file reachable by the Parquet conversion
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The_Sphere_of_Realization_LLM_Training_Dataset_rev6 is The_Sphere_of_Realization_LLM_Training_Dataset_rev5 with files added and the card's `configs` extended, so that the Hugging Face Parquet converter (the "parquet-converter" bot, which builds the dataset viewer and the `refs/convert/parquet` branch from the configurations a card declares) reaches every piece of content in the repository. No file of The_Sphere_of_Realization_LLM_Training_Dataset_rev5 was removed or changed except the OpenAI-layout vision files named below, which were rewritten in place with the same content. Upload the folder of The_Sphere_of_Realization_LLM_Training_Dataset_rev5 and this folder into the same repository; this README replaces the previous one.
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What was added:
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* `images` configuration: the 41 pictures under `images/` are now an image dataset (the pictures themselves become an image column in the Parquet conversion), with `images/metadata.jsonl` giving each picture's vision questions and answers.
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* `coordinates` configuration: the arrays of all 31 `.npz` coordinate files (26,399,936 numbers) as rows of JSON lists — one row per graphic, one entry per array (name, dtype, shape, `values` as numbers — integers exactly, as the dtype column records — and `text_values_json`, a JSON list, for the few arrays of strings) — so the exact coordinates are in the Parquet conversion as numbers, not only as the text of the plot-number files. The `.npz` and STL files stay as they were.
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* `documents` configuration: the 1 Markdown and text files of the folder (The_Sphere_of_Realization_full_text.md) as rows with their full text. A document longer than the viewer's cell limit is still complete in the Parquet file.
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* The undivided files `pretraining/The_Sphere_of_Realization_pretraining_corpus.jsonl`, `sft_alpaca/The_Sphere_of_Realization_sft_alpaca.json`, `sft_openai_chat/The_Sphere_of_Realization_sft_openai_chat.jsonl`, `sft_prompt_completion/The_Sphere_of_Realization_sft_prompt_completion.jsonl`, `sft_sharegpt/The_Sphere_of_Realization_sft_sharegpt.jsonl`, `vision_llava/The_Sphere_of_Realization_vision_llava.json` hold the same rows as their `_train` and `_validation` parts together and stay undeclared, so no row is converted twice.
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* `vision_openai_chat` configuration: `vision_openai_chat/The_Sphere_of_Realization_vision_openai_chat_train.jsonl`, `vision_openai_chat/The_Sphere_of_Realization_vision_openai_chat_validation.jsonl` — rewritten so that every message `content` is a list of parts, which the Parquet conversion requires of one column.
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* `files` configuration: the manifest as a table — every file of the revision with its size and SHA-256.
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New files:
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| File | What it holds |
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|---|---|
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| `images/metadata.jsonl` | one row per picture (41): file name, the vision examples asked about it (questions and answers), and the earlier metadata columns where the folder had them |
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| `coordinates/The_Sphere_of_Realization_coordinates.jsonl.gz` | 31 graphics, every stored array as JSON lists (0.34 GB of text) |
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| `The_Sphere_of_Realization_documents.jsonl` | the 1 text documents of the folder (full texts, reports, audits) as rows |
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| `vision_openai_chat/The_Sphere_of_Realization_vision_openai_chat_train.jsonl` | the OpenAI-layout vision file rewritten so every message `content` is a list of parts (66 string contents converted in 33 rows); the pictures and text are unchanged |
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| `vision_openai_chat/The_Sphere_of_Realization_vision_openai_chat_validation.jsonl` | the OpenAI-layout vision file rewritten so every message `content` is a list of parts (4 string contents converted in 2 rows); the pictures and text are unchanged |
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| `The_Sphere_of_Realization_LLM_Training_Dataset_rev6_files.jsonl` | every file of the revision (118 carried from The_Sphere_of_Realization_LLM_Training_Dataset_rev5, 5 new) with its size and SHA-256, as rows |
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What the bot still does not read, and why it is no loss: the `.npz` coordinate files and the STL files (their numbers are in `coordinates`); the Markdown and text files (their text is in `documents`); the manifest (its rows are in `files`). The pictures are converted through the `images` configuration.
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Built 2026-10-30; every new configuration was loaded with the Hugging Face `datasets` library before this card was written (see the check table).
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### Loading check of the new configurations
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| Configuration | Split | Result | Rows | Columns |
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|---|---|---|---:|---:|
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| `images` | train | loads | 41 | 5 |
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| `coordinates` | train | loads | 31 | 3 |
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| `documents` | train | loads | 1 | 4 |
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| `vision_openai_chat` | train | loads | 33 | 1 |
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| `vision_openai_chat` | validation | loads | 2 | 1 |
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| `files` | train | loads | 123 | 4 |
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The_Sphere_of_Realization_LLM_Training_Dataset_rev6_files.jsonl
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{"path": "README.md", "bytes": 27469, "sha256": "414eea01e1c32712d07d69f9c6e857bede465e8c50744c12b3f6a3cf1dee77f7", "revision": "The_Sphere_of_Realization_LLM_Training_Dataset_rev5"}
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{"path": "The_Sphere_of_Realization_dataset_manifest.json", "bytes": 18119, "sha256": "95c9730bce173e985f9dfaaf16b54218763b272f07d24a20f533b5e37cdeb60f", "revision": "The_Sphere_of_Realization_LLM_Training_Dataset_rev5"}
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{"path": "The_Sphere_of_Realization_documents.jsonl", "bytes": 611019, "sha256": "ce6a938946ebd1748220f3ebb0a45b4a863abae33ff3b30d6f0f10170a008b3b", "revision": "The_Sphere_of_Realization_LLM_Training_Dataset_rev6"}
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{"path": "The_Sphere_of_Realization_full_text.md", "bytes": 609151, "sha256": "5eca21895c82c8bbd3463a8a8e3dd77c1cc82a6c9976a94e2c118d512664d5b9", "revision": "The_Sphere_of_Realization_LLM_Training_Dataset_rev5"}
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{"path": "The_Sphere_of_Realization_plot_index.jsonl", "bytes": 1133672, "sha256": "1890dda534ba769689db75f652913391f4ae1b05fa271272f5919065452966c7", "revision": "The_Sphere_of_Realization_LLM_Training_Dataset_rev5"}
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{"file_name": "sor-plot-0001.png", "book": "The Sphere of Realization", "vision_examples": 0, "questions": "", "answers": ""}
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{"file_name": "sor-plot-0002.png", "book": "The Sphere of Realization", "vision_examples": 0, "questions": "", "answers": ""}
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{"file_name": "sor-plot-0003.png", "book": "The Sphere of Realization", "vision_examples": 2, "questions": "Transcribe the text and mathematics in this picture from The Sphere of Realization (chapter \"1. Step-by-step Solutions to the Phenomenological Velocity (V-Curvature) Equation and Discourse on the V-Curvature Method and Energy As Boundaries\").\n\nTranscribe the text and mathematics in this picture from The Sphere of Realization (chapter \"1. Step-by-step Solutions to the Phenomenological Velocity (V-Curvature) Equation and Discourse on the V-Curvature Method and Energy As Boundaries\").", "answers": "**Input interpretation:**\n\n$$v=\\left(\\sqrt{3548143227025099264\\,\\sin^{2}(\\beta)+89875517873681760\\,\\theta^{2}-1129409066758147072\\,\\theta}\\right)\\Big/\\left(\\sqrt{39.4784\\,\\sin^{2}(\\beta)+\\theta^{2}+\\theta\\times(-12.5664)}\\right)$$\n\n**Result:**\n\n$$v=\\left(\\sqrt{3548143227025099264\\,\\sin^{2}(\\beta)+89875517873681760\\,\\theta^{2}-1129409066758147072\\,\\theta}\\right)\\Big/\\left(\\sqrt{39.4784\\,\\sin^{2}(\\beta)+\\theta^{2}-12.5664\\,\\theta}\\right)$$\n\n**Roots:**\n\n$$\\sqrt{\\left(-8.67927\\times10^{27}\\sin^{2}(\\beta)-2.06332\\times10^{13}\\sqrt{1.94636\\times10^{31}-1.94636\\times10^{31}\\sin^{2}(\\beta)}-9.10286\\times10^{28}\\right)}\\neq 0,\\quad \\theta\\approx 3.56048\\times10^{-16}\\left(4.\\,\\sqrt{1.94636\\times10^{31}-1.94636\\times10^{31}\\sin^{2}(\\beta)}+1.7647\\times10^{16}\\right)$$\n\n$$\\sqrt{\\left(-8.67927\\times10^{27}\\sin^{2}(\\beta)+2.06332\\times10^{13}\\sqrt{1.94636\\times10^{31}-1.94636\\times10^{31}\\sin^{2}(\\beta)}-9.10286\\times10^{28}\\right)}\\neq 0,\\quad \\theta\\approx 3.56048\\times10^{-16}\\left(1.7647\\times10^{16}-4.\\,\\sqrt{1.94636\\times10^{31}-1.94636\\times10^{31}\\sin^{2}(\\beta)}\\right)$$\n\n**Input interpretation:**\n\n$$v=\\left(\\sqrt{3548143227025099264\\,\\sin^{2}(\\beta)+89875517873681760\\,\\theta^{2}-1129409066758147072\\,\\theta}\\right)\\Big/\\left(\\sqrt{39.4784\\,\\sin^{2}(\\beta)+\\theta^{2}+\\theta\\times(-12.5664)}\\right)$$\n\n**Result:**\n\n$$v=\\left(\\sqrt{3548143227025099264\\,\\sin^{2}(\\beta)+89875517873681760\\,\\theta^{2}-1129409066758147072\\,\\theta}\\right)\\Big/\\left(\\sqrt{39.4784\\,\\sin^{2}(\\beta)+\\theta^{2}-12.5664\\,\\theta}\\right)$$\n\n**Roots:**\n\n$$\\sqrt{\\left(-8.67927\\times10^{27}\\sin^{2}(\\beta)-2.06332\\times10^{13}\\sqrt{1.94636\\times10^{31}-1.94636\\times10^{31}\\sin^{2}(\\beta)}-9.10286\\times10^{28}\\right)}\\neq 0,\\quad \\theta\\approx 3.56048\\times10^{-16}\\left(4.\\,\\sqrt{1.94636\\times10^{31}-1.94636\\times10^{31}\\sin^{2}(\\beta)}+1.7647\\times10^{16}\\right)$$\n\n$$\\sqrt{\\left(-8.67927\\times10^{27}\\sin^{2}(\\beta)+2.06332\\times10^{13}\\sqrt{1.94636\\times10^{31}-1.94636\\times10^{31}\\sin^{2}(\\beta)}-9.10286\\times10^{28}\\right)}\\neq 0,\\quad \\theta\\approx 3.56048\\times10^{-16}\\left(1.7647\\times10^{16}-4.\\,\\sqrt{1.94636\\times10^{31}-1.94636\\times10^{31}\\sin^{2}(\\beta)}\\right)$$"}
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{"file_name": "sor-plot-0004.png", "book": "The Sphere of Realization", "vision_examples": 0, "questions": "", "answers": ""}
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{"file_name": "sor-plot-0005.png", "book": "The Sphere of Realization", "vision_examples": 2, "questions": "Describe this picture from The Sphere of Realization (chapter \"4. Difference between Surface Areas of Two Spheres\").\n\nDescribe this picture from The Sphere of Realization (chapter \"4. Difference between Surface Areas of Two Spheres\").", "answers": "A sphere of radius $r$ with a spherical cap of height $h$; the cap's base circle has radius $a$, and $\\theta$ is the angle at the centre between the axis and the radius $r$ drawn to the edge of the base.\n\nA sphere of radius $r$ with a spherical cap of height $h$; the cap's base circle has radius $a$, and $\\theta$ is the angle at the centre between the axis and the radius $r$ drawn to the edge of the base."}
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{"file_name": "sor-plot-0006.png", "book": "The Sphere of Realization", "vision_examples": 2, "questions": "Describe this picture from The Sphere of Realization (chapter \"4. Difference between Surface Areas of Two Spheres\").\n\nDescribe this picture from The Sphere of Realization (chapter \"4. Difference between Surface Areas of Two Spheres\").", "answers": "A sphere of radius $R$ with a spherical cap of height $h$ and base radius $a$; $R-h$ is the distance from the centre to the base, and $\\alpha$ is the angle at the edge of the base between the base and the radius $R$.\n\nA sphere of radius $R$ with a spherical cap of height $h$ and base radius $a$; $R-h$ is the distance from the centre to the base, and $\\alpha$ is the angle at the edge of the base between the base and the radius $R$."}
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{"file_name": "sor-plot-0007.png", "book": "The Sphere of Realization", "vision_examples": 0, "questions": "", "answers": ""}
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{"file_name": "sor-plot-0008.png", "book": "The Sphere of Realization", "vision_examples": 2, "questions": "Which Wolfram Language command in The Sphere of Realization by Parker Emmerson drew this?\n\nWhich Wolfram Language command in The Sphere of Realization by Parker Emmerson drew this?", "answers": "```wolfram\nManipulate[SphericalPlot3D[{-(i Sqrt[a^2 + h^2] (2 π - θ))/(Sqrt[θ] Sqrt[-16 π + 4 θ]), (i Sqrt[a^2 + h^2] (2 π - θ))/(Sqrt[θ] Sqrt[-16 π + 4 θ])}, {a, -1, 1}, {h, -1, 1}, PlotTheme → {\"Classic\", \"ClassicLights\"}], {θ, 0, 2 π}]\n```\n\n```wolfram\nManipulate[SphericalPlot3D[{-(i Sqrt[a^2 + h^2] (2 π - θ))/(Sqrt[θ] Sqrt[-16 π + 4 θ]), (i Sqrt[a^2 + h^2] (2 π - θ))/(Sqrt[θ] Sqrt[-16 π + 4 θ])}, {a, -1, 1}, {h, -1, 1}, PlotTheme → {\"Classic\", \"ClassicLights\"}], {θ, 0, 2 π}]\n```"}
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| 9 |
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{"file_name": "sor-plot-0009.jpg", "book": "The Sphere of Realization", "vision_examples": 4, "questions": "This is from The Sphere of Realization (chapter \"4. Difference between Surface Areas of Two Spheres\"). Give the command that produced it.\n\nWhat does this graph from The Sphere of Realization show, according to the coordinates stored with it?\n\nThis is from The Sphere of Realization (chapter \"4. Difference between Surface Areas of Two Spheres\"). Give the command that produced it.\n\nWhat does this graph from The Sphere of Realization show, according to the coordinates stored with it?", "answers": "```wolfram\nContourPlot3D[{-(i Sqrt[a^2 + h^2] (2 π - θ))/(Sqrt[θ] Sqrt[-16 π + 4 θ]), (i Sqrt[a^2 + h^2] (2 π - θ))/(Sqrt[θ] Sqrt[-16 π + 4 θ])}, {a, -1, 1}, {h, -1, 1}, {θ, -2 π, 2}, PlotTheme → {\"Classic\", \"ClassicLights\"}]\n```\n\nContourPlot3D draws the surface inside the box a from −1 to 1, h from −1 to 1, θ from −2π to 2 on which the expression stays constant at one of the contour levels Mathematica picks by default. The stored points are (x, y, z) = (a, h, θ) on that surface. The command plots 2 expressions together in one picture. The stored surface has 13,038 points joined into 16,532 faces, with 60 mesh lines drawn over it. It reaches from a = −1 to 1, h = −1 to 1, θ = −0.04501 to 2. The surface is one connected piece, open, with its edge in 1 loop.\n\n```wolfram\nContourPlot3D[{-(i Sqrt[a^2 + h^2] (2 π - θ))/(Sqrt[θ] Sqrt[-16 π + 4 θ]), (i Sqrt[a^2 + h^2] (2 π - θ))/(Sqrt[θ] Sqrt[-16 π + 4 θ])}, {a, -1, 1}, {h, -1, 1}, {θ, -2 π, 2}, PlotTheme → {\"Classic\", \"ClassicLights\"}]\n```\n\nContourPlot3D draws the surface inside the box a from −1 to 1, h from −1 to 1, θ from −2π to 2 on which the expression stays constant at one of the contour levels Mathematica picks by default. The stored points are (x, y, z) = (a, h, θ) on that surface. The command plots 2 expressions together in one picture. The stored surface has 13,038 points joined into 16,532 faces, with 60 mesh lines drawn over it. It reaches from a = −1 to 1, h = −1 to 1, θ = −0.04501 to 2. The surface is one connected piece, open, with its edge in 1 loop."}
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| 10 |
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{"file_name": "sor-plot-0010.jpg", "book": "The Sphere of Realization", "vision_examples": 4, "questions": "Which Wolfram Language command in The Sphere of Realization by Parker Emmerson drew this?\n\nDescribe this plot from The Sphere of Realization (chapter \"4. Difference between Surface Areas of Two Spheres\").\n\nWhich Wolfram Language command in The Sphere of Realization by Parker Emmerson drew this?\n\nDescribe this plot from The Sphere of Realization (chapter \"4. Difference between Surface Areas of Two Spheres\").", "answers": "```wolfram\nContourPlot3D[(i Sqrt[a^2 + h^2] (2 π - θ))/(Sqrt[θ] Sqrt[-16 π + 4 θ]), {a, -1, 1}, {h, -1, 1}, {θ, -2 π, 2}, PlotTheme → {\"Classic\", \"ClassicLights\"}]\n```\n\nContourPlot3D draws the surface inside the box a from −1 to 1, h from −1 to 1, θ from −2π to 2 on which the expression stays constant at one of the contour levels Mathematica picks by default. The stored points are (x, y, z) = (a, h, θ) on that surface. The stored surface has 50,010 points joined into 63,358 faces, with 197 mesh lines drawn over it. It reaches from a = −1 to 1, h = −1 to 1, θ = −6.283 to 2. The surface is one connected piece, open, with its edge in 3 loops. The stored points are symmetric under x → −x, and under y → −y, and under (x, y) → (−x, −y).\n\n```wolfram\nContourPlot3D[(i Sqrt[a^2 + h^2] (2 π - θ))/(Sqrt[θ] Sqrt[-16 π + 4 θ]), {a, -1, 1}, {h, -1, 1}, {θ, -2 π, 2}, PlotTheme → {\"Classic\", \"ClassicLights\"}]\n```\n\nContourPlot3D draws the surface inside the box a from −1 to 1, h from −1 to 1, θ from −2π to 2 on which the expression stays constant at one of the contour levels Mathematica picks by default. The stored points are (x, y, z) = (a, h, θ) on that surface. The stored surface has 50,010 points joined into 63,358 faces, with 197 mesh lines drawn over it. It reaches from a = −1 to 1, h = −1 to 1, θ = −6.283 to 2. The surface is one connected piece, open, with its edge in 3 loops. The stored points are symmetric under x → −x, and under y → −y, and under (x, y) → (−x, −y)."}
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| 11 |
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{"file_name": "sor-plot-0011.jpg", "book": "The Sphere of Realization", "vision_examples": 0, "questions": "", "answers": ""}
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| 12 |
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{"file_name": "sor-plot-0012.jpg", "book": "The Sphere of Realization", "vision_examples": 0, "questions": "", "answers": ""}
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| 13 |
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{"file_name": "sor-plot-0013.jpg", "book": "The Sphere of Realization", "vision_examples": 4, "questions": "Which Wolfram Language command in The Sphere of Realization by Parker Emmerson drew this?\n\nWhat does this graph from The Sphere of Realization show, according to the coordinates stored with it?\n\nWhich Wolfram Language command in The Sphere of Realization by Parker Emmerson drew this?\n\nWhat does this graph from The Sphere of Realization show, according to the coordinates stored with it?", "answers": "```wolfram\nContourPlot3D[(i Sqrt[4 a^2 π^2 - 4 a^2 π θ - 16 π x^2 θ + a^2 θ^2 + 4 x^2 θ^2])/(-2 π + θ), {x, -1, 1}, {a, -1, 1}, {θ, -2 π, 2 π}, PlotTheme → {\"Classic\", \"ClassicLights\"}]\n```\n\nContourPlot3D draws the surface inside the box x from −1 to 1, a from −1 to 1, θ from −2π to 2π on which the expression stays constant at one of the contour levels Mathematica picks by default. The stored points are (x, y, z) = (x, a, θ) on that surface. The stored surface has 152,670 points joined into 172,563 faces, with 325 mesh lines drawn over it. It reaches from x = −1 to 1, a = −1 to 1, θ = −6.283 to 6.283. The surface falls into 5 main separate pieces plus 2 small fragments: 1 is closed, with sheets meeting along shared edges; 4 are open, each with its edge in 1 loop. The stored points are symmetric under x → −x, and under y → −y, and under (x, y) → (−x, −y).\n\n```wolfram\nContourPlot3D[(i Sqrt[4 a^2 π^2 - 4 a^2 π θ - 16 π x^2 θ + a^2 θ^2 + 4 x^2 θ^2])/(-2 π + θ), {x, -1, 1}, {a, -1, 1}, {θ, -2 π, 2 π}, PlotTheme → {\"Classic\", \"ClassicLights\"}]\n```\n\nContourPlot3D draws the surface inside the box x from −1 to 1, a from −1 to 1, θ from −2π to 2π on which the expression stays constant at one of the contour levels Mathematica picks by default. The stored points are (x, y, z) = (x, a, θ) on that surface. The stored surface has 152,670 points joined into 172,563 faces, with 325 mesh lines drawn over it. It reaches from x = −1 to 1, a = −1 to 1, θ = −6.283 to 6.283. The surface falls into 5 main separate pieces plus 2 small fragments: 1 is closed, with sheets meeting along shared edges; 4 are open, each with its edge in 1 loop. The stored points are symmetric under x → −x, and under y → −y, and under (x, y) → (−x, −y)."}
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| 14 |
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{"file_name": "sor-plot-0014.jpg", "book": "The Sphere of Realization", "vision_examples": 0, "questions": "", "answers": ""}
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{"file_name": "sor-plot-0015.jpg", "book": "The Sphere of Realization", "vision_examples": 0, "questions": "", "answers": ""}
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| 16 |
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{"file_name": "sor-plot-0016.jpg", "book": "The Sphere of Realization", "vision_examples": 2, "questions": "What does this graph from The Sphere of Realization show, according to the coordinates stored with it?\n\nWhat does this graph from The Sphere of Realization show, according to the coordinates stored with it?", "answers": "This graphic was pasted into an input cell; the plotting function that made it is not recorded with it, so its points are given as plain (x, y, z) positions. The stored surface has 1,848 points joined into 2,231 faces, with 47 mesh lines drawn over it. It reaches from x = −1 to 1, y = −1 to 1, z = 0 to 0.8944. The surface is one connected piece, open, with its edge in 1 loop. The stored points are symmetric under (x, y) → (−x, −y).\n\nThis graphic was pasted into an input cell; the plotting function that made it is not recorded with it, so its points are given as plain (x, y, z) positions. The stored surface has 1,848 points joined into 2,231 faces, with 47 mesh lines drawn over it. It reaches from x = −1 to 1, y = −1 to 1, z = 0 to 0.8944. The surface is one connected piece, open, with its edge in 1 loop. The stored points are symmetric under (x, y) → (−x, −y)."}
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| 17 |
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{"file_name": "sor-plot-0017.jpg", "book": "The Sphere of Realization", "vision_examples": 0, "questions": "", "answers": ""}
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| 18 |
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{"file_name": "sor-plot-0018.jpg", "book": "The Sphere of Realization", "vision_examples": 4, "questions": "This is from The Sphere of Realization (chapter \"4. Difference between Surface Areas of Two Spheres\"). Give the command that produced it.\n\nDescribe this plot from The Sphere of Realization (chapter \"4. Difference between Surface Areas of Two Spheres\").\n\nThis is from The Sphere of Realization (chapter \"4. Difference between Surface Areas of Two Spheres\"). Give the command that produced it.\n\nDescribe this plot from The Sphere of Realization (chapter \"4. Difference between Surface Areas of Two Spheres\").", "answers": "```wolfram\nRevolutionPlot3D[(Sqrt[-((Sqrt[4 π r^2 θ - r^2 θ^2])/(2 π))^2 + 4 r^2])/(Sqrt[5]), {θ, -1 π, 1 π}, {r, -1, 1}]\n```\n\nRevolutionPlot3D takes the expression as a height z and sweeps it around the vertical axis: the variable θ is the distance from that axis and r is the angle of rotation. So for every stored point, z is the value of the expression and √(x² + y²) is |θ|. The stored surface has 1,960 points joined into 2,358 faces, with 31 mesh lines drawn over it. It reaches from x = −3.142 to 3.142, y = −2.644 to 2.644, z = 0 to 1.025. The value of the expression (the height z) runs from 0, reached at distance 3.142 from the axis, to 1.025, reached at distance 3.142. The surface lies between distance 0 and 3.142 from the vertical axis. About 16% of the drawn surface stands as near-vertical walls (a steep jump in the value over a small step in the variables, as drawn in the plot box). The surface is one connected piece, open, with its edge in 1 loop.\n\n```wolfram\nRevolutionPlot3D[(Sqrt[-((Sqrt[4 π r^2 θ - r^2 θ^2])/(2 π))^2 + 4 r^2])/(Sqrt[5]), {θ, -1 π, 1 π}, {r, -1, 1}]\n```\n\nRevolutionPlot3D takes the expression as a height z and sweeps it around the vertical axis: the variable θ is the distance from that axis and r is the angle of rotation. So for every stored point, z is the value of the expression and √(x² + y²) is |θ|. The stored surface has 1,960 points joined into 2,358 faces, with 31 mesh lines drawn over it. It reaches from x = −3.142 to 3.142, y = −2.644 to 2.644, z = 0 to 1.025. The value of the expression (the height z) runs from 0, reached at distance 3.142 from the axis, to 1.025, reached at distance 3.142. The surface lies between distance 0 and 3.142 from the vertical axis. About 16% of the drawn surface stands as near-vertical walls (a steep jump in the value over a small step in the variables, as drawn in the plot box). The surface is one connected piece, open, with its edge in 1 loop."}
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{"file_name": "sor-plot-0019.jpg", "book": "The Sphere of Realization", "vision_examples": 2, "questions": "What does this graph from The Sphere of Realization show, according to the coordinates stored with it?\n\nWhat does this graph from The Sphere of Realization show, according to the coordinates stored with it?", "answers": "This graphic was pasted into an input cell; the plotting function that made it is not recorded with it, so its points are given as plain (x, y, z) positions. The stored surface has 161,635 points joined into 194,134 faces, with 1,386 mesh lines drawn over it. It reaches from x = −0.7806 to 0.9709, y = −1 to 1, z = −1 to 1. The surface falls into 3 main separate pieces plus 1 small fragment: 1 is open, with its edge in 153 loops; 1 is open, with its edge in 2 loops; 1 is open, with its edge in 1 loop. The stored points are symmetric under z → −z.\n\nThis graphic was pasted into an input cell; the plotting function that made it is not recorded with it, so its points are given as plain (x, y, z) positions. The stored surface has 161,635 points joined into 194,134 faces, with 1,386 mesh lines drawn over it. It reaches from x = −0.7806 to 0.9709, y = −1 to 1, z = −1 to 1. The surface falls into 3 main separate pieces plus 1 small fragment: 1 is open, with its edge in 153 loops; 1 is open, with its edge in 2 loops; 1 is open, with its edge in 1 loop. The stored points are symmetric under z → −z."}
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{"file_name": "sor-plot-0020.jpg", "book": "The Sphere of Realization", "vision_examples": 0, "questions": "", "answers": ""}
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{"file_name": "sor-plot-0021.jpg", "book": "The Sphere of Realization", "vision_examples": 4, "questions": "This is from The Sphere of Realization (chapter \"5. Difference between Volumes of Two Spheres: Imaginary 4-D realms.\"). Give the command that produced it.\n\nDescribe this plot from The Sphere of Realization (chapter \"5. Difference between Volumes of Two Spheres: Imaginary 4-D realms.\").\n\nThis is from The Sphere of Realization (chapter \"5. Difference between Volumes of Two Spheres: Imaginary 4-D realms.\"). Give the command that produced it.\n\nDescribe this plot from The Sphere of Realization (chapter \"5. Difference between Volumes of Two Spheres: Imaginary 4-D realms.\").", "answers": "```wolfram\nContourPlot3D[1/2 ((-1))^(2 / 3) ((-3 a^2 h - h^3 + 2 h r^2 + 6 r^3))^(1 / 3), {r, -1, 1}, {a, -1, 1}, {h, -1, 1}, PlotTheme → {\"Classic\", \"ClassicLights\"}]\n```\n\nContourPlot3D draws the surface inside the box r from −1 to 1, a from −1 to 1, h from −1 to 1 on which the expression stays constant at one of the contour levels Mathematica picks by default. The stored points are (x, y, z) = (r, a, h) on that surface. The stored surface has 77,021 points joined into 94,723 faces, with 245 mesh lines drawn over it. It reaches from r = −0.9709 to 0.7806, a = −1 to 1, h = −1 to 1. The surface falls into 3 main separate pieces; each main piece is open, with its edge in 1 loop. The stored points are symmetric under y → −y.\n\n```wolfram\nContourPlot3D[1/2 ((-1))^(2 / 3) ((-3 a^2 h - h^3 + 2 h r^2 + 6 r^3))^(1 / 3), {r, -1, 1}, {a, -1, 1}, {h, -1, 1}, PlotTheme → {\"Classic\", \"ClassicLights\"}]\n```\n\nContourPlot3D draws the surface inside the box r from −1 to 1, a from −1 to 1, h from −1 to 1 on which the expression stays constant at one of the contour levels Mathematica picks by default. The stored points are (x, y, z) = (r, a, h) on that surface. The stored surface has 77,021 points joined into 94,723 faces, with 245 mesh lines drawn over it. It reaches from r = −0.9709 to 0.7806, a = −1 to 1, h = −1 to 1. The surface falls into 3 main separate pieces; each main piece is open, with its edge in 1 loop. The stored points are symmetric under y → −y."}
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{"file_name": "sor-plot-0022.jpg", "book": "The Sphere of Realization", "vision_examples": 4, "questions": "This is from The Sphere of Realization (chapter \"5. Difference between Volumes of Two Spheres: Imaginary 4-D realms.\"). Give the command that produced it.\n\nDescribe this plot from The Sphere of Realization (chapter \"5. Difference between Volumes of Two Spheres: Imaginary 4-D realms.\").\n\nThis is from The Sphere of Realization (chapter \"5. Difference between Volumes of Two Spheres: Imaginary 4-D realms.\"). Give the command that produced it.\n\nDescribe this plot from The Sphere of Realization (chapter \"5. Difference between Volumes of Two Spheres: Imaginary 4-D realms.\").", "answers": "```wolfram\nContourPlot3D[{-h/9 + (2 2^(1 / 3) h^2)/(9 ((2916 a^2 h + 956 h^3 + 7776 x^3 + Sqrt[-256 h^6 + ((2916 a^2 h + 956 h^3 + 7776 x^3))^2]))^(1 / 3)) + (((2916 a^2 h + 956 h^3 + 7776 x^3 + Sqrt[-256 h^6 + ((2916 a^2 h + 956 h^3 + 7776 x^3))^2]))^(1 / 3))/(18 2^(1 / 3)), -h/9 - (2^(1 / 3) (1 + i Sqrt[3]) h^2)/(9 ((2916 a^2 h + 956 h^3 + 7776 x^3 + Sqrt[-256 h^6 + ((2916 a^2 h + 956 h^3 + 7776 x^3))^2]))^(1 / 3)) - ((1 - i Sqrt[3]) ((2916 a^2 h + 956 h^3 + 7776 x^3 + Sqrt[-256 h^6 + ((2916 a^2 h + 956 h^3 + 7776 x^3))^2]))^(1 / 3))/(36 2^(1 / 3)), -h/9 - (2^(1 / 3) (1 - i Sqrt[3]) h^2)/(9 ((2916 a^2 h + 956 h^3 + 7776 x^3 + Sqrt[-256 h^6 + ((2916 a^2 h + 956 h^3 + 7776 x^3))^2]))^(1 / 3)) - ((1 + i Sqrt[3]) ((2916 a^2 h + 956 h^3 + 7776 x^3 + Sqrt[-256 h^6 + ((2916 a^2 h + 956 h^3 + 7776 x^3))^2]))^(1 / 3))/(36 2^(1 / 3))}, {h, -1, 1}, {a, -1, 1}, {x, -1, 1}, PlotTheme → {\"Classic\", \"ClassicLights\"}]\n```\n\nContourPlot3D draws the surface inside the box h from −1 to 1, a from −1 to 1, x from −1 to 1 on which the expression stays constant at one of the contour levels Mathematica picks by default. The stored points are (x, y, z) = (h, a, x) on that surface. The command plots 3 expressions together in one picture. The stored surface has 39,503 points joined into 48,116 faces, with 172 mesh lines drawn over it. It reaches from h = −1 to 1, a = −1 to 1, x = −1 to 0.8119. The surface is one connected piece, open, with its edge in 4 loops. The stored points are symmetric under y → −y.\n\n```wolfram\nContourPlot3D[{-h/9 + (2 2^(1 / 3) h^2)/(9 ((2916 a^2 h + 956 h^3 + 7776 x^3 + Sqrt[-256 h^6 + ((2916 a^2 h + 956 h^3 + 7776 x^3))^2]))^(1 / 3)) + (((2916 a^2 h + 956 h^3 + 7776 x^3 + Sqrt[-256 h^6 + ((2916 a^2 h + 956 h^3 + 7776 x^3))^2]))^(1 / 3))/(18 2^(1 / 3)), -h/9 - (2^(1 / 3) (1 + i Sqrt[3]) h^2)/(9 ((2916 a^2 h + 956 h^3 + 7776 x^3 + Sqrt[-256 h^6 + ((2916 a^2 h + 956 h^3 + 7776 x^3))^2]))^(1 / 3)) - ((1 - i Sqrt[3]) ((2916 a^2 h + 956 h^3 + 7776 x^3 + Sqrt[-256 h^6 + ((2916 a^2 h + 956 h^3 + 7776 x^3))^2]))^(1 / 3))/(36 2^(1 / 3)), -h/9 - (2^(1 / 3) (1 - i Sqrt[3]) h^2)/(9 ((2916 a^2 h + 956 h^3 + 7776 x^3 + Sqrt[-256 h^6 + ((2916 a^2 h + 956 h^3 + 7776 x^3))^2]))^(1 / 3)) - ((1 + i Sqrt[3]) ((2916 a^2 h + 956 h^3 + 7776 x^3 + Sqrt[-256 h^6 + ((2916 a^2 h + 956 h^3 + 7776 x^3))^2]))^(1 / 3))/(36 2^(1 / 3))}, {h, -1, 1}, {a, -1, 1}, {x, -1, 1}, PlotTheme → {\"Classic\", \"ClassicLights\"}]\n```\n\nContourPlot3D draws the surface inside the box h from −1 to 1, a from −1 to 1, x from −1 to 1 on which the expression stays constant at one of the contour levels Mathematica picks by default. The stored points are (x, y, z) = (h, a, x) on that surface. The command plots 3 expressions together in one picture. The stored surface has 39,503 points joined into 48,116 faces, with 172 mesh lines drawn over it. It reaches from h = −1 to 1, a = −1 to 1, x = −1 to 0.8119. The surface is one connected piece, open, with its edge in 4 loops. The stored points are symmetric under y → −y."}
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{"file_name": "sor-plot-0023.jpg", "book": "The Sphere of Realization", "vision_examples": 4, "questions": "Which Wolfram Language command in The Sphere of Realization by Parker Emmerson drew this?\n\nDescribe this plot from The Sphere of Realization (chapter \"5. Difference between Volumes of Two Spheres: Imaginary 4-D realms.\").\n\nWhich Wolfram Language command in The Sphere of Realization by Parker Emmerson drew this?\n\nDescribe this plot from The Sphere of Realization (chapter \"5. Difference between Volumes of Two Spheres: Imaginary 4-D realms.\").", "answers": "```wolfram\nContourPlot3D[-h/9 + (2 2^(1 / 3) h^2)/(9 ((2916 a^2 h + 956 h^3 + 7776 x^3 + Sqrt[-256 h^6 + ((2916 a^2 h + 956 h^3 + 7776 x^3))^2]))^(1 / 3)) + (((2916 a^2 h + 956 h^3 + 7776 x^3 + Sqrt[-256 h^6 + ((2916 a^2 h + 956 h^3 + 7776 x^3))^2]))^(1 / 3))/(18 2^(1 / 3)), {h, -1, 1}, {a, -1, 1}, {x, -1, 1}, PlotTheme → {\"Classic\", \"ClassicLights\"}]\n```\n\nContourPlot3D draws the surface inside the box h from −1 to 1, a from −1 to 1, x from −1 to 1 on which the expression stays constant at one of the contour levels Mathematica picks by default. The stored points are (x, y, z) = (h, a, x) on that surface. The stored surface has 151,137 points joined into 181,347 faces, with 853 mesh lines drawn over it. It reaches from h = −1 to 1, a = −1 to 1, x = −0.8119 to 0.9362. The surface falls into 3 main separate pieces plus 2 small fragments: 1 is open, with its edge in 105 loops; 2 are open, each with its edge in 1 loop. The stored points are symmetric under y → −y.\n\n```wolfram\nContourPlot3D[-h/9 + (2 2^(1 / 3) h^2)/(9 ((2916 a^2 h + 956 h^3 + 7776 x^3 + Sqrt[-256 h^6 + ((2916 a^2 h + 956 h^3 + 7776 x^3))^2]))^(1 / 3)) + (((2916 a^2 h + 956 h^3 + 7776 x^3 + Sqrt[-256 h^6 + ((2916 a^2 h + 956 h^3 + 7776 x^3))^2]))^(1 / 3))/(18 2^(1 / 3)), {h, -1, 1}, {a, -1, 1}, {x, -1, 1}, PlotTheme → {\"Classic\", \"ClassicLights\"}]\n```\n\nContourPlot3D draws the surface inside the box h from −1 to 1, a from −1 to 1, x from −1 to 1 on which the expression stays constant at one of the contour levels Mathematica picks by default. The stored points are (x, y, z) = (h, a, x) on that surface. The stored surface has 151,137 points joined into 181,347 faces, with 853 mesh lines drawn over it. It reaches from h = −1 to 1, a = −1 to 1, x = −0.8119 to 0.9362. The surface falls into 3 main separate pieces plus 2 small fragments: 1 is open, with its edge in 105 loops; 2 are open, each with its edge in 1 loop. The stored points are symmetric under y → −y."}
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{"file_name": "sor-plot-0024.jpg", "book": "The Sphere of Realization", "vision_examples": 2, "questions": "What does this graph from The Sphere of Realization show, according to the coordinates stored with it?\n\nWhat does this graph from The Sphere of Realization show, according to the coordinates stored with it?", "answers": "This graphic was pasted into an input cell; the plotting function that made it is not recorded with it, so its points are given as plain (x, y, z) positions. The stored surface has 211,792 points joined into 251,559 faces, with 1,058 mesh lines drawn over it. It reaches from x = −1 to 1, y = −1 to 1, z = −1.571 to 1.571. The surface falls into 2 main separate pieces plus 8 small fragments: 1 is open, with its edge in 69 loops; 1 is closed, with sheets meeting along shared edges. The stored points are symmetric under y → −y, and under z → −z.\n\nThis graphic was pasted into an input cell; the plotting function that made it is not recorded with it, so its points are given as plain (x, y, z) positions. The stored surface has 211,792 points joined into 251,559 faces, with 1,058 mesh lines drawn over it. It reaches from x = −1 to 1, y = −1 to 1, z = −1.571 to 1.571. The surface falls into 2 main separate pieces plus 8 small fragments: 1 is open, with its edge in 69 loops; 1 is closed, with sheets meeting along shared edges. The stored points are symmetric under y → −y, and under z → −z."}
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| 25 |
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{"file_name": "sor-plot-0025.jpg", "book": "The Sphere of Realization", "vision_examples": 0, "questions": "", "answers": ""}
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| 26 |
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{"file_name": "sor-plot-0026.jpg", "book": "The Sphere of Realization", "vision_examples": 0, "questions": "", "answers": ""}
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| 27 |
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{"file_name": "sor-plot-0027.jpg", "book": "The Sphere of Realization", "vision_examples": 0, "questions": "", "answers": ""}
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{"file_name": "sor-plot-0028.jpg", "book": "The Sphere of Realization", "vision_examples": 4, "questions": "Which Wolfram Language command in The Sphere of Realization by Parker Emmerson drew this?\n\nWhat does this graph from The Sphere of Realization show, according to the coordinates stored with it?\n\nWhat does this graph from The Sphere of Realization show, according to the coordinates stored with it?\n\nWhich Wolfram Language command in The Sphere of Realization by Parker Emmerson drew this?", "answers": "```wolfram\nSphericalPlot3D[(6 π^2 (-4 π + 2 θ) Cos[β] Sin[β] Sqrt[-c^2 (-θ^2 + 8 π^2 (1 + Sqrt[(Cos[β])^2]) - 4 π^2 (Sin[β])^2)])/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^(5 / 2)) - (4 c^2 π^2 θ Cos[β] Sin[β])/(Sqrt[-c^2 (-θ^2 + 8 π^2 (1 + Sqrt[(Cos[β])^2]) - 4 π^2 (Sin[β])^2)] ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^(3 / 2)) + (c^2 (-4 π + 2 θ) (-8 π^2 Cos[β] Sin[β] - (8 π^2 Cos[β] Sin[β])/(Sqrt[(Cos[β])^2])))/(4 Sqrt[-c^2 (-θ^2 + 8 π^2 (1 + Sqrt[(Cos[β])^2]) - 4 π^2 (Sin[β])^2)] ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^(3 / 2)) + (c^4 (θ) (-8 π^2 Cos[β] Sin[β] - (8 π^2 Cos[β] Sin[β])/(Sqrt[(Cos[β])^2])))/(2 ((-c^2 (-θ^2 + 8 π^2 (1 + Sqrt[(Cos[β])^2]) - 4 π^2 (Sin[β])^2)))^(3 / 2) Sqrt[θ (-4 π + θ) + 4 π^2 (Sin[β])^2]), {β, 0, π / 2}, {θ, 0, 2 π}, PlotTheme → \"Orange\"]\n```\n\nSphericalPlot3D takes the expression as a distance from the origin: β is the angle measured down from the vertical axis and θ the angle around it. So for every stored point, its distance from the origin is the absolute value of the expression at those two angles. The stored surface has 2,044 points joined into 2,792 faces, with 35 mesh lines drawn over it. It reaches from x = −1.473 × 10^26 to 1.063 × 10^19, y = −1.296 × 10^17 to 6.61 × 10^19, z = −2.269 × 10^22 to 0. The box Mathematica displays is narrower (x from −9.828 × 10^8 to 1.033 × 10^9, y from −9.172 × 10^8 to 1.007 × 10^9, z from −8.432 × 10^9 to 0), so the parts of the surface beyond it are cut off in the picture. Distances from the origin run from 0 to 1.473 × 10^26; the farthest point lies 90° from the vertical axis and 180° around it, so the absolute value of the expression reaches 1.473 × 10^26 there. The surface is one connected piece, open, with its edge in 1 loop.\n\nSphericalPlot3D takes the expression as a distance from the origin: β is the angle measured down from the vertical axis and θ the angle around it. So for every stored point, its distance from the origin is the absolute value of the expression at those two angles. The stored surface has 2,044 points joined into 2,792 faces, with 35 mesh lines drawn over it. It reaches from x = −1.473 × 10^26 to 1.063 × 10^19, y = −1.296 × 10^17 to 6.61 × 10^19, z = −2.269 × 10^22 to 0. The box Mathematica displays is narrower (x from −9.828 × 10^8 to 1.033 × 10^9, y from −9.172 × 10^8 to 1.007 × 10^9, z from −8.432 × 10^9 to 0), so the parts of the surface beyond it are cut off in the picture. Distances from the origin run from 0 to 1.473 × 10^26; the farthest point lies 90° from the vertical axis and 180° around it, so the absolute value of the expression reaches 1.473 × 10^26 there. The surface is one connected piece, open, with its edge in 1 loop.\n\n```wolfram\nSphericalPlot3D[(6 π^2 (-4 π + 2 θ) Cos[β] Sin[β] Sqrt[-c^2 (-θ^2 + 8 π^2 (1 + Sqrt[(Cos[β])^2]) - 4 π^2 (Sin[β])^2)])/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^(5 / 2)) - (4 c^2 π^2 θ Cos[β] Sin[β])/(Sqrt[-c^2 (-θ^2 + 8 π^2 (1 + Sqrt[(Cos[β])^2]) - 4 π^2 (Sin[β])^2)] ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^(3 / 2)) + (c^2 (-4 π + 2 θ) (-8 π^2 Cos[β] Sin[β] - (8 π^2 Cos[β] Sin[β])/(Sqrt[(Cos[β])^2])))/(4 Sqrt[-c^2 (-θ^2 + 8 π^2 (1 + Sqrt[(Cos[β])^2]) - 4 π^2 (Sin[β])^2)] ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^(3 / 2)) + (c^4 (θ) (-8 π^2 Cos[β] Sin[β] - (8 π^2 Cos[β] Sin[β])/(Sqrt[(Cos[β])^2])))/(2 ((-c^2 (-θ^2 + 8 π^2 (1 + Sqrt[(Cos[β])^2]) - 4 π^2 (Sin[β])^2)))^(3 / 2) Sqrt[θ (-4 π + θ) + 4 π^2 (Sin[β])^2]), {β, 0, π / 2}, {θ, 0, 2 π}, PlotTheme → \"Orange\"]\n```"}
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{"file_name": "sor-plot-0029.jpg", "book": "The Sphere of Realization", "vision_examples": 4, "questions": "Which Wolfram Language command in The Sphere of Realization by Parker Emmerson drew this?\n\nDescribe this plot from The Sphere of Realization (chapter \"7. Discussions on the Nature of Acceleration and Non-Commutation: Constrained Models and Generalized Propositions for Light Speed\").\n\nWhich Wolfram Language command in The Sphere of Realization by Parker Emmerson drew this?\n\nDescribe this plot from The Sphere of Realization (chapter \"7. Discussions on the Nature of Acceleration and Non-Commutation: Constrained Models and Generalized Propositions for Light Speed\").", "answers": "```wolfram\nSphericalPlot3D[(6 π^2 (-4 π + 2 2 (π + Sqrt[π^2 - π^2 (Sin[β])^2])) Cos[β] Sin[β] Sqrt[-c^2 (-θ^2 + 8 π^2 (1 + Sqrt[(Cos[β])^2]) - 4 π^2 (Sin[β])^2)])/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^(5 / 2)) - (4 c^2 π^2 θ Cos[β] Sin[β])/(Sqrt[-c^2 (-θ^2 + 8 π^2 (1 + Sqrt[(Cos[β])^2]) - 4 π^2 (Sin[β])^2)] ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^(3 / 2)) + (c^2 (-4 π + 2 θ) (-8 π^2 Cos[β] Sin[β] - (8 π^2 Cos[β] Sin[β])/(Sqrt[(Cos[β])^2])))/(4 Sqrt[-c^2 (-θ^2 + 8 π^2 (1 + Sqrt[(Cos[β])^2]) - 4 π^2 (Sin[β])^2)] ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^(3 / 2)) + (c^4 θ (-8 π^2 Cos[β] Sin[β] - (8 π^2 Cos[β] Sin[β])/(Sqrt[(Cos[β])^2])))/(2 ((-c^2 (-((2 (π + Sqrt[π^2 - π^2 (Sin[β])^2])))^2 + 8 π^2 (1 + Sqrt[(Cos[β])^2]) - 4 π^2 (Sin[β])^2)))^(3 / 2) Sqrt[θ (-4 π + θ) + 4 π^2 (Sin[β])^2]), {β, 0, π / 2}, {θ, 0, 2 π}, PlotTheme → \"Classic\"]\n```\n\nSphericalPlot3D takes the expression as a distance from the origin: β is the angle measured down from the vertical axis and θ the angle around it. So for every stored point, its distance from the origin is the absolute value of the expression at those two angles. The stored surface has 2,758 points joined into 3,441 faces, with 129 mesh lines drawn over it. It reaches from x = −2.633 × 10^32 to 4.804 × 10^32, y = −2.745 × 10^32 to 2.933 × 10^32, z = −7.841 × 10^32 to 2.453 × 10^13. The box Mathematica displays is narrower (x from −1.049 × 10^31 to 1.216 × 10^31, y from −1.326 × 10^31 to 1.941 × 10^31, z from −4.855 × 10^31 to 2.453 × 10^13), so the parts of the surface beyond it are cut off in the picture. Distances from the origin run from 0 to 8.271 × 10^32; the farthest point lies 161.4° from the vertical axis and 180° around it, so the absolute value of the expression reaches 8.271 × 10^32 there. The surface is one connected piece, open, with its edge in 1 loop.\n\n```wolfram\nSphericalPlot3D[(6 π^2 (-4 π + 2 2 (π + Sqrt[π^2 - π^2 (Sin[β])^2])) Cos[β] Sin[β] Sqrt[-c^2 (-θ^2 + 8 π^2 (1 + Sqrt[(Cos[β])^2]) - 4 π^2 (Sin[β])^2)])/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^(5 / 2)) - (4 c^2 π^2 θ Cos[β] Sin[β])/(Sqrt[-c^2 (-θ^2 + 8 π^2 (1 + Sqrt[(Cos[β])^2]) - 4 π^2 (Sin[β])^2)] ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^(3 / 2)) + (c^2 (-4 π + 2 θ) (-8 π^2 Cos[β] Sin[β] - (8 π^2 Cos[β] Sin[β])/(Sqrt[(Cos[β])^2])))/(4 Sqrt[-c^2 (-θ^2 + 8 π^2 (1 + Sqrt[(Cos[β])^2]) - 4 π^2 (Sin[β])^2)] ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^(3 / 2)) + (c^4 θ (-8 π^2 Cos[β] Sin[β] - (8 π^2 Cos[β] Sin[β])/(Sqrt[(Cos[β])^2])))/(2 ((-c^2 (-((2 (π + Sqrt[π^2 - π^2 (Sin[β])^2])))^2 + 8 π^2 (1 + Sqrt[(Cos[β])^2]) - 4 π^2 (Sin[β])^2)))^(3 / 2) Sqrt[θ (-4 π + θ) + 4 π^2 (Sin[β])^2]), {β, 0, π / 2}, {θ, 0, 2 π}, PlotTheme → \"Classic\"]\n```\n\nSphericalPlot3D takes the expression as a distance from the origin: β is the angle measured down from the vertical axis and θ the angle around it. So for every stored point, its distance from the origin is the absolute value of the expression at those two angles. The stored surface has 2,758 points joined into 3,441 faces, with 129 mesh lines drawn over it. It reaches from x = −2.633 × 10^32 to 4.804 × 10^32, y = −2.745 × 10^32 to 2.933 × 10^32, z = −7.841 × 10^32 to 2.453 × 10^13. The box Mathematica displays is narrower (x from −1.049 × 10^31 to 1.216 × 10^31, y from −1.326 × 10^31 to 1.941 × 10^31, z from −4.855 × 10^31 to 2.453 × 10^13), so the parts of the surface beyond it are cut off in the picture. Distances from the origin run from 0 to 8.271 × 10^32; the farthest point lies 161.4° from the vertical axis and 180° around it, so the absolute value of the expression reaches 8.271 × 10^32 there. The surface is one connected piece, open, with its edge in 1 loop."}
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{"file_name": "sor-plot-0030.jpg", "book": "The Sphere of Realization", "vision_examples": 0, "questions": "", "answers": ""}
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{"file_name": "sor-plot-0031.jpg", "book": "The Sphere of Realization", "vision_examples": 4, "questions": "This is from The Sphere of Realization (chapter \"7. Discussions on the Nature of Acceleration and Non-Commutation: Constrained Models and Generalized Propositions for Light Speed\"). Give the command that produced it.\n\nWhat does this graph from The Sphere of Realization show, according to the coordinates stored with it?\n\nThis is from The Sphere of Realization (chapter \"7. Discussions on the Nature of Acceleration and Non-Commutation: Constrained Models and Generalized Propositions for Light Speed\"). Give the command that produced it.\n\nWhat does this graph from The Sphere of Realization show, according to the coordinates stored with it?", "answers": "```wolfram\nContourPlot3D[(Sqrt[θ] Sqrt[-4 π + θ] (4 π γ + γ^2 - Sqrt[γ] Sqrt[((2 π + γ))^2] Sqrt[4 π + γ] Sin[β] - 4 π^2 (Sin[β])^2 - 4 π γ (Sin[β])^2 - γ^2 (Sin[β])^2 - (π Sqrt[γ] Sqrt[((2 π + γ))^2] (Sin[β])^3)/(Sqrt[4 π + γ]) + (π Sqrt[((2 π + γ))^2] Sqrt[4 π + γ] (Sin[β])^3)/(Sqrt[γ]) + Sqrt[γ] Sqrt[((2 π + γ))^2] Sqrt[4 π + γ] (Sin[β])^3)) / (π^2 (-16 π γ - 4 γ^2 + 16 π^2 (Sin[β])^2 + 16 π γ (Sin[β])^2 + 4 γ^2 (Sin[β])^2) √((192 π^4 (Sin[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3) + (64 π^3 θ (Sin[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3) + (16 π^2 θ^2 (Sin[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3) + (4 π θ^3 (Sin[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3) + (θ^4 (Sin[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3) - (64 π^4 (Sin[β])^4)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3) + (4 π^2 θ^2 (Sin[β])^4)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3) - (112 π^2 (Sin[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2) - (20 π θ (Sin[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2) - (2 θ^2 (Sin[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2) - (7 (Sin[β])^2)/(8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) + (36864 π^8 (Cos[β])^2 (Sin[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^5) - (18432 π^7 θ (Cos[β])^2 (Sin[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^5) - (36864 π^8 (Cos[β])^2 (Sin[β])^4)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^5) + (18432 π^7 θ (Cos[β])^2 (Sin[β])^4)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^5) - (147456 π^10 (Cos[β])^2 (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^5) + (147456 π^10 (Cos[β])^4 (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^5) + (294912 π^10 (Cos[β])^2 (Sin[β])^4)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^5) - (147456 π^10 (Cos[β])^4 (Sin[β])^4)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^5) - (147456 π^10 (Cos[β])^2 (Sin[β])^6)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^5) + (1536 π^6 (Sin[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) - (768 π^5 θ (Sin[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) + (6144 π^6 (Cos[β])^2 (Sin[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) - (3840 π^5 θ (Cos[β])^2 (Sin[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) - (1536 π^6 (Sin[β])^4)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) + (768 π^5 θ (Sin[β])^4)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) + (1536 π^6 (Cos[β])^2 (Sin[β])^4)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) - (12288 π^8 (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) - (24576 π^8 (Cos[β])^2 (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) - (18432 π^7 θ (Cos[β])^2 (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) + (36864 π^8 (Cos[β])^4 (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) + (24576 π^8 (Sin[β])^4)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) + (24576 π^8 (Cos[β])^2 (Sin[β])^4)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) + (18432 π^7 θ (Cos[β])^2 (Sin[β])^4)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) - (12288 π^8 (Sin[β])^6)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) + (256 π^4 (Sin[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^3) - (256 π^3 θ (Sin[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^3) - (384 π^4 (Cos[β])^2 (Sin[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^3) + (256 π^4 (Sin[β])^4)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^3) - (3328 π^6 (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^3) - (1792 π^5 θ (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^3) + (1024 π^6 (Cos[β])^2 (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^3) - (4864 π^5 θ (Cos[β])^2 (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^3) + (3328 π^6 (Sin[β])^4)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^3) + (1792 π^5 θ (Sin[β])^4)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^3) + (2816 π^6 (Cos[β])^2 (Sin[β])^4)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^3) - (64 π^2 (Sin[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) - (12 π θ (Sin[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) + (16 π^2 (Sin[β])^4)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) - (1792 π^5 θ (Sin[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) + (1536 π^6 (Sin[β])^4)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) + (1536 π^5 θ (Sin[β])^4)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) - (1024 π^6 (Sin[β])^6)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) - (192 π^4 (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) - (832 π^3 θ (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) - (704 π^4 (Cos[β])^2 (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) + (576 π^4 (Sin[β])^4)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) + (128 π^3 θ (Sin[β])^4)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) - (4 (Sin[β])^2)/(θ (-4 π + θ) + 4 π^2 (Sin[β])^2) + (768 π^5 θ (Sin[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) - (768 π^6 (Sin[β])^4)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) - (512 π^5 θ (Sin[β])^4)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) + (256 π^6 (Sin[β])^6)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) - (384 π^4 (Sin[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) - (576 π^3 θ (Sin[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) + (704 π^4 (Sin[β])^4)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) + (96 π^3 θ (Sin[β])^4)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) - (144 π^2 (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) - (52 π θ (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) + (28 π^2 (Sin[β])^4)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) + (64 π^4 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3) + (4 π θ^3 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3) + (θ^4 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3) - (64 π^4 (Sin[β])^2 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3) + (4 π^2 θ^2 (Sin[β])^2 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3) - (16 π^2 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2) - (4 π θ (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2) - (3 (Tan[β])^2)/(8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) - (4 π θ (Tan[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) + (8 π^2 (Sin[β])^2 (Tan[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) - (256 π^5 θ (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) + (512 π^6 (Sin[β])^2 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) + (256 π^5 θ (Sin[β])^2 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) - (512 π^6 (Sin[β])^4 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) - (64 π^3 θ (Tan[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) + (64 π^3 θ (Sin[β])^2 (Tan[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) - (2 (Tan[β])^2)/(θ (-4 π + θ) + 4 π^2 (Sin[β])^2) + (256 π^5 θ (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) - (256 π^6 (Sin[β])^2 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) - (256 π^5 θ (Sin[β])^2 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) + (256 π^6 (Sin[β])^4 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) - (128 π^4 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) - (64 π^3 θ (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) + (192 π^4 (Sin[β])^2 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) + (32 π^3 θ (Sin[β])^2 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) - (20 π θ (Tan[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) + (12 π^2 (Sin[β])^2 (Tan[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)))), {θ, 0, 2 π}, {β, 0, π / 2}, {γ, 0, 8 π}]\n```\n\nContourPlot3D draws the surface inside the box θ from 0 to 2π, β from 0 to π/2, γ from 0 to 8π on which the expression stays constant at one of the contour levels Mathematica picks by default. The stored points are (x, y, z) = (θ, β, γ) on that surface. The stored surface has 41,037 points joined into 43,220 faces, with 2,761 mesh lines drawn over it. It reaches from θ = 3.793 to 6.283, β = 1.122 × 10^−7 to 0.01841, γ = 1.795 × 10^���6 to 25.13. The surface falls into 3 main separate pieces; each main piece is open, with its edge in 1 loop.\n\n```wolfram\nContourPlot3D[(Sqrt[θ] Sqrt[-4 π + θ] (4 π γ + γ^2 - Sqrt[γ] Sqrt[((2 π + γ))^2] Sqrt[4 π + γ] Sin[β] - 4 π^2 (Sin[β])^2 - 4 π γ (Sin[β])^2 - γ^2 (Sin[β])^2 - (π Sqrt[γ] Sqrt[((2 π + γ))^2] (Sin[β])^3)/(Sqrt[4 π + γ]) + (π Sqrt[((2 π + γ))^2] Sqrt[4 π + γ] (Sin[β])^3)/(Sqrt[γ]) + Sqrt[γ] Sqrt[((2 π + γ))^2] Sqrt[4 π + γ] (Sin[β])^3)) / (π^2 (-16 π γ - 4 γ^2 + 16 π^2 (Sin[β])^2 + 16 π γ (Sin[β])^2 + 4 γ^2 (Sin[β])^2) √((192 π^4 (Sin[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3) + (64 π^3 θ (Sin[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3) + (16 π^2 θ^2 (Sin[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3) + (4 π θ^3 (Sin[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3) + (θ^4 (Sin[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3) - (64 π^4 (Sin[β])^4)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3) + (4 π^2 θ^2 (Sin[β])^4)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3) - (112 π^2 (Sin[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2) - (20 π θ (Sin[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2) - (2 θ^2 (Sin[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2) - (7 (Sin[β])^2)/(8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) + (36864 π^8 (Cos[β])^2 (Sin[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^5) - (18432 π^7 θ (Cos[β])^2 (Sin[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^5) - (36864 π^8 (Cos[β])^2 (Sin[β])^4)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^5) + (18432 π^7 θ (Cos[β])^2 (Sin[β])^4)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^5) - (147456 π^10 (Cos[β])^2 (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^5) + (147456 π^10 (Cos[β])^4 (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^5) + (294912 π^10 (Cos[β])^2 (Sin[β])^4)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^5) - (147456 π^10 (Cos[β])^4 (Sin[β])^4)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^5) - (147456 π^10 (Cos[β])^2 (Sin[β])^6)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^5) + (1536 π^6 (Sin[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) - (768 π^5 θ (Sin[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) + (6144 π^6 (Cos[β])^2 (Sin[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) - (3840 π^5 θ (Cos[β])^2 (Sin[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) - (1536 π^6 (Sin[β])^4)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) + (768 π^5 θ (Sin[β])^4)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) + (1536 π^6 (Cos[β])^2 (Sin[β])^4)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) - (12288 π^8 (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) - (24576 π^8 (Cos[β])^2 (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) - (18432 π^7 θ (Cos[β])^2 (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) + (36864 π^8 (Cos[β])^4 (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) + (24576 π^8 (Sin[β])^4)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) + (24576 π^8 (Cos[β])^2 (Sin[β])^4)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) + (18432 π^7 θ (Cos[β])^2 (Sin[β])^4)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) - (12288 π^8 (Sin[β])^6)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^4) + (256 π^4 (Sin[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^3) - (256 π^3 θ (Sin[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^3) - (384 π^4 (Cos[β])^2 (Sin[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^3) + (256 π^4 (Sin[β])^4)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^3) - (3328 π^6 (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^3) - (1792 π^5 θ (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^3) + (1024 π^6 (Cos[β])^2 (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^3) - (4864 π^5 θ (Cos[β])^2 (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^3) + (3328 π^6 (Sin[β])^4)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^3) + (1792 π^5 θ (Sin[β])^4)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^3) + (2816 π^6 (Cos[β])^2 (Sin[β])^4)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^3) - (64 π^2 (Sin[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) - (12 π θ (Sin[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) + (16 π^2 (Sin[β])^4)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) - (1792 π^5 θ (Sin[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) + (1536 π^6 (Sin[β])^4)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) + (1536 π^5 θ (Sin[β])^4)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) - (1024 π^6 (Sin[β])^6)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) - (192 π^4 (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) - (832 π^3 θ (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) - (704 π^4 (Cos[β])^2 (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) + (576 π^4 (Sin[β])^4)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) + (128 π^3 θ (Sin[β])^4)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) - (4 (Sin[β])^2)/(θ (-4 π + θ) + 4 π^2 (Sin[β])^2) + (768 π^5 θ (Sin[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) - (768 π^6 (Sin[β])^4)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) - (512 π^5 θ (Sin[β])^4)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) + (256 π^6 (Sin[β])^6)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) - (384 π^4 (Sin[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) - (576 π^3 θ (Sin[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) + (704 π^4 (Sin[β])^4)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) + (96 π^3 θ (Sin[β])^4)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) - (144 π^2 (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) - (52 π θ (Sin[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) + (28 π^2 (Sin[β])^4)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) + (64 π^4 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3) + (4 π θ^3 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3) + (θ^4 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3) - (64 π^4 (Sin[β])^2 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3) + (4 π^2 θ^2 (Sin[β])^2 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3) - (16 π^2 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2) - (4 π θ (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2) - (3 (Tan[β])^2)/(8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) - (4 π θ (Tan[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) + (8 π^2 (Sin[β])^2 (Tan[β])^2)/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) - (256 π^5 θ (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) + (512 π^6 (Sin[β])^2 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) + (256 π^5 θ (Sin[β])^2 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) - (512 π^6 (Sin[β])^4 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) - (64 π^3 θ (Tan[β])^2)/((8 π^2 - ��^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) + (64 π^3 θ (Sin[β])^2 (Tan[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^2) - (2 (Tan[β])^2)/(θ (-4 π + θ) + 4 π^2 (Sin[β])^2) + (256 π^5 θ (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) - (256 π^6 (Sin[β])^2 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) - (256 π^5 θ (Sin[β])^2 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) + (256 π^6 (Sin[β])^4 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^3 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) - (128 π^4 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) - (64 π^3 θ (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) + (192 π^4 (Sin[β])^2 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) + (32 π^3 θ (Sin[β])^2 (Tan[β])^2)/(((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2))^2 (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) - (20 π θ (Tan[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)) + (12 π^2 (Sin[β])^2 (Tan[β])^2)/((8 π^2 - θ^2 + 8 π^2 Sqrt[(Cos[β])^2] - 4 π^2 (Sin[β])^2) (θ (-4 π + θ) + 4 π^2 (Sin[β])^2)))), {θ, 0, 2 π}, {β, 0, π / 2}, {γ, 0, 8 π}]\n```\n\nContourPlot3D draws the surface inside the box θ from 0 to 2π, β from 0 to π/2, γ from 0 to 8π on which the expression stays constant at one of the contour levels Mathematica picks by default. The stored points are (x, y, z) = (θ, β, γ) on that surface. The stored surface has 41,037 points joined into 43,220 faces, with 2,761 mesh lines drawn over it. It reaches from θ = 3.793 to 6.283, β = 1.122 × 10^−7 to 0.01841, γ = 1.795 × 10^−6 to 25.13. The surface falls into 3 main separate pieces; each main piece is open, with its edge in 1 loop."}
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{"file_name": "sor-plot-0032.png", "book": "The Sphere of Realization", "vision_examples": 2, "questions": "This is from The Sphere of Realization (chapter \"8. Liberation: Liberating Constraints on the Difference Equation and the Resulting Equalities with V-Curvature (Components for Worm Holes) and Methods for Solving Non-Elementary Integrals\"). Give the command that produced it.\n\nThis is from The Sphere of Realization (chapter \"8. Liberation: Liberating Constraints on the Difference Equation and the Resulting Equalities with V-Curvature (Components for Worm Holes) and Methods for Solving Non-Elementary Integrals\"). Give the command that produced it.", "answers": "```wolfram\nManipulate[ContourPlot3D[-1/4 z^2 θ^2 (δ Cos[β] Log[α + δ Cos[β]] + α (Log[δ] (-1 + Log[α + δ Cos[β]] - Log[1 + (δ Cos[β])/α]) + Log[1 + (δ Cos[β])/α]) - α PolyLog[2, -(δ Cos[β])/α]), {δ, 0, 2 π} , {α, 0, 2 π}, {θ, 0, 2 π}], {z, 0, 50000}, {β, 0, π / 2}]\n```\n\n```wolfram\nManipulate[ContourPlot3D[-1/4 z^2 θ^2 (δ Cos[β] Log[α + δ Cos[β]] + α (Log[δ] (-1 + Log[α + δ Cos[β]] - Log[1 + (δ Cos[β])/α]) + Log[1 + (δ Cos[β])/α]) - α PolyLog[2, -(δ Cos[β])/α]), {δ, 0, 2 π} , {α, 0, 2 π}, {θ, 0, 2 π}], {z, 0, 50000}, {β, 0, π / 2}]\n```"}
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{"file_name": "sor-plot-0033.png", "book": "The Sphere of Realization", "vision_examples": 2, "questions": "This is from The Sphere of Realization (chapter \"8. Liberation: Liberating Constraints on the Difference Equation and the Resulting Equalities with V-Curvature (Components for Worm Holes) and Methods for Solving Non-Elementary Integrals\"). Give the command that produced it.\n\nThis is from The Sphere of Realization (chapter \"8. Liberation: Liberating Constraints on the Difference Equation and the Resulting Equalities with V-Curvature (Components for Worm Holes) and Methods for Solving Non-Elementary Integrals\"). Give the command that produced it.", "answers": "```wolfram\nManipulate[ContourPlot3D[-1/4 z^2 θ^2 (δ Sqrt[1 - ((r α + r δ - z θ) (-r α + r δ + z θ))/(r^2 δ^2)] Log[α + δ Sqrt[1 - ((r α + r δ - z θ) (-r α + r δ + z θ))/(r^2 δ^2)]] + α (Log[δ] (-1 + Log[α + δ Sqrt[1 - ((r α + r δ - z θ) (-r α + r δ + z θ))/(r^2 δ^2)]] - Log[1 + (δ Sqrt[1 - ((r α + r δ - z θ) (-r α + r δ + z θ))/(r^2 δ^2)])/α]) + Log[1 + (δ Sqrt[1 - ((r α + r δ - z θ) (-r α + r δ + z θ))/(r^2 δ^2)])/α]) - α PolyLog[2, -(δ Sqrt[1 - ((r α + r δ - z θ) (-r α + r δ + z θ))/(r^2 δ^2)])/α]), {δ, 0, 2 π} , {α, 0, 2 π}, {θ, 0, 2 π}], {r, 0, 5}, {z, 0, 5}]\n```\n\n```wolfram\nManipulate[ContourPlot3D[-1/4 z^2 θ^2 (δ Sqrt[1 - ((r α + r δ - z θ) (-r α + r δ + z θ))/(r^2 δ^2)] Log[α + δ Sqrt[1 - ((r α + r δ - z θ) (-r α + r δ + z θ))/(r^2 δ^2)]] + α (Log[δ] (-1 + Log[α + δ Sqrt[1 - ((r α + r δ - z θ) (-r α + r δ + z θ))/(r^2 δ^2)]] - Log[1 + (δ Sqrt[1 - ((r α + r δ - z θ) (-r α + r δ + z θ))/(r^2 δ^2)])/α]) + Log[1 + (δ Sqrt[1 - ((r α + r δ - z θ) (-r α + r δ + z θ))/(r^2 δ^2)])/α]) - α PolyLog[2, -(δ Sqrt[1 - ((r α + r δ - z θ) (-r α + r δ + z θ))/(r^2 δ^2)])/α]), {δ, 0, 2 π} , {α, 0, 2 π}, {θ, 0, 2 π}], {r, 0, 5}, {z, 0, 5}]\n```"}
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{"file_name": "sor-plot-0034.png", "book": "The Sphere of Realization", "vision_examples": 2, "questions": "This is from The Sphere of Realization (chapter \"8. Liberation: Liberating Constraints on the Difference Equation and the Resulting Equalities with V-Curvature (Components for Worm Holes) and Methods for Solving Non-Elementary Integrals\"). Give the command that produced it.\n\nThis is from The Sphere of Realization (chapter \"8. Liberation: Liberating Constraints on the Difference Equation and the Resulting Equalities with V-Curvature (Components for Worm Holes) and Methods for Solving Non-Elementary Integrals\"). Give the command that produced it.", "answers": "```wolfram\nManipulate[ContourPlot3D[0.25` s^2 √(1.` α^2 (Log[δ])^2 - 2.` α δ Cos[β] Log[δ] Log[α + δ Cos[β]] - 2.` α^2 (Log[δ])^2 Log[α + δ Cos[β]] + 1.` δ^2 (Cos[β])^2 (Log[α + δ Cos[β]])^2 + 2.` α δ Cos[β] Log[δ] (Log[α + δ Cos[β]])^2 + 1.` α^2 (Log[δ])^2 (Log[α + δ Cos[β]])^2 - 2.` α^2 Log[δ] Log[1.` + (δ Cos[β])/α] + 2.` α^2 (Log[δ])^2 Log[1.` + (δ Cos[β])/α] + 2.` α δ Cos[β] Log[α + δ Cos[β]] Log[1.` + (δ Cos[β])/α] + 2.` α^2 Log[δ] Log[α + δ Cos[β]] Log[1.` + (δ Cos[β])/α] - 2.` α δ Cos[β] Log[δ] Log[α + δ Cos[β]] Log[1.` + (δ Cos[β])/α] - 2.` α^2 (Log[δ])^2 Log[α + δ Cos[β]] Log[1.` + (δ Cos[β])/α] + 1.` α^2 (Log[1.` + (δ Cos[β])/α])^2 - 2.` α^2 Log[δ] (Log[1.` + (δ Cos[β])/α])^2 + 1.` α^2 (Log[δ])^2 (Log[1.` + (δ Cos[β])/α])^2 + 2.` α^2 Log[δ] PolyLog[2.`, -(1.` δ Cos[β])/α] - 2.` α δ Cos[β] Log[α + δ Cos[β]] PolyLog[2.`, -(1.` δ Cos[β])/α] - 2.` α^2 Log[δ] Log[α + δ Cos[β]] PolyLog[2.`, -(1.` δ Cos[β])/α] - 2.` α^2 Log[1.` + (δ Cos[β])/α] PolyLog[2.`, -(1.` δ Cos[β])/α] + 2.` α^2 Log[δ] Log[1.` + (δ Cos[β])/α] PolyLog[2.`, -(1.` δ Cos[β])/α] + 1.` α^2 (PolyLog[2.`, -(1.` δ Cos[β])/α])^2), {α, 0, 2 π}, {δ, 0, 2 π}, {β, 0, π / 2}], {s, 1, 8}]\n```\n\n```wolfram\nManipulate[ContourPlot3D[0.25` s^2 √(1.` α^2 (Log[δ])^2 - 2.` α δ Cos[β] Log[δ] Log[α + δ Cos[β]] - 2.` α^2 (Log[δ])^2 Log[α + δ Cos[β]] + 1.` δ^2 (Cos[β])^2 (Log[α + δ Cos[β]])^2 + 2.` α δ Cos[β] Log[δ] (Log[α + δ Cos[β]])^2 + 1.` α^2 (Log[δ])^2 (Log[α + δ Cos[β]])^2 - 2.` α^2 Log[δ] Log[1.` + (δ Cos[β])/α] + 2.` α^2 (Log[δ])^2 Log[1.` + (δ Cos[β])/α] + 2.` α δ Cos[β] Log[α + δ Cos[β]] Log[1.` + (δ Cos[β])/α] + 2.` α^2 Log[δ] Log[α + δ Cos[β]] Log[1.` + (δ Cos[β])/α] - 2.` α δ Cos[β] Log[δ] Log[α + δ Cos[β]] Log[1.` + (δ Cos[β])/α] - 2.` α^2 (Log[δ])^2 Log[α + δ Cos[β]] Log[1.` + (δ Cos[β])/α] + 1.` α^2 (Log[1.` + (δ Cos[β])/α])^2 - 2.` α^2 Log[δ] (Log[1.` + (δ Cos[β])/α])^2 + 1.` α^2 (Log[δ])^2 (Log[1.` + (δ Cos[β])/α])^2 + 2.` α^2 Log[δ] PolyLog[2.`, -(1.` δ Cos[β])/α] - 2.` α δ Cos[β] Log[α + δ Cos[β]] PolyLog[2.`, -(1.` δ Cos[β])/α] - 2.` α^2 Log[δ] Log[α + δ Cos[β]] PolyLog[2.`, -(1.` δ Cos[β])/α] - 2.` α^2 Log[1.` + (δ Cos[β])/α] PolyLog[2.`, -(1.` δ Cos[β])/α] + 2.` α^2 Log[δ] Log[1.` + (δ Cos[β])/α] PolyLog[2.`, -(1.` δ Cos[β])/α] + 1.` α^2 (PolyLog[2.`, -(1.` δ Cos[β])/α])^2), {α, 0, 2 π}, {δ, 0, 2 π}, {β, 0, π / 2}], {s, 1, 8}]\n```"}
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{"file_name": "sor-plot-0035.png", "book": "The Sphere of Realization", "vision_examples": 0, "questions": "", "answers": ""}
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{"file_name": "sor-plot-0036.jpg", "book": "The Sphere of Realization", "vision_examples": 2, "questions": "Describe this plot from The Sphere of Realization (chapter \"9. Warp Variable Nature\").\n\nDescribe this plot from The Sphere of Realization (chapter \"9. Warp Variable Nature\").", "answers": "This graphic was pasted into an input cell; the plotting function that made it is not recorded with it, so its points are given as plain (x, y, z) positions. The stored surface has 14,576 points joined into 19,068 faces, with 188 mesh lines drawn over it. It reaches from x = −4.817 × 10^21 to 3.621 × 10^14, y = −5.404 × 10^14 to 4.063 × 10^7, z = −1.073 × 10^28 to 6.175 × 10^14. The box Mathematica displays is narrower (x from −2.142 × 10^4 to 2.098 × 10^4, y from −1.88 × 10^4 to 1.879 × 10^4, z from −5 × 10^4 to 4.759 × 10^4), so the parts of the surface beyond it are cut off in the picture. The surface falls into 2 main separate pieces: 1 is open, with its edge in 3 loops; 1 is open, with its edge in 1 loop. The stored points are symmetric under reflection through the origin.\n\nThis graphic was pasted into an input cell; the plotting function that made it is not recorded with it, so its points are given as plain (x, y, z) positions. The stored surface has 14,576 points joined into 19,068 faces, with 188 mesh lines drawn over it. It reaches from x = −4.817 × 10^21 to 3.621 × 10^14, y = −5.404 × 10^14 to 4.063 × 10^7, z = −1.073 × 10^28 to 6.175 × 10^14. The box Mathematica displays is narrower (x from −2.142 × 10^4 to 2.098 × 10^4, y from −1.88 × 10^4 to 1.879 × 10^4, z from −5 × 10^4 to 4.759 × 10^4), so the parts of the surface beyond it are cut off in the picture. The surface falls into 2 main separate pieces: 1 is open, with its edge in 3 loops; 1 is open, with its edge in 1 loop. The stored points are symmetric under reflection through the origin."}
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{"file_name": "sor-plot-0037.jpg", "book": "The Sphere of Realization", "vision_examples": 0, "questions": "", "answers": ""}
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{"file_name": "sor-plot-0038.jpg", "book": "The Sphere of Realization", "vision_examples": 2, "questions": "Describe this plot from The Sphere of Realization (chapter \"9. Warp Variable Nature\").\n\nDescribe this plot from The Sphere of Realization (chapter \"9. Warp Variable Nature\").", "answers": "This graphic was pasted into an input cell; the plotting function that made it is not recorded with it, so its points are given as plain (x, y, z) positions. The stored surface has 12,005 points joined into 15,942 faces, with 162 mesh lines drawn over it. It reaches from x = −4.413 × 10^19 to 9.716 × 10^13, y = −4.951 × 10^12 to 1.09 × 10^7, z = −9.832 × 10^25 to 1.067 × 10^14. The box Mathematica displays is narrower (x from −4.773 × 10^4 to 4.754 × 10^4, y from −3.912 × 10^4 to 3.833 × 10^4, z from −8.083 × 10^4 to 7.95 × 10^4), so the parts of the surface beyond it are cut off in the picture. The surface falls into 2 main separate pieces: 1 is open, with its edge in 3 loops; 1 is open, with its edge in 1 loop. The stored points are symmetric under reflection through the origin.\n\nThis graphic was pasted into an input cell; the plotting function that made it is not recorded with it, so its points are given as plain (x, y, z) positions. The stored surface has 12,005 points joined into 15,942 faces, with 162 mesh lines drawn over it. It reaches from x = −4.413 × 10^19 to 9.716 × 10^13, y = −4.951 × 10^12 to 1.09 × 10^7, z = −9.832 × 10^25 to 1.067 × 10^14. The box Mathematica displays is narrower (x from −4.773 × 10^4 to 4.754 × 10^4, y from −3.912 × 10^4 to 3.833 × 10^4, z from −8.083 × 10^4 to 7.95 × 10^4), so the parts of the surface beyond it are cut off in the picture. The surface falls into 2 main separate pieces: 1 is open, with its edge in 3 loops; 1 is open, with its edge in 1 loop. The stored points are symmetric under reflection through the origin."}
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{"file_name": "sor-plot-0039.jpg", "book": "The Sphere of Realization", "vision_examples": 2, "questions": "What does this graph from The Sphere of Realization show, according to the coordinates stored with it?\n\nWhat does this graph from The Sphere of Realization show, according to the coordinates stored with it?", "answers": "This graphic was pasted into an input cell; the plotting function that made it is not recorded with it, so its points are given as plain (x, y, z) positions. The stored surface has 13,770 points joined into 18,452 faces, with 158 mesh lines drawn over it. It reaches from x = −4.413 × 10^19 to 9.716 × 10^13, y = −4.951 × 10^12 to 1.09 × 10^7, z = −9.832 × 10^25 to 1.067 × 10^14. The box Mathematica displays is narrower (x from −4.008 × 10^4 to 3.973 × 10^4, y from −2.836 × 10^4 to 2.774 × 10^4, z from −6.84 × 10^4 to 6.687 × 10^4), so the parts of the surface beyond it are cut off in the picture. The surface falls into 2 main separate pieces: 1 is open, with its edge in 3 loops; 1 is open, with its edge in 1 loop. The stored points are symmetric under reflection through the origin.\n\nThis graphic was pasted into an input cell; the plotting function that made it is not recorded with it, so its points are given as plain (x, y, z) positions. The stored surface has 13,770 points joined into 18,452 faces, with 158 mesh lines drawn over it. It reaches from x = −4.413 × 10^19 to 9.716 × 10^13, y = −4.951 × 10^12 to 1.09 × 10^7, z = −9.832 × 10^25 to 1.067 × 10^14. The box Mathematica displays is narrower (x from −4.008 × 10^4 to 3.973 × 10^4, y from −2.836 × 10^4 to 2.774 × 10^4, z from −6.84 × 10^4 to 6.687 × 10^4), so the parts of the surface beyond it are cut off in the picture. The surface falls into 2 main separate pieces: 1 is open, with its edge in 3 loops; 1 is open, with its edge in 1 loop. The stored points are symmetric under reflection through the origin."}
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| 40 |
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{"file_name": "sor-plot-0040.jpg", "book": "The Sphere of Realization", "vision_examples": 2, "questions": "Describe this plot from The Sphere of Realization (chapter \"9. Warp Variable Nature\").\n\nDescribe this plot from The Sphere of Realization (chapter \"9. Warp Variable Nature\").", "answers": "This graphic was pasted into an input cell; the plotting function that made it is not recorded with it, so its points are given as plain (x, y, z) positions. The stored surface has 15,722 points joined into 20,411 faces, with 172 mesh lines drawn over it. It reaches from x = −5.814 × 10^21 to 4.015 × 10^14, y = −6.523 × 10^14 to 4.505 × 10^7, z = −1.295 × 10^28 to 7.004 × 10^14. The box Mathematica displays is narrower (x from −7015 to 4261, y from −7957 to 7593, z from −2.188 × 10^4 to 2.092 × 10^4), so the parts of the surface beyond it are cut off in the picture. The surface falls into 2 main separate pieces: 1 is open, with its edge in 2 loops; 1 is open, with its edge in 1 loop.\n\nThis graphic was pasted into an input cell; the plotting function that made it is not recorded with it, so its points are given as plain (x, y, z) positions. The stored surface has 15,722 points joined into 20,411 faces, with 172 mesh lines drawn over it. It reaches from x = −5.814 × 10^21 to 4.015 × 10^14, y = −6.523 × 10^14 to 4.505 × 10^7, z = −1.295 × 10^28 to 7.004 × 10^14. The box Mathematica displays is narrower (x from −7015 to 4261, y from −7957 to 7593, z from −2.188 × 10^4 to 2.092 × 10^4), so the parts of the surface beyond it are cut off in the picture. The surface falls into 2 main separate pieces: 1 is open, with its edge in 2 loops; 1 is open, with its edge in 1 loop."}
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| 41 |
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{"file_name": "sor-plot-0041.jpg", "book": "The Sphere of Realization", "vision_examples": 2, "questions": "Describe this plot from The Sphere of Realization (chapter \"9. Warp Variable Nature\").\n\nDescribe this plot from The Sphere of Realization (chapter \"9. Warp Variable Nature\").", "answers": "This graphic was pasted into an input cell; the plotting function that made it is not recorded with it, so its points are given as plain (x, y, z) positions. The stored surface has 16,333 points joined into 22,229 faces, with 156 mesh lines drawn over it. It reaches from x = −1.77 × 10^16 to 1.539 × 10^14, y = −1.986 × 10^9 to 1.727 × 10^7, z = −3.944 × 10^22 to 1.933 × 10^14. The box Mathematica displays is narrower (x from −1.675 × 10^4 to 1.665 × 10^4, y from −1.056 × 10^4 to 1.065 × 10^4, z from −3.023 × 10^4 to 2.917 × 10^4), so the parts of the surface beyond it are cut off in the picture. The surface falls into 2 main separate pieces: 1 is open, with its edge in 3 loops; 1 is open, with its edge in 1 loop.\n\nThis graphic was pasted into an input cell; the plotting function that made it is not recorded with it, so its points are given as plain (x, y, z) positions. The stored surface has 16,333 points joined into 22,229 faces, with 156 mesh lines drawn over it. It reaches from x = −1.77 × 10^16 to 1.539 × 10^14, y = −1.986 × 10^9 to 1.727 × 10^7, z = −3.944 × 10^22 to 1.933 × 10^14. The box Mathematica displays is narrower (x from −1.675 × 10^4 to 1.665 × 10^4, y from −1.056 × 10^4 to 1.065 × 10^4, z from −3.023 × 10^4 to 2.917 × 10^4), so the parts of the surface beyond it are cut off in the picture. The surface falls into 2 main separate pieces: 1 is open, with its edge in 3 loops; 1 is open, with its edge in 1 loop."}
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{"messages": [{"role": "system", "content": "You are an assistant trained on The Sphere of Realization: The Mathematical Path of Harmonious Being (2nd Edition) by Parker Emmerson (Yaohushuason). Answer with the book's own text, Wolfram Language computations and results, in the book's notation."}, {"role": "user", "content": [{"type": "text", "text": "What does this graph from The Sphere of Realization show, according to the coordinates stored with it?"}, {"type": "image_url", "image_url": {"url": 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"detail": "high"}}]}, {"role": "assistant", "content": "ContourPlot3D draws the surface inside the box x from −1 to 1, a from −1 to 1, θ from −2π to 2π on which the expression stays constant at one of the contour levels Mathematica picks by default. The stored points are (x, y, z) = (x, a, θ) on that surface. The stored surface has 152,670 points joined into 172,563 faces, with 325 mesh lines drawn over it. It reaches from x = −1 to 1, a = −1 to 1, θ = −6.283 to 6.283. The surface falls into 5 main separate pieces plus 2 small fragments: 1 is closed, with sheets meeting along shared edges; 4 are open, each with its edge in 1 loop. The stored points are symmetric under x → −x, and under y → −y, and under (x, y) → (−x, −y)."}]}
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{"messages": [{"role": "system", "content": "You are an assistant trained on The Sphere of Realization: The Mathematical Path of Harmonious Being (2nd Edition) by Parker Emmerson (Yaohushuason). Answer with the book's own text, Wolfram Language computations and results, in the book's notation."}, {"role": "user", "content": [{"type": "text", "text": "Which Wolfram Language command in The Sphere of Realization by Parker Emmerson drew this?"}, {"type": "image_url", "image_url": {"url": 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"detail": "high"}}]}, {"role": "assistant", "content": "```wolfram\nSphericalPlot3D[(6 π^2 (-4 π + 2 θ) Cos[β] Sin[β] Sqrt[-c^2 (-θ^2 + 8 π^2 (1 + Sqrt[(Cos[β])^2]) - 4 π^2 (Sin[β])^2)])/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^(5 / 2)) - (4 c^2 π^2 θ Cos[β] Sin[β])/(Sqrt[-c^2 (-θ^2 + 8 π^2 (1 + Sqrt[(Cos[β])^2]) - 4 π^2 (Sin[β])^2)] ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^(3 / 2)) + (c^2 (-4 π + 2 θ) (-8 π^2 Cos[β] Sin[β] - (8 π^2 Cos[β] Sin[β])/(Sqrt[(Cos[β])^2])))/(4 Sqrt[-c^2 (-θ^2 + 8 π^2 (1 + Sqrt[(Cos[β])^2]) - 4 π^2 (Sin[β])^2)] ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^(3 / 2)) + (c^4 (θ) (-8 π^2 Cos[β] Sin[β] - (8 π^2 Cos[β] Sin[β])/(Sqrt[(Cos[β])^2])))/(2 ((-c^2 (-θ^2 + 8 π^2 (1 + Sqrt[(Cos[β])^2]) - 4 π^2 (Sin[β])^2)))^(3 / 2) Sqrt[θ (-4 π + θ) + 4 π^2 (Sin[β])^2]), {β, 0, π / 2}, {θ, 0, 2 π}, PlotTheme → \"Orange\"]\n```"}]}
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{"messages": [{"role": "system", "content": [{"type": "text", "text": "You are an assistant trained on The Sphere of Realization: The Mathematical Path of Harmonious Being (2nd Edition) by Parker Emmerson (Yaohushuason). Answer with the book's own text, Wolfram Language computations and results, in the book's notation."}]}, {"role": "user", "content": [{"type": "text", "text": "What does this graph from The Sphere of Realization show, according to the coordinates stored with it?", "image_url": null}, {"type": "image_url", "image_url": 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nFcjml0PUg5dzkYJrpbaZIBCXF1KFVbhSf8AWHPI5Na8sd3HaYItWZovmAncbCP+A/yq5a29uXvEaEiWO9nydgK48xuD71DdWNs0cqRWluAqHc/lKQOOg45/pVc6bEm0tzM0s376faCP7OxMEZGI2I+6OCQR2qHWra8FqjSXUTkXEICCLofMX1bJq3p1rZR6LayJFGkaQRmSRmIUfKOvNV9ZsrOXSY5IJmiT7TFmZ52Xd+8UYVScjjnP+Q1LUhz9wW4OpLcLbeZGXZdxRYQeOxyTgciqU2nXMmrWgv7t1ZopSyW67Ao+Xuck/pWmF0iziCJdyzHJwsFw7PnH3m5wB2zmqV1KZdXsIhNeIPImQiDe5OdufmYH/wAdxVKQSmupoy2ENixtpbqWPO10jVcEjsfkGc1nRfaLnVL8CeaNCkQBklYsy/NgYz+PJ/CrEYeH91bi+iO7BjECZHHGThaigi1T+2r8wwHCJDuaZUB6N6N3+tCe43PYkGnh4pBLNczAnndcOox/uhv071Q0KwtTp6F43lxJKduWwAHYZx3+lWrqfU0kWSBbC8k3neiuxKkdN3b6Y9Kr6JHrk2khRJa+W0koESBhyZG5LZ7U76bidRcy0JNVtrKKzuN1vBEZIiY18sFj29Pf9a+3PBI/4ozQ/wDsHW//AKLWviW60rV7fSbp1SyKKgMpjLb2BPUkjJr7b8Ef8iZofGP+Jdb/APota68K9zys0d+XSxsUUUV1nkhRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABXMfFbn4da77WbGunrmfip/wAk51//AK8ZD+lAHzL8Tbi7j+JniCPT4obovcuZIt4V8iRhgE9+f/1VzC6pFJcrBqFleWAZSA+1XB9vlPFa3xlu47T4j6rJdQIqSX11HG87KEc+YDgEkYI57g9awLfXLiZd+CJDkL5z+fHJxg4Ma549z+deRUjeTdj36VW0Ermh4fv2i0hfJ3q8ZP34mG4Z5xxycVQ1T+yr3StTeUxJciLdE0cgBY+p9T0qPwxqUcUbeTqFjYKcb0dSUY7Fzj5wQfwHetTWEvG0i4+1vGsTgbDHa5WQFhyDz09+frWbTUi/aXgPlu9S+x28zRw3MIU/vIlywOO6k5/LNMunE76PKZA0v2shmRt38JAyO3UVY0nTra8sLNYrlWYxqfMgijXZx0OFznPvVTXdDjt7mzIl1C2mSUDzUnbcBkcgD5eCfTvUq17FOUrXN6S2MoZZogS454yG9j/hWBcaekWvtF9n8yI2YY7OSjeZ1UHr7j8quro0lsIHup5V8x8CUTBoyfQ7hwT2ycds1WTSwniMxqLVwbEErNbg8eZ7EfWlFDnO/QIdP01lM8klksyEiMT7Qc91cHkKfzHUVDoKaQ0t+hMUUiXZzGHLBRtXuvVeuCK1JdPvIiJ7G1sI5Bx8hKAj/aXB6diTkH2qnpkzSte3Hky21xDcyfcZXZTsThxkZTj+oqtWiOazRaaOx3SmK4vwqKciPzjz6+lZ2j2lvNpVjJ/xNi7W0edqALwo5y/H5Vs2er20tobjyLmJsH7QpiZwHHBO4ZH0qHwve2KaFYwyy7VMEZ/egqGbAJGSMYFTaViuZNoo6uupLp9zCPNCEKU3+XuADKOce9Wbo601nJ5KWYiV9uZSd/BHXbx+VaXihrdtPfyZrWeX5FYxSAgLvHXHT61LJG+bi2YOqKBgDkjJ9e/1oSdtgvrudR4H/tD/AIV38RBcC18z7BBsEbsUz+865H0rlbhfELWluoTSioZgAHkOOR9K9U+GGoaPafC/xRb3tsLv7OjSXeDg3Mbx7UGe2NpX2xnvXDw25ljtrGYss0cADuB1OwHd71tXjaELHPh5XqVDkYodat9fuUkiG63to9wt1T7u+TkAnrTLrT/OaSeY6k7ltzHESnJ/i2oCfrgVrq7Qa9f3LXkcKi0hWXfhuC0vr9KtXl7pnmRr/aVg+E+SVORuIzg45GcDisHe51Jq25y+kNaCKWO4n1SVzcTAsvmIVPmNhjgDP0pb3+xxZzjzhHGqEtI7vk8e/X8avadqscVtcSxRyowu55GMVu+2P962DyMfhRHP/bFpd3slpdSWiQNLDtdIxHgcu/zc9+McD3p2d7k8ytYwdOtNLvtLtZJDZ20cUUTKGdS2QoyzDqfUDt356S3Ecg014YYilpcXFvvkMe1v9av3M888c9Px6XLBLm+0vTfM02MW620ey3mmBDsFHzsApwMdB7ZPpUevaa3kxlo9OWQ3MO4rB9z94uORgH8vWq1uTzaGnb2cVvG0aRC3hPJUffceh55qhfK0euafgJCot5ye56pnI7H61LLpEUmBDfs90zgCKH5eOcsOflA9T69zxVKbwxI/iaztLy/uZDNBLgC5YhMbAFJAHf8ADpUrzZbm+i7E0VzMLlorWzjmjfCiTovXA+bPX6ZrPSOKTXtTfUprdiqR4US4VeDwOefqa3X8NrHCZC1wEGAxkZHY+/K/hWLpsLQ61qP2F0ZyIwnkWu8tlemVPXt165qk072JlKWly3pd9Zo5CpGGUYXywWGPQYFZenaoken4MdyXE0pjjSE5bMjfl17mtTUZtQs1kt7i9s7Wdj+8to4/3irjONxfA4xz78VgaJftbaGhiWFI97Hbb5MjMT/E7IVwB71UVcUqrTSNHUbm6utOeZrFraCPG95pFBYkgYUKeTX2r4Kx/wAIZohHT+z7f/0WtfDM+qxzNDD9njDzTpGCZ0aZ/mBP8YI6HACgYr7l8EHd4L0M+unW/wD6LWu3CK1zzcfPm5dTZooortPPCiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACua+Kf8AyTjxD7afKf8Ax010tc18U/8Akm/iP/sGzn/xw0AfPXj+106/8Yaut/ZR3aw6lcMqyLuAJYg4B9jXJ3ngzTIbZ7nTme3jm5CJJIvksBnI2sAD9Qa6fxnqLw+MfEKGwuJ1XVJgslvEzFfnPDcAfkTWb/aS3Ec32a3uDLFBuZGKx9O7KTux+HevHmpKbse5TcXBX7HDJ4a8QQ/vG1G3u0niidI57dI2b5ByHAOCMAe+OcVSuvt+mWRS6j1TSmLrsZQGhY7hnJjPH/fJPPtXpOnvfT6bYwtZJ8kH/LW4I/hBB4UnFV9US/NqhufscTS3EWGw0hILjvkY+mKtTd9SXBW0PP8Aw/4psJbC3kuru9tLyLKRTxKr9MgAg/1rZv8AxFqz29k1zF9o8mbzPN8rbK4yD0DYI/3R+FdVeaFFf2klrdTQG2OVGy2QFMHO4EA4zzyOfeuX1z4dRTW0Vvp+o6ramZXeES3OY+MEnGOM47c01yN6kv2kVobGnX95JO7f2ZPPbtGAypLGwUkdNrMCDx0OKx9RuLe08RwAibT0NruS3uYyASJMkIwJAzye4FVl0LW47lxcC7v5xHmJ1uZBLKP90HDAfUD1xms6+kbTPEtnDrFtNbIsXzrJGoZRhuQcFGyT68cimqa6CdV21PQINe0a4xJFqNrukYjymb5yR1DL/e/Ss+wMLz6nc2kU73KXbBcQMQysi8MTgYP5jqKqwW2mX0WI7Iah5qhY7i3kiMnsWCsd35e2KzvDsFp/wkWoWctgqSbl2CG4MUe4g/xA5UnHQ9xikqSSZTqu6OgSQJaNqFjp12u1D50cm0LJt4YYySSOxFTeFzcvoGluttDtNsvlvLOTk7eSFUdAOxNZmpaPJYB722tr6AF/mjS6DAZ9dxznv15A5rG8FwanFpMKixjuIZm/dzTyFfLJJGAQxwDjrjqafs9Be0akdLr9rc3OiTGWOxRHAYyfZySCXAxkscVYk0qBAm64aOPcys8UYQnA6554rF8Ti5Xw5fF9JuoR5RlV4LzciAEDoSP5Vp3M9lFGUuNN1+KNED581iBnjOd5GKOR2DnVz0T4c6FdJ8LvHLQrdTxzWUcasZmZnZFd2CnPGFdemOa5668NW9xC14l9evlYniaWXzBggevOMH1qL4ffEi/8NeFPEOnWWq232aG7ijh+1nc1t5yEvKpydwBUnacjLDp0OdoWoNNotta2/wDbF+gDqkkRckhT8oOHAHA9B9K0q03yxsZUJ+9K42HTGt/Ek6g2tpcxWUXlvJbgs7B5PukNWnLHrVtLIs4sjHIAdw3AMTyp+6cfmKz7qS4uNfmf/hHNXkJsIJGSTUAjYDygkZkJwfTtg1R1IarbypbHSJIpHQgrNetNmM9H6j7u4YA6nFZezbN+dJCaZNf3Pn2M9jM1kLidZj5g2sWlf5QSR8oPJwe/Puwuslncu1jdpbeW3kwJEWU7QfnYrnODyAeAKoadp4d5bLzNSl0+K4nR4oZVTzCZW+TJYjoNxA55qbW7XTdP0u9i/seeZ3tpPKd71nZSqkgjB24HfpQ4K4lN2Lmk6tp8GjWQkvFgcwRgyTHGcKPuE4/LpWTq+uWtzbeTbXkSxm5gRfLjaR3YSglkxxkfjmn6NbaRLo8UzeHxcOLVU/dShYozgHe5PDN6+nT3rD8X32iRtZWunzJNdG7haUBI3RW54XaHYjocZ+tNUlcTqvlOuOoSQ2O3T9LvFAO95XWOHzOeuWYM2fesK/1vVZPEVl9ntzG6RSsCX3DaSmeQcHkY4J5qrLpd/dxLKbW6EsrYh81mhR37BVX5QOMnIBHrTYfBl7da5b3Ota3crmF0jihkbMQHP3ucj6UKnFbidSb2Lt54lidGj1e5vp51/wBVbRQJHExPPLknHHr9cVz+n67JqWq6iyXWoHMoSK2tQzEkdyAQuP09q7Ow8F2+nnz7VryQqCQ00KzYPfb0x7nrU1pa3dvqdz9mFlBG0cbMxhZN/L8nGefwpLlWyG1NtXOWTR9evI5ZY7ZLIKhUPepHJLk99qjGSe5ORx1q9pfh64g0/wD4mGqTXQiZ4VjUNCkfJBIVGHP1rpZJdRLGFbSBsqpXypsY5znkCqtpqcttDcC5tZUHnysX81CN2enLY96TcmtClGKepNa6D4Wgsrd7bSIHuYPm8+VMuzYIyTnnrxX1t4H48FaF/wBg63/9FLXyEdZxbRNb2txmQ4Mklu0cZBHclc/pX154HJbwXoZOOdOt+n/XJa6MJFq9zlx0k1GxtUUUV2nnhRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFc38R9Uv9I8MT3umTrFeL/qg1uZRI20kJx0yQOT9ByRQB0lcv8AFmQxfDHxPIEaQrpVwdqjk/u2rD8W+KvE1hHqUukWUt6hs7eS28vT5H+zvumE28A5c/u1AAxgsM8c10HxJbzPhn4jfBXOj3LYIwR+5Y0AeGa/AJdb1iRp40EuoXDAtg8+YcfQYzWTr1jbtpkjXQJHkMFkHUYU9COau6nPclrxpbEz273M0gkgk3Mv7xuq4z+RNVNW1PTbjQZIxdR/u7d2KMwV1BU8YJzwexFebKN5NnsRklBLyK6QX1hZ20bRxXaNChjLyFJcADgnBB+vHoabfzWCJb2nmp+8uYTslOHjPmDIKnn1rTGoWZiggeeKd4UACQHfnOMFduc49KpazMLuzVG067lIuIoA8sIVT+8Xg7yCPypKI3LTQ0vsy2upCFQRAeSQeORVXUIj/bttGJFSMebEjAdPlHHSqv2PUyrHS/KgWRQvkXU7SKdv90L0H49O1U7ma/Gr20eoyLYSCZjlYMoflAyGJPt1xVKAnUNM2izRsl4geNpPL2hchQO4Pb61ky28lvqMqzxzTwCCPcckTQje+3A/jGPocepreuNNWa2nkN1qEueGVpioJBHOEwP0qrFpGnPreoB7OKQJbwlSwLc5fnPPtTUSW7mDq+g+ENRk+1rOlvcK/mfaLKYq23+9hOM+tcnH4fvV1jVbyGW81FHmA86eM7mAAwXDfeXOR6857cepvp5ikgu7JlgnlAV0I/dSeu5ff1xx156VW8OyxzPeWcqm1vY7mVwp4PJ5weh6dRnirV0RJJs4GHU722tTDN4egnmHyb5GlEi8/wC6cnHIIYHHuKy/CJjj8P21tqU+63ugWiZJSNnJbYC5wCDkgjI/GvZLq0W9tpllhRnQFZE6q/Gfz6cmuTg8I2P9j2U+hpa2kphVZIpU3pI2Acrn+POTuzzyDxVohpmDrSTp4Z1WzuZtYaeOBjJGs6CIoW+ThWwwIx075rWOj2uyOM6Tf+fMd2Xud2E6jGZDisLxHp6/2TPFPobafqFowBigndUl3Hhv7oDdhu4PritW5uYrdmFte6zLfMSskMvztE3dfkfOfrng5p8pPNqSafJGLXWfImigjGsWbbLpMyxjbKNydc/Tnkj6VuWmm2Nxo/n/ANm6v51tKzvLwu9Gc8j51I49AMVykcBuLbWLqd7m0t476AGGcIJXCiYKys6gr3BYjv7V01nfancadb6gJdStGZCjLlkjkQEgEPlUBx2AIq5xukTCdmyjd6daWuu3UsUespbLZwTQA32GKl5QUG5znPYdeOnNZNy72EURPmT3lyj+SLqQFok7qxV9wxkEYGWP6RO2l3XiZ7vfdLawW0LI17du0oYtJyirncByB8uOR682NL0KbUHZ7rTxbmV8yoqjdICchSeq9Mk9SeOlRylc19jE0rUJ7WGW3t9Ihvo4550WWWWbIbeTtT5WIbnJ6HHU1VubPV9Ut7iG3splmlhcsYlZlC4PXkkDsBxjOT3rvdHtINLgktrW28m4+0TI785UeY3AHUc+nWrF9cwQr9nVpDGx+YhCXkIGeByeOelKw+mpx2jeH9MlsbBtT1K5usxIscFxKYljG0ZwvGB1wc1tx3EEOmS2OhwWUNsJoo0YYKBvMHzLgcn1OR+NS2ET3WmRyXayeSIESKEyEEDaPmYg9/7vTnnnoms2Vp9mtY1srZAskRHlxBCMuO4walq5SdloXobOC1y7xxSlf3fnFixYk5Ldf/rUwHfrFs0jQnPmICcY+6MYpFso4Vt3je5jwW27ZSQDnjhsiqM5uLfV7aKK+M9wJX/cmAMxBXvtx+uBUchfOdBaz38iSRyfMq52bT17duvQVVsdbS2vb4zxQw7reMsC2OjPyAazTHrTBHnFvHCjMWSBts2DwM7sgfh+dJbpFb6hIp0mYMsEaOyqJCpLP82cksaXIV7RmnBqYltJprWyhaNnzFcXQKsvTPygE7c+4z7VBoNtpq2dzc3CJd3punAbfsTk5IXOAoOcdO1SjUbN0a3abyNx2lJlCnI/iO4DrVezeze1vFV4PLNy7IS2BjCk85x3qeQrn1Nqx0u3ngQSziI7Nyosgyrc8e4r6V8E4Hg3RADnGn2//ota+UZJob3TljtLS5cpHvEjMETuBjIy34Cvqb4cnd4B0Anr/Z0A/wDHBXRh42ucmLldI36KKK6jiCiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACuf+JQz8OvEv/YIu/8A0S1dBWB8SP8AknniT/sE3X/olqAPARp0cm5vtNyFuZpN6i5ZACXbkBcD+pqpr2i6Y+hXvm6bEs0UTnzfL3MTjhtx5zx+FXdN0uC4sontzc2c7lpxJHLtUlmJzjofXpzUerLqdrpt2ytb3TS20isjJsIwCMqRkHg9P1rj5dT07+6KNPmsyBZXbeU0eEWctKobsM5ypPtxVbUrm5tGthe2L248+ENMG8xMCRTgkDIPvitQ6hFDst7yzvLJk6AIzKPptBBBpmqXNsUtp7WWOeOWeDa8I6gOD1PHb0zT5ROSJbGWC6TzINlxEdw2K24xHv8An1xVa+hUXdisiKI2WVQScMEK/wAQqzPpa3lwDFYyRvJH5izqRvjYDhc47+/FZ2oxX9lfWPm3aPEtvIXiupEjIPy5CsAATjse/emog5libS5LEG5sJTCibWCjHlOehyDyMj0PvzVOC8MXiC9jurdleZYogQxZG+/0bAwcdiB0qxeajpt1ELaSz1NXbAimlVYosr6OWwevI71LaxXl/qt/hdNNrLDCHw5uckbiOmBuHr2xVqJLkW3QmO1EEMxAkwzMMKnBxnHPPrWLA9sI7iPUvJWZbyY4t2JmjbccOMZJ46HH88Vq3OgRxtzqN1CX+9GzfunHYeUP8c8VU8OaglnC9nPZxWrtPOQ0bBIpCJDkDv6HB/pT5SXLUii1CS3jMj6fcSo/ym8IEUT56Mwf5lJ4ycEZ71Pp9pdvoll9uuoLaJrePaETLHCj+Ju/0X863o45XiczxQCzkiIG3Hzf4jFZmn6e0GkWbW8aG1e2i8y2uXKpu2jayEZKHPoMH2qlElyOe8RaBZ3OmyzR3N1e/ZmCtbTSttkBYEr9e/Ixn0qTT9NgSWTU9HVNKt93l3NtFlGLDgHlSFYdDxz7VuXl3HJot5buvk3aMgaCXAkzuXoByw9+RWjDbTJK+oWhWaO4OZovOCNvAChkY8cgAYP5+uiiRc8s1yC0kS4OpahdCa4vGDRzW6+cQnmRp5TAcNnO44POD2rat9OEmhacl6upadFdQ7biW3u1UTYAyojAGxTnuc4PSjWNT02G4ljuYlFxNdzuyTMytIWeQIGVgSAvI4z7ZzUOmWGrapplvZSKVcwqjyMNryJjGUU8nI7nngnHSqcdCE9Ro0/TrrxSVs7a1ubeOKJIpo4hbtIx3gHKEDCnuc569Oulepr9i7+a8gdTsLTRBmBwMDeuORx1yaSKG20/xBM1zewvGiRqz84GC/AJ5zx2qQzXE8EYLzaZYSTOV/eFzIDwc4A2HGeeT7Cs3EtMxdD1meyvmkudPuLpklmYtFJ5m0mQ4L55GTntk+lSpeWxv5rhxDFc3K4CLuQID/AmccH6ZOfpjT8LadGtvPCII5UM02CvRj5hGdw5PHrV/V4reKIW6kzyP8qW0yqwOPvc9lA7+wHtUuJSehn6Za6bLo0YS+dCIISBLFgu5XG1cdee9VvFNrLp9rbedFvuN8bLFEdzFA68n+7ycZJqLT/DE8um20rSXFvH5MbbNPlygJQY3A9T9MU6/h1GC1dVntJyHjGZVMUrYcYBJznHJ6etS4lKQ1LC9ni8y5keGzkB+SHPDDrl8evHGPrSiyhivrJYYwuRIJNnzF/l+96mtC11+OzuoRfSXjxxk/6NbIs8ZyME/K2B16kDpUGr3TavqlsmnwwRWnzrFKNjyqNvdEJ29xk59cdKXKNSRLJiGRzc7YVABYu+xXHbJbgYrMjuGfWbj7LA92RAmWiXAC7nySw4z6Vt/wBk2dvcvPrMly8ccIaGRxvVj/dC4Cg9Of8ACpU01k1aQvPCxktE2iQFW++4/h/PPvU8pV2YzQXV45a4dLeI4UJE+XcA9CzYx68fnVay0bTZGvHksY5naf5pJYiXGETo2SRjJOK2WuFMUkcS3VyYmI/0dDOjHP3emM/jVC0fUJprpxDBYsbjErOSGAMcfAA4z75pco7oc9gpRWjvLr5TtXbcbyCO+1/5V9KfC4k/DvQcsWIsoxkjBOBjtXzuulSTxuLqe9uiTkbGDRFcYxtUjPrySa+hfhSix/DnQo0QIq2igKBjGO2O1XSVrmGIeiOnooorc5QooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigArA+Iwz8PvEn/YJuv/AEU1b9YfxCGfAXiEHp/Zdz/6KagDwDQbbXF0mxkihhuYo7aKRU2BWfcuflY8ZHoevqKk8RzXI0W5mvob63TaBygVeo4+90qTRbXxDFoenR/28k8X2WPMQiEcg+ReN6qeB06Z96brEH2Wxme70QtIQpS63vckHeOCWAKn9Kxtqd19BbbV/DjORbpfXkkcnn48hkKe4aTbkcdiaZe2X2qe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"text": null}]}, {"role": "assistant", "content": [{"type": "text", "text": "ContourPlot3D draws the surface inside the box x from −1 to 1, a from −1 to 1, θ from −2π to 2π on which the expression stays constant at one of the contour levels Mathematica picks by default. The stored points are (x, y, z) = (x, a, θ) on that surface. The stored surface has 152,670 points joined into 172,563 faces, with 325 mesh lines drawn over it. It reaches from x = −1 to 1, a = −1 to 1, θ = −6.283 to 6.283. The surface falls into 5 main separate pieces plus 2 small fragments: 1 is closed, with sheets meeting along shared edges; 4 are open, each with its edge in 1 loop. The stored points are symmetric under x → −x, and under y → −y, and under (x, y) → (−x, −y)."}]}]}
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{"messages": [{"role": "system", "content": [{"type": "text", "text": "You are an assistant trained on The Sphere of Realization: The Mathematical Path of Harmonious Being (2nd Edition) by Parker Emmerson (Yaohushuason). Answer with the book's own text, Wolfram Language computations and results, in the book's notation."}]}, {"role": "user", "content": [{"type": "text", "text": "Which Wolfram Language command in The Sphere of Realization by Parker Emmerson drew this?", "image_url": null}, {"type": "image_url", "image_url": 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"text": null}]}, {"role": "assistant", "content": [{"type": "text", "text": "```wolfram\nSphericalPlot3D[(6 π^2 (-4 π + 2 θ) Cos[β] Sin[β] Sqrt[-c^2 (-θ^2 + 8 π^2 (1 + Sqrt[(Cos[β])^2]) - 4 π^2 (Sin[β])^2)])/(((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^(5 / 2)) - (4 c^2 π^2 θ Cos[β] Sin[β])/(Sqrt[-c^2 (-θ^2 + 8 π^2 (1 + Sqrt[(Cos[β])^2]) - 4 π^2 (Sin[β])^2)] ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^(3 / 2)) + (c^2 (-4 π + 2 θ) (-8 π^2 Cos[β] Sin[β] - (8 π^2 Cos[β] Sin[β])/(Sqrt[(Cos[β])^2])))/(4 Sqrt[-c^2 (-θ^2 + 8 π^2 (1 + Sqrt[(Cos[β])^2]) - 4 π^2 (Sin[β])^2)] ((θ (-4 π + θ) + 4 π^2 (Sin[β])^2))^(3 / 2)) + (c^4 (θ) (-8 π^2 Cos[β] Sin[β] - (8 π^2 Cos[β] Sin[β])/(Sqrt[(Cos[β])^2])))/(2 ((-c^2 (-θ^2 + 8 π^2 (1 + Sqrt[(Cos[β])^2]) - 4 π^2 (Sin[β])^2)))^(3 / 2) Sqrt[θ (-4 π + θ) + 4 π^2 (Sin[β])^2]), {β, 0, π / 2}, {θ, 0, 2 π}, PlotTheme → \"Orange\"]\n```"}]}]}
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