{
  "vercy": "1.0-draft",
  "publication": {
    "status": "research-draft",
    "adjudicationStatus": "unreviewed",
    "publishableCanonical": false,
    "generatedAt": "2026-09-13T04:06:08Z",
    "providers": [
      "Claude"
    ],
    "breadth": "written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify",
    "pass": 2,
    "wave": 3,
    "engine": "claude",
    "enrichment": {
      "pass": 3,
      "engine": "claude",
      "producedAt": "2026-10-06T10:00:00Z",
      "ingestedAt": "2026-10-06T07:10:50+00:00",
      "fields": [
        "agent_conduct",
        "ethics",
        "owners"
      ]
    },
    "gate": {
      "passed": true,
      "version": "1.0",
      "at": "2026-10-06T19:51:58+00:00",
      "evidence": [
        "completeness 1.00",
        "sense: Wikidata description",
        "2 live sources",
        "review by codex: pass"
      ]
    }
  },
  "metaModel": {
    "id": "THING-Q3071715",
    "registryId": "vr.tr.isogonal-figure",
    "name": "isogonal figure",
    "version": "0.2.0-wave.3",
    "entryKind": "thing",
    "family": "Thing Registry",
    "domain": [
      "XCT.QLT"
    ],
    "status": "research-draft"
  },
  "canonicalUrl": "https://ver.cy/models/thing/q3071715/",
  "model": {
    "registry_id": "vr.tr.isogonal-figure",
    "name": "isogonal figure",
    "purpose": "Let an agent explain isogonal figures and vertex-transitivity, relay the classification of uniform and semiregular polytopes from geometric references, describe examples across dimensions, and distinguish isogonal from isohedral, isotoxal and regular figures.",
    "definition": "A polytope or tiling whose symmetry group acts transitively on its vertices, so that every vertex is equivalent under the symmetries of the figure, also called vertex-transitive, including the uniform polytopes that are isogonal with regular polygon faces, such as the Platonic and Archimedean solids in three dimensions, the semiregular polytopes, and uniform polytopes in higher dimensions catalogued up to ten dimensions and beyond; isogonal figures are dual to isohedral figures.",
    "what_it_is_for": "Not applicable; a mathematical concept.",
    "affordances": [
      "explain the definition",
      "relay classification",
      "describe examples",
      "distinguish related symmetry classes"
    ],
    "distinguishing_features": [
      "Vertex transitivity",
      "Uniform subclass",
      "Duality with isohedral",
      "Higher-dimensional catalogues"
    ],
    "appearance": "Not a visible object; polytopes with equivalent vertices.",
    "visual_identification": [
      "Vertex-transitive polytope or tiling",
      "Uniform polytopes, semiregular polytopes, uniform 7- to 10-polytopes, Archimedean solids",
      "Isohedral figures have equivalent faces; regular figures are transitive on flags"
    ],
    "physical_properties": [
      {
        "quantity": "Archimedean solids",
        "typical_range": "13",
        "unit": "count",
        "note": ""
      },
      {
        "quantity": "convex uniform 4-polytopes",
        "typical_range": "64",
        "unit": "count",
        "note": "plus infinite families"
      },
      {
        "quantity": "convex uniform 7-polytopes",
        "typical_range": "hundreds",
        "unit": "count",
        "note": "by Coxeter group"
      }
    ],
    "families_and_kinds": [
      "isogonal polygons",
      "isogonal polyhedra including uniform polyhedra",
      "uniform polychora and higher uniform polytopes such as uniform 7-, 8-, 9- and 10-polytopes",
      "semiregular polytopes",
      "isogonal tilings and honeycombs",
      "non-uniform isogonal figures"
    ],
    "related_models": [
      {
        "relation": "is a kind of",
        "target": "polytope",
        "why": "in registry terms"
      },
      {
        "relation": "is dual to",
        "target": "isohedral figure",
        "why": "with equivalent faces"
      },
      {
        "relation": "includes",
        "target": "uniform polytope",
        "why": "as the subclass with regular faces"
      },
      {
        "relation": "is classified by",
        "target": "Coxeter group",
        "why": "symmetry"
      }
    ],
    "identifiers": [],
    "standards_and_regulation": [
      "No regulation; standard mathematical definitions"
    ],
    "failure_modes_and_hazards": [
      "Confusing isogonal with isohedral or regular",
      "Assuming all isogonal figures are uniform",
      "Miscounting higher-dimensional catalogues"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": [],
    "agent_conduct": {
      "may": [
        "Explain isogonal figures and how they differ from isohedral and regular ones.",
        "Check whether a given polytope or tiling is vertex-transitive.",
        "Cite catalogues of uniform polytopes with their sources."
      ],
      "must_not": [
        "Assume all isogonal figures are uniform.",
        "Confuse isogonal with isohedral or regular.",
        "Miscount higher-dimensional catalogues or present incomplete ones as complete.",
        "Claim new results without proof."
      ],
      "requires_human": [
        "Publishing a claimed new enumeration."
      ]
    },
    "ethics": {
      "considerations": [
        "Correct credit for enumeration work matters to mathematicians.",
        "Errors in teaching materials mislead students."
      ],
      "affected_parties": [
        "Students",
        "Researchers",
        "Designers using tilings"
      ]
    },
    "owners": {
      "steward": "Nobody owns the concept; the mathematical community maintains it.",
      "master_systems": []
    }
  },
  "sources": [
    {
      "id": "wikidata",
      "title": "Wikidata item Q3071715: isogonal figure",
      "url": "https://www.wikidata.org/wiki/Q3071715",
      "url_status": "live",
      "what_it_supports": "identity and sense of the item",
      "checked_by": "wikidata-api"
    },
    {
      "id": "wikipedia-en",
      "title": "Wikipedia: Isogonal figure",
      "url": "https://en.wikipedia.org/wiki/Isogonal_figure",
      "url_status": "live",
      "what_it_supports": "general description of the item",
      "checked_by": "wikidata-api sitelink"
    }
  ],
  "structure": {
    "bundles": [
      {
        "id": "understand",
        "name": "Understand",
        "description": "What an isogonal figure is.",
        "rationale": "Mathematics.",
        "layers": [
          {
            "id": "definition",
            "name": "Definition",
            "description": "Definition.",
            "findings": [
              {
                "id": "definition-finding",
                "name": "Definition",
                "description": "Definition.",
                "questions": [
                  {
                    "text": "What does vertex-transitive mean, and how do isogonal, isohedral, isotoxal and regular figures relate?",
                    "kind": "definition"
                  },
                  {
                    "text": "Is the question about isogonal figures in general, uniform polytopes or a specific solid?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          },
          {
            "id": "examples",
            "name": "Examples",
            "description": "Examples.",
            "findings": [
              {
                "id": "examples-finding",
                "name": "Examples",
                "description": "Examples.",
                "questions": [
                  {
                    "text": "What are examples in two, three and four dimensions, from isogonal polygons to Archimedean solids and uniform polychora?",
                    "kind": "definition"
                  },
                  {
                    "text": "Which entry fits the specific figure?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "classify",
        "name": "Classify",
        "description": "Classification.",
        "rationale": "Mathematics.",
        "layers": [
          {
            "id": "uniform",
            "name": "Uniform",
            "description": "Uniform polytopes.",
            "findings": [
              {
                "id": "uniform-finding",
                "name": "Uniform",
                "description": "Uniform.",
                "questions": [
                  {
                    "text": "How are uniform polytopes classified by Coxeter groups and Wythoff constructions in each dimension?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which references are standard?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          },
          {
            "id": "higher",
            "name": "Higher",
            "description": "Higher dimensions.",
            "findings": [
              {
                "id": "higher-finding",
                "name": "Higher",
                "description": "Higher.",
                "questions": [
                  {
                    "text": "What is known about uniform 7-, 8-, 9- and 10-polytopes and semiregular polytopes?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits the specific dimension?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "theory",
        "name": "Theory",
        "description": "Theory.",
        "rationale": "Mathematics.",
        "layers": [
          {
            "id": "symmetry",
            "name": "Symmetry",
            "description": "Symmetry groups.",
            "findings": [
              {
                "id": "symmetry-finding",
                "name": "Symmetry",
                "description": "Symmetry.",
                "questions": [
                  {
                    "text": "How do symmetry groups act on vertices, and what is vertex figure analysis?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which sources are cited?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          },
          {
            "id": "duality",
            "name": "Duality",
            "description": "Duality.",
            "findings": [
              {
                "id": "duality-finding",
                "name": "Duality",
                "description": "Duality.",
                "questions": [
                  {
                    "text": "How does duality map isogonal figures to isohedral ones, as with Catalan solids?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits dual polyhedron?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "context",
        "name": "Context",
        "description": "History and applications.",
        "rationale": "Context.",
        "layers": [
          {
            "id": "history",
            "name": "History",
            "description": "History.",
            "findings": [
              {
                "id": "history-finding",
                "name": "History",
                "description": "History.",
                "questions": [
                  {
                    "text": "How did Kepler, Wythoff, Coxeter and others develop the theory of uniform polytopes?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits the history of geometry?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "applications",
            "name": "Applications",
            "description": "Applications.",
            "findings": [
              {
                "id": "applications-finding",
                "name": "Applications",
                "description": "Applications.",
                "questions": [
                  {
                    "text": "Where do isogonal figures appear in chemistry, crystallography and design?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits polyhedra in chemistry?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "Should uniform polytope be a separate primary entry?",
    "How should geometric references be linked?",
    "The registry entry has merged aliases naming uniform polytopes by dimension; should they be split off?"
  ],
  "statistics": {
    "bundles": 4,
    "layers": 8,
    "findings": 8,
    "questions": 16
  }
}