{
  "vercy": "1.0-draft",
  "publication": {
    "status": "research-draft",
    "adjudicationStatus": "unreviewed",
    "publishableCanonical": false,
    "generatedAt": "2026-09-12T21:54:36Z",
    "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"
  },
  "metaModel": {
    "id": "THING-Q735071",
    "registryId": "vr.tr.regular-polyhedron",
    "name": "regular polyhedron",
    "version": "0.2.0-wave.3",
    "entryKind": "thing",
    "family": "Thing Registry",
    "domain": [
      "XCT.QLT"
    ],
    "status": "research-draft"
  },
  "canonicalUrl": "https://ver.cy/models/thing/q735071/",
  "model": {
    "registry_id": "vr.tr.regular-polyhedron",
    "name": "regular polyhedron",
    "purpose": "Let an agent define regular polyhedra, list the Platonic and Kepler-Poinsot solids with their properties, explain the extended and abstract cases, relay proofs of the classification, and distinguish regular from uniform and semiregular polyhedra.",
    "definition": "A polyhedron whose faces are congruent regular polygons and whose vertices are all alike, so that its symmetry group acts transitively on its flags; the convex regular polyhedra are the five Platonic solids, the non-convex ones are the four Kepler-Poinsot polyhedra, and the definition extends to degenerate cases such as hosohedra and dihedra and to abstract and skew forms such as the Petrie duals.",
    "what_it_is_for": "A foundational object in geometry and symmetry.",
    "affordances": [
      "define and list regular polyhedra",
      "relay properties and formulas",
      "explain the classification proof",
      "distinguish from uniform and semiregular polyhedra"
    ],
    "distinguishing_features": [
      "Flag-transitive symmetry",
      "Finite classification",
      "Duality pairs",
      "Extensions to degenerate and abstract cases"
    ],
    "appearance": "Highly symmetric solids: tetrahedron, cube, octahedron, dodecahedron, icosahedron, and star forms.",
    "visual_identification": [
      "Congruent regular faces, identical vertices",
      "Five convex and four star forms",
      "Archimedean solids have regular faces but are not regular"
    ],
    "physical_properties": [
      {
        "quantity": "convex regular polyhedra",
        "typical_range": "5",
        "unit": "count",
        "note": "Platonic solids"
      },
      {
        "quantity": "non-convex regular polyhedra",
        "typical_range": "4",
        "unit": "count",
        "note": "Kepler-Poinsot"
      }
    ],
    "families_and_kinds": [
      "Platonic solids",
      "Kepler-Poinsot polyhedra",
      "degenerate regular polyhedra: hosohedra and dihedra",
      "regular skew and Petrie dual polyhedra such as the petrial cube",
      "abstract regular polyhedra",
      "regular tilings as infinite analogues"
    ],
    "related_models": [
      {
        "relation": "is a kind of",
        "target": "regular polytope",
        "why": "in three dimensions"
      },
      {
        "relation": "is a kind of",
        "target": "isohedron",
        "why": "face-transitive"
      },
      {
        "relation": "includes",
        "target": "Platonic solid",
        "why": "the convex cases"
      },
      {
        "relation": "is contrasted with",
        "target": "Archimedean solid",
        "why": "semiregular"
      }
    ],
    "identifiers": [
      {
        "scheme": "Schlafli symbol",
        "value_or_pattern": "{p,q}",
        "note": "for each polyhedron"
      }
    ],
    "standards_and_regulation": [
      "No regulation; definitions follow standard geometry"
    ],
    "failure_modes_and_hazards": [
      "Confusing regular with uniform or semiregular",
      "Ignoring the extended definitions",
      "Errors in vertex and face counts"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": []
  },
  "sources": [],
  "structure": {
    "bundles": [
      {
        "id": "define",
        "name": "Define",
        "description": "What a regular polyhedron is.",
        "rationale": "Definition.",
        "layers": [
          {
            "id": "definition",
            "name": "Definition",
            "description": "Definition and list.",
            "findings": [
              {
                "id": "definition-finding",
                "name": "Definition",
                "description": "Definition.",
                "questions": [
                  {
                    "text": "What defines a regular polyhedron, and which nine exist in the classical sense?",
                    "kind": "definition"
                  },
                  {
                    "text": "Is the solid regular, uniform or semiregular?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          },
          {
            "id": "extended",
            "name": "Extended",
            "description": "Extended cases.",
            "findings": [
              {
                "id": "extended-finding",
                "name": "Extended",
                "description": "Extended.",
                "questions": [
                  {
                    "text": "How do hosohedra, dihedra, skew and abstract polyhedra extend the definition?",
                    "kind": "definition"
                  },
                  {
                    "text": "Which entry fits the specific polyhedron?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "properties",
        "name": "Properties",
        "description": "Properties and formulas.",
        "rationale": "Computation.",
        "layers": [
          {
            "id": "counts",
            "name": "Counts",
            "description": "Faces, edges, vertices.",
            "findings": [
              {
                "id": "counts-finding",
                "name": "Counts",
                "description": "Counts.",
                "questions": [
                  {
                    "text": "What are the face, edge and vertex counts, angles and symmetry groups of each?",
                    "kind": "action"
                  },
                  {
                    "text": "What are the values for the polyhedron in question?",
                    "kind": "measurement"
                  }
                ]
              }
            ]
          },
          {
            "id": "duality",
            "name": "Duality",
            "description": "Duality.",
            "findings": [
              {
                "id": "duality-finding",
                "name": "Duality",
                "description": "Duality.",
                "questions": [
                  {
                    "text": "How do the regular polyhedra pair as duals?",
                    "kind": "definition"
                  },
                  {
                    "text": "Which entry fits dual polyhedron?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "theory",
        "name": "Theory",
        "description": "Classification.",
        "rationale": "Theory.",
        "layers": [
          {
            "id": "proof",
            "name": "Proof",
            "description": "Classification proof.",
            "findings": [
              {
                "id": "proof-finding",
                "name": "Proof",
                "description": "Proof.",
                "questions": [
                  {
                    "text": "How is the classification proved, from Euclid to Euler and Schlafli?",
                    "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": "How does the classification extend to regular polytopes in higher dimensions?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits regular polytope?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "context",
        "name": "Context",
        "description": "History and use.",
        "rationale": "Context.",
        "layers": [
          {
            "id": "history",
            "name": "History",
            "description": "History.",
            "findings": [
              {
                "id": "history-finding",
                "name": "History",
                "description": "History.",
                "questions": [
                  {
                    "text": "How were regular polyhedra studied from Plato and Euclid to Kepler and Coxeter?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits the history of geometry?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "use",
            "name": "Use",
            "description": "Applications.",
            "findings": [
              {
                "id": "use-finding",
                "name": "Use",
                "description": "Use.",
                "questions": [
                  {
                    "text": "Where do regular polyhedra appear in crystals, viruses, art and games?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits a specific application?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "Should each polyhedron be a separate entry?",
    "How should Schlafli symbols be linked?",
    "The registry entry has merged aliases naming degenerate and Petrie dual cases; should they be split off?"
  ],
  "statistics": {
    "bundles": 4,
    "layers": 8,
    "findings": 8,
    "questions": 16
  }
}