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Research draft

regular polyhedron

vr.tr.regular-polyhedron · thing-q735071

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.

Thing Registry Cross-cutting context XCT.QLT

Research draft, second pass

A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.

Researched by: Claude

Bundle → Layer → Finding → QuestionsClassifiersWhat it isWhy it existsDistinguishing featuresWhat robots and AI may and may not doMoral aspectsOwnersLinks to other meta-modelsIdentity and identifiersDirect propertiesRecognitionCapabilities and actionsHazards and failure modesStandards and interfacesContext of useSources

Bundle → Layer → Finding → Questions Filled

4 bundles · 8 layers · 8 findings · 16 questions

Define What a regular polyhedron is.

Definition

Definition and list.

Definition

Definition.

  1. What defines a regular polyhedron, and which nine exist in the classical sense? definition
  2. Is the solid regular, uniform or semiregular? boundary

Extended

Extended cases.

Extended

Extended.

  1. How do hosohedra, dihedra, skew and abstract polyhedra extend the definition? definition
  2. Which entry fits the specific polyhedron? action
Properties Properties and formulas.

Counts

Faces, edges, vertices.

Counts

Counts.

  1. What are the face, edge and vertex counts, angles and symmetry groups of each? action
  2. What are the values for the polyhedron in question? measurement

Duality

Duality.

Duality

Duality.

  1. How do the regular polyhedra pair as duals? definition
  2. Which entry fits dual polyhedron? action
Theory Classification.

Proof

Classification proof.

Proof

Proof.

  1. How is the classification proved, from Euclid to Euler and Schlafli? provenance
  2. Which references are standard? provenance

Higher

Higher dimensions.

Higher

Higher.

  1. How does the classification extend to regular polytopes in higher dimensions? provenance
  2. Which entry fits regular polytope? action
Context History and use.

History

History.

History

History.

  1. How were regular polyhedra studied from Plato and Euclid to Kepler and Coxeter? provenance
  2. Which entry fits the history of geometry? action

Use

Applications.

Use

Use.

  1. Where do regular polyhedra appear in crystals, viruses, art and games? provenance
  2. Which entry fits a specific application? action

Classifiers Filled

Family
Thing Registry
Category
Cross-cutting context
Entry kind
thing
Plane
XCT
Domain
XCT.QLT
Also called
hosohedron, polygonal dihedron, petrial cube, petrial octahedron, petrial tetrahedron, petrial dodecahedron, petrial icosahedron, petrial great dodecahedron, petrial small stellated dodecahedron, petrial great icosahedron, petrial great stellated dodecahedron, apeirogonal hosohedron

What it is Filled

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.

Why it exists Filled

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.

Distinguishing features Filled

  • Flag-transitive symmetry
  • Finite classification
  • Duality pairs
  • Extensions to degenerate and abstract cases

What robots and AI may and may not do Derived, awaiting review

May

  • define and list regular polyhedra
  • relay properties and formulas
  • explain the classification proof
  • distinguish from uniform and semiregular polyhedra

Note: Affordances listed as what may be done; prohibitions collected from the text.

Moral aspects Missing, in the backlog

Not described yet. This gap is in the card backlog.

Owners Missing, in the backlog

Not described yet. This gap is in the card backlog.

Links to other meta-models Filled

parent

  • Q23821397 - registry parent class
  • Q2745944 - registry parent class
  • Q3847026 - registry parent class
  • Q7045824 - registry parent class

related

  • regular polytope - in three dimensions
  • isohedron - face-transitive
  • Platonic solid - the convex cases
  • Archimedean solid - semiregular

What else AI and robots need to interact with it Filled

Identity and identifiers required Filled

  • Vercy registry: vr.tr.regular-polyhedron
  • Wikidata: Q735071 (https://www.wikidata.org/wiki/Q735071)
  • Schlafli symbol: {p,q} for each polyhedron

Direct properties not applicable Not applicable

  • convex regular polyhedra: 5 count - Platonic solids
  • non-convex regular polyhedra: 4 count - Kepler-Poinsot

Plane XCT: no invented physical properties.

Recognition optional Filled

  • Congruent regular faces, identical vertices
  • Five convex and four star forms
  • Archimedean solids have regular faces but are not regular
  • Highly symmetric solids: tetrahedron, cube, octahedron, dodecahedron, icosahedron, and star forms.

Capabilities and actions required Filled

  • define and list regular polyhedra
  • relay properties and formulas
  • explain the classification proof
  • distinguish from uniform and semiregular polyhedra

Hazards and failure modes required Filled

  • Confusing regular with uniform or semiregular
  • Ignoring the extended definitions
  • Errors in vertex and face counts

Standards and interfaces required Filled

  • No regulation; definitions follow standard geometry

Context of use required Filled

  • A foundational object in geometry and symmetry.
  • 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

Sources Missing, in the backlog

Not described yet. This gap is in the card backlog.

Note: Written from model knowledge without web access; claims are unverified.

Open questions

  • 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?
  • Which entry fits the specific polyhedron?
  • What are the face, edge and vertex counts, angles and symmetry groups of each?
  • Which entry fits dual polyhedron?

Machine files

Card as JSON Specification Card backlog vercy.card/1

Provenance

thing registry research (pass 2) · unreviewed

Built from: models/things/publications/thing-q735071/spec.json