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regular polytope

vr.tr.regular-polytope · thing-q2745944

Let an agent explain regular polytopes and their classification, relay the enumeration in each dimension and the extension to honeycombs and abstract polytopes from geometry sources, describe the hyperbolic honeycombs the registry aliases name, and distinguish regular polytopes from uniform and semiregular polytopes and from regular polygons and polyhedra alone.

Thing Registry Cross-cutting context XCT.QLT
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

Understand What a regular polytope is.

Definition

Definition.

Definition

Definition.

  1. What is a regular polytope, and how does it differ from uniform and semiregular polytopes and from regular polygons and polyhedra? definition
  2. Is the question about a dimension, a specific polytope, a honeycomb or abstract theory? boundary

Examples

Examples.

Examples

Examples.

  1. What are the 120-cell, the regular honeycombs such as the order-8 tetrahedral honeycomb and the edgeless map? definition
  2. Which entry fits the specific object? action
Classification Classification.

Dimensions

By dimension.

Dimensions

Dimensions.

  1. How are regular polytopes enumerated in each dimension, and why only three from dimension five? provenance
  2. Which references are standard? provenance

Honeycombs

Honeycombs.

Honeycombs

Honeycombs.

  1. How are regular tessellations and honeycombs classified in Euclidean and hyperbolic spaces? provenance
  2. Which sources are cited? provenance
Theory Theory.

Symmetry

Symmetry groups.

Symmetry

Symmetry.

  1. How do Coxeter groups and Schlafli symbols describe regular polytopes? provenance
  2. Which entry fits Coxeter group? action

Abstract

Abstract polytopes.

Abstract

Abstract.

  1. What are abstract regular polytopes and degenerate cases such as the edgeless map? provenance
  2. Which entry fits abstract polytope? action
Context History and applications.

History

History.

History

History.

  1. How did Schlafli, Coxeter and others develop the theory? provenance
  2. Which entry fits the history of geometry? action

Applications

Applications.

Applications

Applications.

  1. Where do regular polytopes appear in crystallography, physics and art? provenance
  2. Which entry fits polytope visualisation? action

Classifiers Filled

Family
Thing Registry
Category
Cross-cutting context
Entry kind
thing
Plane
XCT
Domain
XCT.QLT
Other names and narrower kinds
Order-8 tetrahedral honeycomb, Order-9 tetrahedral honeycomb, edgeless map, 120-cell honeycomb, order-5 5-cell honeycomb, Order-7 tetrahedral honeycomb, regular tessellation, Infinite-order tetrahedral honeycomb, pentagonal polytope, regular convex polytope, dodecahedral pentagonal polytope, icosahedral pentagonal polytope

What it is Filled

A polytope whose symmetry group acts transitively on its flags, so that all its faces, edges and vertices of each kind are alike, generalising the regular polygons and the five Platonic solids to higher dimensions, with six convex regular 4-polytopes such as the 120-cell and three convex regular polytopes in each dimension five and above, extended to regular tessellations and honeycombs of Euclidean and hyperbolic space, such as the order-5 5-cell honeycomb, the 120-cell honeycomb and the order-7, order-8 and order-9 tetrahedral honeycombs, and to abstract regular polytopes including degenerate cases such as the edgeless map.

Why it exists Filled

Let an agent explain regular polytopes and their classification, relay the enumeration in each dimension and the extension to honeycombs and abstract polytopes from geometry sources, describe the hyperbolic honeycombs the registry aliases name, and distinguish regular polytopes from uniform and semiregular polytopes and from regular polygons and polyhedra alone.

Distinguishing features Filled

  • Flag transitivity
  • Complete classification
  • Schlafli symbols
  • Extension to honeycombs

What robots and AI may and may not do Filled

Must not

  • Claim a polytope is regular without checking flag transitivity.
  • Attribute discoveries to the wrong mathematicians.
  • Present conjecture as proven classification.

Only with a human decision

  • Publishing a new classification result.

May

  • State the definition of a regular polytope and list known families with their properties.
  • Compute or check symmetry and face counts for a stated polytope.

Moral aspects Filled

  • Credit for classification belongs to those who proved it.
  • Errors in geometry teaching spread widely through students.

Who is affected

  • Students
  • Researchers
  • Readers

Owners Filled

Steward

Nobody owns the concept; the mathematical community maintains it through peer review.

Links to other meta-models Filled

parent

  • Q41074997 - registry parent class
  • Q747980 - registry parent class

related

  • polytope - in registry terms
  • symmetrical object - in registry terms
  • Schlafli symbol - in each dimension
  • H. S. M. Coxeter - in Regular Polytopes

What else AI and robots need to interact with it Filled

Identity and identifiers required Filled

  • Vercy registry: vr.tr.regular-polytope
  • Wikidata: Q2745944 (https://www.wikidata.org/wiki/Q2745944)

Direct properties not applicable Not applicable

  • regular polyhedra: 5 count - Platonic solids
  • regular 4-polytopes: 6 count
  • regular polytopes, dimension 5 and above: 3 count - per dimension
  • Schlafli symbol: notation for regular polytopes note

Plane XCT: no invented physical properties.

Recognition optional Filled

  • Flag-transitive polytope
  • Platonic solids, 120-cell, regular honeycombs such as order-8 tetrahedral, edgeless map
  • Uniform polytopes are vertex-transitive only; semiregular have several face types; polygons and polyhedra are dimensions two and three
  • Not a visible object; highly symmetric figures such as the Platonic solids.

Capabilities and actions required Filled

  • explain the concept
  • relay classification
  • describe honeycombs
  • distinguish related objects

Hazards and failure modes required Filled

  • Confusing regular with uniform polytopes
  • Registry aliases naming specific hyperbolic honeycombs
  • Misreading Schlafli symbols

Standards and interfaces required Filled

  • No regulation

Context of use required Filled

  • Not applicable; a mathematical object.
  • regular polygons
  • Platonic solids and Kepler-Poinsot polyhedra
  • regular 4-polytopes such as the 120-cell
  • simplex, hypercube and cross-polytope families
  • regular Euclidean and hyperbolic honeycombs
  • abstract regular polytopes including the edgeless map

Sources Filled

  1. Wikidata item Q2745944: regular polytope - identity and sense of the item
  2. Wikipedia: Regular polytope - general description of the item

Open questions

  • Should regular honeycomb and abstract polytope be separate primary entries?
  • How should geometry sources be linked?
  • The registry entry has merged aliases naming specific honeycombs and a degenerate map; should they be split off?

Machine files

Card as JSON Specification Card backlog vercy.card/1

Provenance

thing registry research (pass 2) · unreviewed

Built from: models/things/publications/thing-q2745944/spec.json