isogonal figure
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.
Bundle → Layer → Finding → Questions Filled
4 bundles · 8 layers · 8 findings · 16 questions
Understand What an isogonal figure is.
Definition
Definition.
Definition
Definition.
- What does vertex-transitive mean, and how do isogonal, isohedral, isotoxal and regular figures relate? definition
- Is the question about isogonal figures in general, uniform polytopes or a specific solid? boundary
Examples
Examples.
Examples
Examples.
- What are examples in two, three and four dimensions, from isogonal polygons to Archimedean solids and uniform polychora? definition
- Which entry fits the specific figure? action
Classify Classification.
Uniform
Uniform polytopes.
Uniform
Uniform.
- How are uniform polytopes classified by Coxeter groups and Wythoff constructions in each dimension? provenance
- Which references are standard? provenance
Higher
Higher dimensions.
Higher
Higher.
- What is known about uniform 7-, 8-, 9- and 10-polytopes and semiregular polytopes? provenance
- Which entry fits the specific dimension? action
Theory Theory.
Symmetry
Symmetry groups.
Symmetry
Symmetry.
- How do symmetry groups act on vertices, and what is vertex figure analysis? provenance
- Which sources are cited? provenance
Duality
Duality.
Duality
Duality.
- How does duality map isogonal figures to isohedral ones, as with Catalan solids? provenance
- Which entry fits dual polyhedron? action
Context History and applications.
History
History.
History
History.
- How did Kepler, Wythoff, Coxeter and others develop the theory of uniform polytopes? provenance
- Which entry fits the history of geometry? action
Applications
Applications.
Applications
Applications.
- Where do isogonal figures appear in chemistry, crystallography and design? provenance
- Which entry fits polyhedra in chemistry? action
Classifiers Filled
- Family
- Thing Registry
- Category
- Cross-cutting context
- Entry kind
- thing
- Plane
- XCT
- Domain
- XCT.QLT
- Other names and narrower kinds
- uniform polytope, semiregular polytope, uniform 7-polytope, uniform 8-polytope, uniform 9-polytope, uniform 10-polytope, bitruncated 120-cell, permutohedron, hyperrectangle, noble polyhedron, rectified 120-cell, tritetragonal tiling
What it is Filled
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.
Why it exists Filled
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.
Distinguishing features Filled
- Vertex transitivity
- Uniform subclass
- Duality with isohedral
- Higher-dimensional catalogues
What robots and AI may and may not do Filled
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.
Only with a human decision
- Publishing a claimed new enumeration.
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.
Moral aspects Filled
- Correct credit for enumeration work matters to mathematicians.
- Errors in teaching materials mislead students.
Who is affected
- Students
- Researchers
- Designers using tilings
Owners Filled
Steward
Nobody owns the concept; the mathematical community maintains it.
Links to other meta-models Filled
parent
- Q747980 - registry parent class
related
- polytope - in registry terms
- isohedral figure - with equivalent faces
- uniform polytope - as the subclass with regular faces
- Coxeter group - symmetry
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.isogonal-figure
- Wikidata: Q3071715 (https://www.wikidata.org/wiki/Q3071715)
Direct properties not applicable Not applicable
- Archimedean solids: 13 count
- convex uniform 4-polytopes: 64 count - plus infinite families
- convex uniform 7-polytopes: hundreds count - by Coxeter group
Plane XCT: no invented physical properties.
Recognition optional Filled
- 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
- Not a visible object; polytopes with equivalent vertices.
Capabilities and actions required Filled
- explain the definition
- relay classification
- describe examples
- distinguish related symmetry classes
Hazards and failure modes required Filled
- Confusing isogonal with isohedral or regular
- Assuming all isogonal figures are uniform
- Miscounting higher-dimensional catalogues
Standards and interfaces required Filled
- No regulation; standard mathematical definitions
Context of use required Filled
- Not applicable; a mathematical concept.
- 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
Sources Filled
- Wikidata item Q3071715: isogonal figure - identity and sense of the item
- Wikipedia: Isogonal figure - general description of the item
Open questions
- 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?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q3071715/spec.json