great icosidodecahedron
Enable an agent to recognize a great icosidodecahedron, validate a proposed realization, and choose geometric operations that preserve its defining face and incidence structure.
Bundle → Layer → Finding → Questions Filled
5 bundles · 9 layers · 14 findings · 23 questions
Identity and disambiguation Establish whether a named or depicted candidate denotes this particular uniform polyhedron.
Catalogue identity
Resolve the registry identity against explicitly named polyhedron catalogues.
Scheme-qualified identification
Record the numbering scheme beside each identifier and verify the associated face description.
- Which consulted catalogue associates the candidate with the great icosidodecahedron? provenance
- Does the identifier include its scheme, and does the referenced entry agree with the triangular and pentagrammic face inventory? definition
Near-neighbour exclusion
Separate this solid from similarly named or visually related polyhedra.
Face-based disambiguation
Use complete face boundaries and local incidences to resolve candidates that names or silhouettes cannot distinguish.
- Are the twelve five-vertex faces traced as pentagrams rather than convex pentagons? boundary
- What face or vertex-configuration evidence excludes the great icosahedron and great icosicosidodecahedron? boundary
Faces and incidences Capture the polygon cycles and their attachment independently of visible intersections.
Polygon boundaries
Identify complete triangular and pentagrammic face cycles.
Pentagram traversal
Retain the five-edge star traversal of each pentagram instead of replacing it with its outline or a triangulated fill.
- Does each pentagram cycle connect every second vertex of its supporting regular pentagon? definition
- Can every displayed fragment be traced back to an original triangle or pentagram face? provenance
Local and global incidence
Check attachment consistency across face cycles, edges, and vertices.
Incidence consistency
Validate inventory and cyclic adjacency without counting geometric crossings as additional incidences.
- Do the original face cycles yield 30 vertices and 60 edges, with exactly two incident faces per edge? measurement
- Does each vertex alternate two triangles and two pentagrams, excluding incidental surface crossings? measurement
Metric realization Determine whether coordinates realize the intended regular-faced, vertex-transitive geometry.
Regularity and normalization
Evaluate polygon regularity and scale-independent metric checks.
Metric validation
Check full face geometry alongside common edge length and circumradius.
- Are all face cycles planar and regular, with one common edge length across triangles and pentagrams? measurement
- After normalizing edge length, does the measured circumradius agree with 1/phi within the declared tolerance? measurement
Symmetry and equivalence
Test symmetry on the complete face-bearing realization.
Face-preserving vertex transitivity
Require symmetry transformations to preserve edges and polygon cycles, not just the vertex cloud.
- Can verified geometric symmetries carry any vertex to every other vertex while preserving the face cycles? measurement
- Can two candidate realizations be aligned by rigid motion and uniform scaling with matching polygon cycles? action
Intersection-aware operations Control interpretation and transformation of the intersecting realization.
Surface interpretation
Make boundary, fill, and interior conventions explicit before computation.
Crossings and interiors
Distinguish original polygon incidences from intersections introduced by the embedding or display.
- Which apparent junctions are original vertices, and which are crossings of edges or face interiors? boundary
- Which winding, overlap, and orientation conventions will define the requested area or volume? definition
- Will mesh conversion retain original face identities and mark any subdivision vertices as derived? action
Related-object operations
Track operations that produce associated but distinct mathematical objects.
Hull, skeleton, and dual
Record the operation, its convention, and a correspondence back to the original realization.
- Does hull or skeleton extraction explicitly record which original face information it discards? boundary
- Which duality convention and normalization are used, and how are original faces mapped to dual vertices? action
Evidence and external alignment What the world already says about this thing, gathered so the model can be checked against it.
Reported evidence
Findings from the breadth pass, kept separate from the structural claims.
Check these first
Recalled without web access and unsourced; every item is a lead to verify.
- Which of these check these first hold for the sense of great icosidodecahedron this model covers, and on what evidence? provenance
Identifiers and schemes
Recalled without web access and unsourced; every item is a lead to verify.
- Which of these identifiers and schemes hold for the sense of great icosidodecahedron this model covers, and on what evidence? provenance
Real-world use
Recalled without web access and unsourced; every item is a lead to verify.
- Which of these real-world use hold for the sense of great icosidodecahedron this model covers, and on what evidence? provenance
Typical measurements
Recalled without web access and unsourced; every item is a lead to verify.
- Which of these typical measurements hold for the sense of great icosidodecahedron this model covers, and on what evidence? provenance
Failure modes and hazards
Recalled without web access and unsourced; every item is a lead to verify.
- Which of these failure modes and hazards hold for the sense of great icosidodecahedron this model covers, and on what evidence? provenance
Neighbouring kinds and how to tell them apart
Recalled without web access and unsourced; every item is a lead to verify.
- Which of these neighbouring kinds and how to tell them apart hold for the sense of great icosidodecahedron this model covers, and on what evidence? provenance
Classifiers Filled
- Family
- Thing Registry
- Category
- Cross-cutting context
- Entry kind
- thing
- Plane
- XCT
- Domain
- XCT.QLT
What it is Filled
The great icosidodecahedron is a nonconvex uniform polyhedron with 20 equilateral triangular faces and 12 regular pentagram faces, with two faces of each type meeting alternately at every vertex.
In scope
- Identification through regular triangular and pentagrammic faces and their arrangement
- Vertex, edge, and face incidence validation
- Geometric realization, scale normalization, and symmetry checks
- Interpretation of self-intersections and polygon interiors
- Relations to its convex hull, skeleton, and dual
Out of scope
- The general classification of uniform polyhedra
- Independent models of the convex icosidodecahedron and great icosahedron
- Independent models of the great icosicosidodecahedron and other similarly named solids
- Fabrication materials, joints, and structural performance
- General rendering algorithms and mesh file formats
Why it exists Filled
Enable an agent to recognize a great icosidodecahedron, validate a proposed realization, and choose geometric operations that preserve its defining face and incidence structure.
Distinguishing features Filled
- Require 20 equilateral triangular faces and 12 regular pentagrammic faces; replacing the pentagrams with convex pentagons fails identification. [MathWorld](https://mathworld.wolfram.com/GreatIcosidodecahedron.html)
- Require the cyclic vertex configuration (3, 5/2, 3, 5/2), rather than merely matching the face inventory. [MathWorld](https://mathworld.wolfram.com/GreatIcosidodecahedron.html)
- Cross-check catalogue identifiers with their schemes: Maeder U54 and Wenninger 94 identify this solid. [MathWorld](https://mathworld.wolfram.com/GreatIcosidodecahedron.html)
- Do not accept an icosidodecahedral skeleton or convex hull alone as identification; both require the intended face structure to resolve the realization. [MathWorld](https://mathworld.wolfram.com/GreatIcosidodecahedron.html)
What robots and AI may and may not do Filled
Must not
- Present an incorrect model or wrong face counts as accurate.
- Confuse it with the icosidodecahedron or the great dodecahedron in teaching material.
- Copy published images or models without credit.
Only with a human decision
- Publishing new claims about its properties under a person's name.
May
- Explain the great icosidodecahedron's faces, vertex figure and relation to other uniform polyhedra.
- Generate accurate models or images of it from verified coordinates.
Moral aspects Filled
- Accuracy and credit matter in mathematical teaching and publication.
Who is affected
- Students
- Mathematicians
- Model makers
Owners Filled
Steward
Nobody owns it; it is a concept of geometry.
Links to other meta-models Filled
parent
- Q2878957 - registry parent class
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.great-icosidodecahedron
- Wikidata: Q3101027 (https://www.wikidata.org/wiki/Q3101027)
Direct properties not applicable Not applicable
- Face inventory: 20 regular {3} faces and 12 regular {5/2} faces
- Incidence counts: 30 vertices, 60 edges, 32 faces
- Vertex configuration: (3, 5/2, 3, 5/2), cyclically
- Normalized circumradius: R/a = 1/phi, where a is edge length and phi = (1 + sqrt(5))/2
- Realization validation: unverified | exact match | numerical match within declared tolerance | inconsistent
- Polygon interior convention: boundary only | even-odd fill | nonzero-winding fill | explicitly specified alternative
- Related constructions: convex hull: icosidodecahedron; dual: great rhombic triacontahedron
Plane XCT: no invented physical properties.
Recognition optional Filled
- Twenty triangles and twelve pentagrams intersecting, with triangles and pentagrams alternating around each vertex.
Capabilities and actions required Filled
- Validate a coordinate-and-face representation against the defining inventory and vertex configuration
- Normalize edge length and compare realizations up to rigid motion and scale
- Display pentagram boundaries and face intersections without promoting crossings to vertices
- Extract the skeleton or convex hull while retaining a relation to the original solid
- Construct a dual after declaring the geometric duality convention
- Reject or qualify area and volume requests whose interior convention is unspecified
Hazards and failure modes required Filled
- Self-intersecting faces are easily mis-rendered by software.
Standards and interfaces not applicable Not applicable
Not applicable
No specific standard or regulation applies to this abstract polyhedron as a mathematical object.
Context of use required Filled
- The great icosidodecahedron is studied in geometry as a nonconvex uniform polyhedron.
- Its models and diagrams support investigation of symmetry, face intersections and the distinction between geometric intersections and combinatorial incidence.
Sources Filled
- Wikidata item Q3101027: great icosidodecahedron - identity and sense of the item
- Wikipedia: Great icosidodecahedron - general description of the item
Open questions
- Which consulted coordinate-and-face dataset should serve as the canonical executable reference for this registry entry?
- Which numerical tolerances are appropriate for recognizing imported approximations without accepting geometrically different realizations?
- Which polygon-fill and spatial-interior conventions should be supported for area and volume, and how should their results be labelled?
- Which explicit polarity and normalization should govern construction of the great rhombic triacontahedron dual?
- Does an existing Vercy world-model publication already cover this identity and therefore need to be linked instead of duplicated?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q3101027/spec.json