Small ditrigonary icosidodecahedron
Small ditrigonary icosidodecahedron | |
---|---|
Rank | 3 |
Type | Uniform |
Space | Spherical |
Notation | |
Bowers style acronym | Sidtid |
Coxeter diagram | x5/2o3o3*a () |
Elements | |
Faces | 20 triangles, 12 pentagrams |
Edges | 60 |
Vertices | 20 |
Vertex figure | Ditrigon, edge lengths 1 and (√5–1)/2 |
Measures (edge length 1) | |
Circumradius | |
Volume | |
Dihedral angle | |
Central density | 2 |
Number of external pieces | 72 |
Level of complexity | 5 |
Related polytopes | |
Army | Doe, edge length |
Regiment | Sidtid |
Dual | Small triambic icosahedron |
Conjugate | Great ditrigonary icosidodecahedron |
Convex core | Truncated icosahedron |
Abstract & topological properties | |
Flag count | 240 |
Euler characteristic | -8 |
Orientable | Yes |
Genus | 5 |
Properties | |
Symmetry | H_{3}, order 120 |
Convex | No |
Nature | Tame |
The small ditrigonary icosidodecahedron, or sidtid, is a quasiregular uniform polyhedron. It consists of 20 equilateral triangles and 12 pentagrams, with three of each joining at a vertex.
It can be constructed as a holosnub dodecahedron. The pentagrammic faces lie in the same planes as the pentagons of the convex hull dodecahedron, and the triangles are the dodecahedron's vertex figures.
This polyhedron is the vertex figure of the small ditrigonary hexacosihecatonicosachoron.
Vertex coordinates[edit | edit source]
A small ditrigonary icosidodecahedron of side length 1 has vertex coordinates given by all permutations of
and even permutations of
The first set of vertices correspond to those of an inscribed unit cube. This relates to the fact that a uniform compound of 5 cubes has the same vertices and edges as this polyhedron.
Representations[edit | edit source]
A small ditrigonary icosidodecahedron has the following Coxeter diagrams:
- x5/2o3o3*a
- ß5o3o (as holosnub)
Related polyhedra[edit | edit source]
The small ditrigonary icosidodecahedron is the colonel of a three-member regiment that also includes the ditrigonary dodecadodecahedron and the great ditrigonary icosidodecahedron. This regiment also contains the rhombihedron, the uniform compound of 5 cubes. The pentagrammic cuploid and pentagonal cuploid are contained within the edge structure.
Name | OBSA | CD diagram | Picture |
---|---|---|---|
Small ditrigonary icosidodecahedron | sidtid | x5/2o3o3*a () | |
(degenerate, double cover of id) | x5/2x3o3*a () | ||
(degenerate, double cover of ike) | o5/2o3x3*a () | ||
Small icosicosidodecahedron | siid | x5/2o3x3*a () | |
(degenerate, double cover of ti) | x5/2x3x3*a () | ||
Small snub icosicosidodecahedron | seside | s5/2s3s3*a () |
External links[edit | edit source]
- Bowers, Jonathan. "Polyhedron Category 3: Quasiregulars" (#33).
- Klitzing, Richard. "sidtid".
- Wikipedia Contributors. "Small ditrigonal icosidodecahedron".
- McCooey, David. "Small Ditrigonal Icosidodecahedron"