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From Wikipedia, the free encyclopedia
In geometry, the small complex icosidodecahedron is a degenerate uniform star polyhedron. Its edges are doubled, making it degenerate. The star has 32 faces (20 triangles and 12 pentagons), 60 (doubled) edges and 12 vertices and 4 sharing faces. The faces in it are considered as two overlapping edges as topological polyhedron.
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Small complex icosidodecahedron | |
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Type | Uniform star polyhedron |
Elements | F = 32, E = 60 (30x2) V = 12 (χ = −16) |
Faces by sides | 20{3}+12{5} |
Coxeter diagram | |
Wythoff symbol | 5 | 3/2 5 |
Symmetry group | Ih, [5,3], *532 |
Index references | U-, C-, W- |
Dual polyhedron | Small complex icosidodecacron |
Vertex figure | (3/2.5)5 (3.5)5/3 |
Bowers acronym | Cid |
A small complex icosidodecahedron can be constructed from a number of different vertex figures.
A very similar figure emerges as a geometrical truncation of the great stellated dodecahedron, where the pentagram faces become doubly-wound pentagons ({5/2} --> {10/2}), making the internal pentagonal planes, and the three meeting at each vertex become triangles, making the external triangular planes.
The small complex icosidodecahedron can be seen as a compound of the icosahedron {3,5} and the great dodecahedron {5,5/2} where all vertices are precise and edges coincide. The small complex icosidodecahedron resembles an icosahedron, because the great dodecahedron is completely contained inside the icosahedron.
Icosahedron | Great dodecahedron | Compound |
Its two-dimensional analogue would be the compound of a regular pentagon, {5}, representing the icosahedron as the n-dimensional pentagonal polytope, and regular pentagram, {5/2}, as the n-dimensional star. These shapes would share vertices, similarly to how its 3D equivalent shares edges.
Pentagon | Pentagram | Compound |
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