Small complex icosidodecahedron

Small complex icosidodecahedron
File:Small complex icosidodecahedron.png
Type	Uniform star polyhedron
Elements	F = 32, E = 60 (30x2) V = 12 (χ = −16)
Faces by sides	20{3}+12{5}
Coxeter diagram	File:CDel label5.pngFile:CDel branch 01rd.pngFile:CDel split2-5t.pngFile:CDel node.png
Wythoff symbol	5 | 3/2 5
Symmetry group	Ih, [5,3], *532
Index references	U-, C-, W-
Dual polyhedron	Small complex icosidodecacron
Vertex figure	File:Small complex icosidodecahedron verf.png (3/2.5)5 (3.5)5/3
Bowers acronym	Cid

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.

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.

As a compound

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.

Compound polyhedron

File:Icosahedron.png	File:Great dodecahedron.png	File:Small complex icosidodecahedron.png
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.

Compound polygon

File:Pentagon.svg	File:Star polygon 5-2.svg	File:Complete graph K5.svg
Pentagon	Pentagram	Compound

See also

• Great complex icosidodecahedron

References

• (Table 6, degenerate cases)