| Pentakis icosidodecahedron | |
|---|---|
| Type | Conway polyhedron |
| Faces | 80 triangles (2 types) |
| Edges | 120 (2 types) |
| Vertices | 42 (2 types) |
| Vertex configurations | (12) 35 (30) 36 |
| Symmetry group | Icosahedral (Ih) |
| Dual polyhedron | Truncated rhombic triacontahedron |
| Properties | convex, equilateral-faced |
The pentakis icosidodecahedron is a convex polyhedron with 80 triangular faces, 120 edges, and 42 vertices. It a dual of the truncated rhombic triacontahedron.
Its name comes from a topological construction from the icosidodecahedron with the kis operator applied to the pentagonal faces. In this construction, all the vertices are assumed to be the same distance from the center, while in general icosahedral symmetry can be maintained the 12 order-5 vertices at a different distance from the center as the other 30.
It can also be topologically constructed from the icosahedron, dividing each triangular face into 4 triangles by adding mid-edge vertices. From this construction, all 80 triangles will be equilateral.
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The Pentakis Icosidodecahedron is not the Kleetope of the Icosidodecahedron. In fact, the kleetope is a triangular hecatonicosahedron with 2 types of triangles. The polyhedron mentioned previously is called the "Tripentakis Icosidodecahedron". The solid has 62 vertices and 180 edges. It shouldn't be surprising that the solid has order-3 vertices, order-5 vertices, and order-8 vertices, which are the only 3 vertex types. It can also be obtained by raising low pyramids on each equilateral triangular face on the pentakis icosidodecahedron.
This polyhedron can be confused with a slightly smaller Catalan solid, the pentakis dodecahedron, which has only 60 triangles, 90 edges, and 32 vertices.
It represents the exterior envelope of a vertex-centered orthogonal projection of the 600-cell, one of six convex regular 4-polytopes, into 3 dimensions.