# Pentakis icosidodecahedron

Pentakis icosidodecahedron
Type Conway polyhedron
Faces 80 triangles
(20 equilateral; 60 isosceles)
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

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 even with 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, but faces will be coplanar.

## Related polytopes

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.