Rhombic icosahedron
The rhombic icosahedron is a polyhedron shaped like an oblate sphere. Its 20 faces are congruent golden rhombi, of which three, four, or five meet at each vertex. It has 5 faces meeting at each of its 2 poles, and 10 faces following its equator. It has D5d, , symmetry, order 20.
Even though all its faces are congruent, the rhombic icosahedron is not face-transitive, since one may distinguish whether a particular face is near the equator or a pole by examining the types of vertices surrounding that face.
Zonohedron
The rhombic icosahedron is a zonohedron, that is dual to an irregular-faced pentagonal gyrobicupola. It has 5 sets of 8 parallel edges, described as 85 belts.| The edges of the rhombic icosahedron exist in 5 parallel sets, seen in this wireframe orthogonal projection. |
The rhombic icosahedron forms the convex hull of the vertex-first projection of a 5-cube to 3 dimensions. The 32 vertices of a 5-cube map into 22 exterior vertices of the rhombic icosahedron, with the remaining 10 interior vertices forming a pentagonal antiprism. This is the same way one can obtain a Bilinski dodecahedron from a 4-cube and a rhombic triacontahedron from a 6-cube.
Related polyhedra
The rhombic icosahedron can be derived from the rhombic triacontahedron by removing 10 middle faces.A rhombic triacontahedron as an elongated rhombic icosahedron | The rhombic icosahedron shares its 5-fold symmetry orthogonal projection with the rhombic triacontahedron |