Rhombic icosahedron

Rhombic icosahedron
Type	Zonohedron
Faces	20 congruent golden rhombi
Edges	40
Vertices	22
Symmetry group	D5d = D5v, [2+,10], (2*5)
Dual polyhedron	irregular-faced pentagonal gyrobicupola
Properties	convex

The rhombic icosahedron is a polyhedron shaped like an oblate sphere. Its 20 faces are congruent golden rhombi;^[1] 3, 4, or 5 faces meet at each vertex. It has 5 faces (green on top figure) meeting at each of its 2 poles; these 2 vertices lie on its axis of 5-fold symmetry, which is perpendicular to 5 axes of 2-fold symmetry through the midpoints of opposite equatorial edges (example on top figure: most left-hand and most right-hand mid-edges). Its other 10 faces follow its equator, 5 above and 5 below it; each of these 10 rhombi has 2 of its 4 sides lying on this zig-zag skew decagon equator. The rhombic icosahedron has 22 vertices. It has D_5d, [2⁺,10], (2*5) symmetry group, of order 20; thus it has a center of symmetry (since 5 is odd).

Even though all its faces are congruent, the rhombic icosahedron is not face-transitive, since one can distinguish whether a particular face is near the equator or near a pole by examining the types of vertices surrounding this face.