The Small Rhombicosidodecahedron

A semi-regular polyhedron with (in order) a triangle, a square, a pentagon, and another square surrounding each vertex.

The small rhombicosidodecahedron is one of the thirteen archimedian solids. Each of its faces belongs to one of three circubscribing solids: the dodecahedron, the icosahedron, and the rhombic triacontahedron.

 VertexSymbol WythoffSymbol No. ofVertices No. ofEdges {3}Faces {4}Faces {5}Faces SymmetryGroup Dual Polyhedron 3.4.5.4 3 5 | 2 60 120 20 30 12 A5×C2 TrapezoidalHexecontahedron

Edge ratios:

• e/rho = sqrt(2)tan(18^@) = sqrt((10-4sqrt(5))/5)
• e/R = 2sqrt((11-4sqrt(5))/41)
• e/(e_12) = (1+sqrt(5))/6
• e/(e_20) = (1+3sqrt(5))/22
• e/(d_30) = (3-sqrt(5))/2

where e is the edge length, rho is the inter-radius, R is the circum-radius, e_12 is the edge of the circumscribing dodecahedron, e_20 is the edge of the circumscribing icosahedron, and d_30 is the long face diagonal of the circumscribing rhombic triacontahedron.