Icosidodecahedron

In general, an icosidodecahedron is a 32-faced polyhedron.

"The" (regular) icosidodecahedron is the 32-faced Archimedean solid with faces . It is one of the two convex quasiregular polyhedra. It is also uniform polyhedron and Wenninger model . It has Schläfli symbol and Wythoff symbol .

It is implemented in the Wolfram Language as PolyhedronData["Icosidodecahedron"].

Several symmetric projections of the icosidodecahedron are illustrated above. The dual polyhedron is the rhombic triacontahedron. The polyhedron vertices of an icosidodecahedron of polyhedron edge length are , , , , , . The 30 polyhedron vertices of an octahedron 5-compound form an icosidodecahedron (Ball and Coxeter 1987). Faceted versions include the small icosihemidodecahedron and small dodecahemidodecahedron.

The icosidodecahedron constructed in origami is illustrated above (Kasahara and Takahama 1987, pp. 48-49). This construction uses 120 sonobè units, each made from a single sheet of origami paper.

The faces of the icosidodecahedron consist of 20 triangles and 12 pentagons. Furthermore, its 60 edges are bisected perpendicularly by those of the reciprocal rhombic triacontahedron (Ball and Coxeter 1987).

Five octahedra of edge length can be inscribed on the vertices of an icosidodecahedron of unit edge length, resulting in the beautiful octahedron 5-compound.

The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are

			(1)
			(2)
			(3)
			(4)
			(5)
			(6)

The surface area and volume for an icosidodecahedron are given by

			(7)
			(8)

The distance to the centers of the triangular and pentagonal faces are

			(9)
			(10)

The dihedral angle between triangular and pentagonal faces is

			(11)
			(12)