Small Triambic Icosahedron

The small triambic icosahedron is the dual polyhedron of the small ditrigonal icosidodecahedron , and is Wenninger model . It can be constructed by augmentation of a unit edge-length icosahedron by a pyramid with height , giving a solid with edge lengths and 1.

Wenninger (1989, p. 49) calls this solid the triakis octahedron (which is a term more commonly used for the dual of one of the Archimedean solids).

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

The convex hull of the small ditrigonal icosidodecahedron is a regular dodecahedron whose dual is the icosahedron, so the dual of the small ditrigonal icosidodecahedron (i.e., the small triambic icosahedron) is one of the icosahedron stellations (Wenninger 1983, p. 42). In fact, it is the second icosahedron stellation in the enumeration of Maeder (1994).

The small triambic icosahedron has 32 vertices, 90 edges, and 70 faces. Its vertices are arranged in two concentric groups of 12 (indicated in red in the above illustration) and 20 (blue).

For a small triambic icosahedron with edge lengths

			(1)
			(2)

the vertices determine a regular icosahedron and regular dodecahedron with circumradii

			(3)
			(4)

respectively.

The solid in the right figure is the dodecahedron-small triambic icosahedron compound.

The surface area and volume of the small triambic icosahedron are

			(5)
			(6)

The solid small triambic icosahedron has moment of inertia tensor

(7)

for uniform density solid of mass .