Triakis icosahedron

In geometry, the triakis icosahedron (or kisicosahedron) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated dodecahedron.

The triakis icosahedron has three symmetry positions, two on vertices, and one on a midedge: The Triakis icosahedron has five special orthogonal projections, centered on a vertex, on two types of edges, and two types of faces: hexagonal and pentagonal. The last two correspond to the A₂ and H₂Coxeter planes.

It can be seen as an icosahedron with triangular pyramids augmented to each face; that is, it is the Kleetope of the icosahedron. This interpretation is expressed in the name, triakis.

This interpretation can also apply to other similar nonconvex polyhedra with pyramids of different heights:

The triakis icosahedron has numerous stellations, including this one.

The triakis icosahedron is a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have (*n32) reflectional symmetry.

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