Triakis icosahedron

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

Orthogonal projections

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.

Kleetope

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.

Other triakis icosahedra

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

First stellation of icosahedron, or small triambic icosahedron (sometimes called a triakis icosahedron)

Great stellated dodecahedron (with very tall pyramids)

Great dodecahedron (with inverted pyramids)