# Triakis icosahedron

Triakis icosahedron Type Catalan solid
Coxeter diagram     Conway notation kI
Face type V3.10.10
isosceles triangle
Faces 60
Edges 90
Vertices 32
Vertices by type 20{3}+12{10}
Symmetry group Ih, H3, [5,3], (*532)
Rotation group I, [5,3]+, (532)
Dihedral angle 160°36′45″
arccos(−24 + 155/61)
Properties convex, face-transitive Truncated dodecahedron
(dual polyhedron) Net

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 H2 Coxeter 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.

If the icosahedron is augmented by tetrahedral without removing the center icosahedron, one gets the net of an icosahedral pyramid.

## Related polyhedra

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.