Triakis icosahedron

Triakis icosahedron

(Click here for rotating model)
TypeCatalan solid
Coxeter diagram
Conway notationkI
Face typeV3.10.10

isosceles triangle
Vertices by type20{3}+12{10}
Symmetry groupIh, H3, [5,3], (*532)
Rotation groupI, [5,3]+, (532)
Dihedral angle160°36′45″
arccos(−24 + 155/61)
Propertiesconvex, face-transitive

Truncated dodecahedron
(dual polyhedron)


In geometry, the triakis icosahedron (or kisicosahedron[1]) 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.

Orthogonal projections of wireframe modes
[2] [6] [10]


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:


The triakis icosahedron has numerous stellations, including this one.

Related polyhedra

Spherical triakis icosahedron

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.

See also


  1. Conway, Symmetries of things, p.284

External links

This article is issued from Wikipedia - version of the 5/19/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.