Triakis icosahedron

In geometry, the triakis icosahedron (or kisicosahedron[1]) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated dodecahedron.

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)


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.

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

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.

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
  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)
  • Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9.
  • Wenninger, Magnus (1983). Dual Models. Cambridge University Press. ISBN 978-0-521-54325-5. MR 0730208. (The thirteen semiregular convex polyhedra and their duals, Page 19, Triakisicosahedron)
  • The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Triakis icosahedron )
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