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Triakis icosahedron

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Triakis icosahedron
(Click here for rotating model)
Type Catalan solid
Coxeter diagram CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node.png
Conway notation kI
Face type V3.10.10
DU26 facets.png

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.png
Truncated dodecahedron
(dual polyhedron)
Triakis icosahedron Net
3d model of a 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.

Orthogonal projections[edit]

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]
Image Dual dodecahedron t12 exx.png Dual dodecahedron t12 A2.png Dual dodecahedron t12 H3.png
Dodecahedron t01 exx.png Dodecahedron t01 A2.png Dodecahedron t01 H3.png


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.

Tetrahedra augmented icosahedron.png

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

Other triakis icosahedra[edit]

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


Stellation of triakis icosahedron.png
The triakis icosahedron has numerous stellations, including this one.

Related polyhedra[edit]

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[edit]


  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 [1] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Triakis icosahedron )

External links[edit]

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