# 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"
$\arccos ( -\frac{24 + 15\sqrt{5}}{61} )$
Properties convex, face-transitive

Truncated dodecahedron
(dual polyhedron)

Net

In geometry, the triakis icosahedron 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.

Projective Image Dual symmetry image [2] [6] [10]

## 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:

## Stellations

The triakis icosahedron has numerous stellations, including this one.

## Related polyhedra

Spherical triakis icosahedron
Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532) [5,3]+, (532)
{5,3} t{5,3} r{5,3} 2t{5,3}=t{3,5} 2r{5,3}={3,5} rr{5,3} tr{5,3} sr{5,3}
Duals to uniform polyhedra
V5.5.5 V3.10.10 V3.5.3.5 V5.6.6 V3.3.3.3.3 V3.4.5.4 V4.6.10 V3.3.3.3.5

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.

Dimensional family of truncated polyhedra and tilings: 3.2n.2n
Symmetry
*n32
[n,3]
Spherical Euclidean Compact hyperbolic Paracompact
*232
[2,3]
D3h
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
P6m
*732
[7,3]

*832
[8,3]...

*∞32
[∞,3]

Truncated
figures

3.4.4

3.6.6

3.8.8

3.10.10

3.12.12

3.14.14

3.16.16

3.∞.∞
Coxeter
Schläfli

t{2,3}

t{3,3}

t{4,3}

t{5,3}

t{6,3}

t{7,3}

t{8,3}

t{∞,3}
Uniform dual figures
Triakis
figures

V3.4.4

V3.6.6

V3.8.8

V3.10.10

V3.12.12

V3.14.14

V3.16.16

V3.∞.∞
Coxeter