Triakis icosahedron

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Triakis icosahedron
Triakisicosahedron.jpg
(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 ( -\frac{24 + 15\sqrt{5}}{61} )
Properties convex, face-transitive
Truncated dodecahedron.png
Truncated dodecahedron
(dual polyhedron)
Triakis icosahedron Net
Net

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
Projective
symmetry
[2] [6] [10]
Image Dual dodecahedron t12 exx.png Dual dodecahedron t12 A2.png Dual dodecahedron t12 H3.png
Dual
image
Dodecahedron t01 exx.png Dodecahedron t01 A2.png Dodecahedron t01 H3.png

Kleetope[edit]

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

Other triakis icosahedra[edit]

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

Stellations[edit]

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

Related polyhedra[edit]

Spherical triakis icosahedron
Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532) [5,3]+, (532)
Uniform polyhedron-53-t0.png Uniform polyhedron-53-t01.png Uniform polyhedron-53-t1.png Uniform polyhedron-53-t12.png Uniform polyhedron-53-t2.png Uniform polyhedron-53-t02.png Uniform polyhedron-53-t012.png Uniform polyhedron-53-s012.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
{5,3} t{5,3} r{5,3} t{3,5} {3,5} rr{5,3} tr{5,3} sr{5,3}
Duals to uniform polyhedra
Icosahedron.svg Triakisicosahedron.jpg Rhombictriacontahedron.svg Pentakisdodecahedron.jpg Dodecahedron.svg Deltoidalhexecontahedron.jpg Disdyakistriacontahedron.jpg Pentagonalhexecontahedronccw.jpg
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.

*n32 symmetry mutation of truncated tilings: 3.2n.2n
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
[12i,3] [9i,3] [6i,3]
Truncated
figures
Spherical triangular prism.png Uniform tiling 332-t01-1-.png Uniform tiling 432-t01.png Uniform tiling 532-t01.png Uniform tiling 63-t01.png H2 tiling 237-3.png H2 tiling 238-3.png H2 tiling 23i-3.png H2 tiling 23j12-3.png H2 tiling 23j9-3.png H2 tiling 23j6-3.png
Config. 3.4.4 3.6.6 3.8.8 3.10.10 3.12.12 3.14.14 3.16.16 3.∞.∞ 3.24i.24i 3.18i.18i 3.12i.12i
Triakis
figures
Spherical trigonal bipyramid.png Spherical triakis tetrahedron.png Spherical triakis octahedron.png Spherical triakis icosahedron.png Tiling Dual Semiregular V3-12-12 Triakis Triangular.svg Ord7 triakis triang til.png Ord8 triakis triang til.png Ord-infin triakis triang til.png
Config. V3.4.4 V3.6.6 V3.8.8 V3.10.10 V3.12.12 V3.14.14 V3.16.16 V3.∞.∞

See also[edit]

References[edit]

  1. ^ Conway, Symmetries of things, p.284

External links[edit]