# Truncated icosidodecahedron

Truncated icosidodecahedron

Type Archimedean solid
Uniform polyhedron
Elements F = 62, E = 180, V = 120 (χ = 2)
Faces by sides 30{4}+20{6}+12{10}
Schläfli symbols tr{5,3}
t0,1,2{5,3}
Wythoff symbol 2 3 5 |
Coxeter diagram
Symmetry group Ih, H3, [5,3], (*532), order 120
Rotation group I, [5,3]+, (532), order 60
Dihedral Angle 6-10:142.62°
4-10:148.28°
4-6:159.095°
References U28, C31, W16
Properties Semiregular convex zonohedron

Colored faces

4.6.10
(Vertex figure)

Disdyakis triacontahedron
(dual polyhedron)

Net

In geometry, the truncated icosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.

It has 30 square faces, 20 regular hexagonal faces, 12 regular decagonal faces, 120 vertices and 180 edges – more than any other nonprismatic uniform polyhedron. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated icosidodecahedron is a zonohedron.

## Other names

Alternate interchangeable names include:

The name truncated icosidodecahedron, originally given by Johannes Kepler, is somewhat misleading. If one truncates an icosidodecahedron by cutting the corners off, one does not get this uniform figure: instead of squares the truncation has golden rectangles. However, the resulting figure is topologically equivalent to this and can always be deformed until the faces are regular.

 Icosidodecahedron A literal geometric truncation of the icosidodecahedron produces rectangular faces rather than squares.

The alternative name great rhombicosidodecahedron (as well as rhombitruncated icosidodecahedron) refers to the fact that the 30 square faces lie in the same planes as the 30 faces of the rhombic triacontahedron which is dual to the icosidodecahedron. Compare to small rhombicosidodecahedron.

One unfortunate point of confusion is that there is a nonconvex uniform polyhedron of the same name. See nonconvex great rhombicosidodecahedron.

### Variations

Within Icosahedral symmetry there are unlimited geometric variations of the truncated icosidodecahedron with isogonal faces. The truncated dodecahedron, rhombicosidodecahedron, and truncated icosahedron as degenerate limiting cases.

## Area and volume

The surface area A and the volume V of the truncated icosidodecahedron of edge length a are:

\begin{align} A & = 30 \left [ 1 + \sqrt{ 2 \left ( 4 + \sqrt{5} + \sqrt{15+6\sqrt{6}} \right ) } \right ] a^2 \\ & \approx 175.031045a^2 \\ V & = ( 95 + 50\sqrt{5} ) a^3 \approx 206.803399a^3. \\ \end{align}

If a set of all 13 Archimedean solids were constructed with all edge lengths equal, the truncated icosidodecahedron would be the largest.

## Cartesian coordinates

Cartesian coordinates for the vertices of a truncated icosidodecahedron with edge length 2τ − 2, centered at the origin, are all the even permutations of:4

(±1/τ, ±1/τ, ±(3+τ)),
(±2/τ, ±τ, ±(1+2τ)),
(±1/τ, ±τ2, ±(−1+3τ)),
(±(-1+2τ), ±2, ±(2+τ)) and
(±τ, ±3, ±2τ),

where τ = (1 + √5)/2 is the golden ratio.

## Orthogonal projections

The truncated icosidodecahedron has seven special orthogonal projections, centered on a vertex, on three types of edges, and three types of faces: square, hexagonal and decagonal. The last two correspond to the A2 and H2 Coxeter planes.

Orthogonal projections
Centered by Vertex Edge
4-6
Edge
4-10
Edge
6-10
Face
square
Face
hexagon
Face
decagon
Image
Projective
symmetry
[2]+ [2] [2] [2] [2] [6] [10]

## Spherical tiling

The truncated icosidodecahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.

 Spherical tiling Stereographic projections (face-centered) Decagon-centered Hexagon-centered square-centered

## Related polyhedra and tilings

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

This polyhedron can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.

Dimensional family of omnitruncated polyhedra and tilings: 4.6.2n
Symmetry
*n32
[n,3]
Spherical Euclidean Hyperbolic
*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]

Coxeter
Schläfli

tr{2,3}

tr{3,3}

tr{4,3}

tr{5,3}

tr{6,3}

tr{7,3}

tr{8,3}

tr{∞,3}
Omnitruncated
figure
Vertex figure 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6.∞
Dual figures
Coxeter
Omnitruncated
duals
Face
configuration
V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞

## Notes

1. ^ Wenninger, (Model 16, p. 30)
2. ^ Williamson (Section 3-9, p. 94)
3. ^ Cromwell (p. 82)
4. ^

## References

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