Truncated icosidodecahedron

Truncated icosidodecahedron
(Click here for rotating model)
Type	Archimedean solid Uniform polyhedron
Elements	F = 62, E = 180, V = 120 (χ = 2)
Faces by sides	30{4}+20{6}+12{10}
Conway notation	bD or taD
Schläfli symbols	tr{5,3} or $t\begin{Bmatrix} 5 \\ 3 \end{Bmatrix}$
	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 convex 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:

• Truncated icosidodecahedron (Johannes Kepler)
• Rhombitruncated icosidodecahedron (Magnus Wenninger)
• Great rhombicosidodecahedron (Robert Williams, Peter Cromwell)
• Omnitruncated dodecahedron or icosahedron (Norman Johnson)

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.

Area and volume

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

$A = 30a^2 \left (1 + \sqrt{3} + \sqrt{5 + 2\sqrt{5}} \right) \approx 174.2920303a^2.$

$V = ( 95 + 50\sqrt{5} ) a^3 \approx 206.803399a^3.$

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 2ϕ − 2, centered at the origin, are all the even permutations of:

(±1/ϕ, ±1/ϕ, ±(3+ϕ)),

(±2/ϕ, ±ϕ, ±(1+2ϕ)),

(±1/ϕ, ±ϕ², ±(−1+3ϕ)),

(±(-1+2ϕ), ±2, ±(2+ϕ)) and

(±ϕ, ±3, ±2ϕ),

where ϕ = (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]
Dual image

Spherical tilings and Schlegel diagrams

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.

Schlegel diagrams are similar, with a perspective projection and straight edges.

Orthographic projection	Stereographic projections
	Decagon-centered	Hexagon-centered	Square-centered

Geometric 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.

Truncated icosidodecahedral graph

Truncated icosidodecahedral graph
5-fold symmetry
Vertices	120
Edges	180
Radius	15
Diameter	15
Girth	4
Automorphisms	120 (A5×2)
Chromatic number	2
Properties	Cubic, Hamiltonian, regular, zero-symmetric

In the mathematical field of graph theory, a truncated icosidodecahedral graph (or great rhombicosidodecahedral graph) is the graph of vertices and edges of the truncated icosidodecahedron, one of the Archimedean solids. It has 120 vertices and 180 edges, and is a zero-symmetric and cubic Archimedean graph.

Schlegel diagram graphs

3-fold symmetry

2-fold symmetry

Related polyhedra and tilings

Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532)							[5,3]+, (532)
{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
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.

*n32 symmetry mutations of omnitruncated tilings: 4.6.2n
Sym. *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]	[3i,3]
Figures
Config.	4.6.4	4.6.6	4.6.8	4.6.10	4.6.12	4.6.14	4.6.16	4.6.∞	4.6.24i	4.6.18i	4.6.12i	4.6.6i
Duals
Config.	V4.6.4	V4.6.6	V4.6.8	V4.6.10	V4.6.12	V4.6.14	V4.6.16	V4.6.∞	V4.6.24i	V4.6.18i	V4.6.12i	V4.6.6i

References

• Cromwell, P.; Polyhedra, CUP hbk (1997), pbk. (1999).
• Eric W. Weisstein, GreatRhombicosidodecahedron (Archimedean solid) at MathWorld.
• Richard Klitzing, 3D convex uniform polyhedra, x3x5x - grid

External links

• Weisstein, Eric W., "Great rhombicosidodecahedron", MathWorld.
• * Weisstein, Eric W., "Great rhombicosidodecahedral graph", MathWorld.
• Editable printable net of a truncated icosidodecahedron with interactive 3D view
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra