Truncated icosidodecahedron

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Great rhombicosidodecahedron
(Click here for rotating model)
Type	Archimedean solid
Elements	F=62, E=180, V=120 (χ=2)
Faces by sides	30{4}+20{6}+12{10}
Schläfli symbol	<math>t\begin{Bmatrix} 3 \\ 5 \end{Bmatrix}</math>
Wythoff symbol	2 3 5 \|
Symmetry group	I_h
Index references	U₂₈, C₃₁, W₁₆
Dual	Disdyakis triacontahedron
Properties	Semiregular convex zonohedron
Vertex figure 4.6.10

The truncated icosidodecahedron is an Archimedean solid. It has 30 regular square faces, 20 regular hexagonal faces, 12 regular decagonal faces, 120 vertices and 180 edges. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated icosidodecahedron is a zonohedron.

1 Other names
2 Cartesian coordinates
3 See also
4 References
5 External links

[edit] Other names

Alternate interchangeable names include:

Great rhombicosidodecahedron
Rhombitruncated icosidodecahedron
Omnitruncated icosidodecahedron

The name truncated icosidodecahedron, originally given by Johannes Kepler, is somewhat misleading. If you truncate an icosidodecahedron by cutting the corners off, you do not get this uniform figure: some of the faces will be rectangles. However, the resulting figure is topologically equivalent to this and can always be deformed until the faces are regular.

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 uniform great rhombicosidodecahedron.

[edit] Cartesian coordinates

Cartesian coordinates for the vertices of a truncated icosidodecahedron centered at the origin are all the even permutations of

(±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.

[edit] See also

[edit] References

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)