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Feel free to surf to my site - online psychic; mouse click for source, 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:

• 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.

File:Icosidodecahedron.png Icosidodecahedron

File:Nonuniform truncated icosidodecahedron.png 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.

File:Truncated dodecahedron.png

File:Great truncated icosidodecahedron convex hull.png

File:Nonuniform truncated icosidodecahedron.png

File:Truncated dodecadodecahedron convex hull.png

File:Icositruncated dodecadodecahedron convex hull.png

Area and volume

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

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

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:

(±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	File:Dodecahedron t012 v.png	File:Dodecahedron t012 e46.png	File:Dodecahedron t012 e4x.png	File:Dodecahedron t012 e6x.png	File:Dodecahedron t012 f4.png	File:Dodecahedron t012 A2.png	File:Dodecahedron t012 H3.png
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.

File:Uniform tiling 532-t012.png	File:Truncated icosidodecahedron stereographic projection decagon.png Decagon-centered	File:Truncated icosidodecahedron stereographic projection hexagon.png Hexagon-centered	File:Truncated icosidodecahedron stereographic projection square.png square-centered
Spherical tiling	Stereographic projections (face-centered)

Related polyhedra and tilings

This polyhedron can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram Template:CDD. 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.

See also

• Spinning great rhombicosidodecahedron
• Dodecahedron
• Great truncated icosidodecahedron
• Icosahedron
• Truncated cuboctahedron

Notes

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• Template:The Geometrical Foundation of Natural Structure (book)
• Cromwell, P.; Polyhedra, CUP hbk (1997), pbk. (1999).
• Template:Mathworld2
• Template:KlitzingPolytopes

External links

• Editable printable net of a truncated icosidodecahedron with interactive 3D view
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra