Truncated icosidodecahedron


Truncated icosidodecahedron
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}
Schläfli symbol t0,1,2{5,3}
Wythoff symbol 2 3 5 |
Coxeter-Dynkin CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Symmetry Ih, [5,3], (*532)
Dihedral Angle hexagon-decagon: 142.62°
square-decagon: 148.28°
hexagon-square: 159.095°
References U28, C31, W16
Properties Semiregular convex zonohedron
Truncated icosidodecahedron color
Colored faces
Truncated icosidodecahedron
4.6.10
(Vertex figure)
Disdyakistriacontahedron.jpg
Disdyakis triacontahedron
(dual polyhedron)
Truncated icosidodecahedron Net
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.

Contents

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.png
Icosidodecahedron
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.

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}

Truncated icosidodecahedron is the largest solid among all 13 Archimedean solids with the same side lengths.

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 Dodecahedron t012 v.png Dodecahedron t012 e46.png Dodecahedron t012 e4x.png Dodecahedron t012 e6x.png Dodecahedron t012 f4.png Dodecahedron t012 A2.png 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.

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

See also

Notes

  1. ^ Wenninger, (Model 16, p. 30)
  2. ^ Williamson (Section 3-9, p. 94)
  3. ^ Cromwell (p. 82)
  4. ^ Weisstein, Eric W., "Icosahedral group" from MathWorld.

References

External links


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