Truncated icosidodecahedron

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Truncated icosidodecahedron
Truncatedicosidodecahedron.jpg
(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 CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png
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
Great rhombicosidodecahedron.png
Colored faces
Great rhombicosidodecahedron vertfig.png
4.6.10
(Vertex figure)
Disdyakistriacontahedron.jpg
Disdyakis triacontahedron
(dual polyhedron)
Truncated icosidodecahedron flat.svg
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[edit]

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[edit]

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[edit]

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

(±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[edit]

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]
Dual
image
Dual dodecahedron t012 v.png Dual dodecahedron t012 e46.png Dual dodecahedron t012 e4x.png Dual dodecahedron t012 e6x.png Dual dodecahedron t012 f4.png Dual dodecahedron t012 A2.png Dual dodecahedron t012 H3.png

Spherical tilings and Schlegel diagrams[edit]

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
Uniform tiling 532-t012.png Truncated icosidodecahedron stereographic projection decagon.png Truncated icosidodecahedron stereographic projection hexagon.png Truncated icosidodecahedron stereographic projection square.png
Truncated icosidodecahedron ortho skew.png Truncated icosidodecahedron schlegel-decacenter-color.png Truncated icosidodecahedron schlegel-hexacenter-color.png Truncated icosidodecahedron schlegel-squarecenter-color.png

Geometric variations[edit]

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 dodecahedron.png Great truncated icosidodecahedron convex hull.png Nonuniform truncated icosidodecahedron.png Great rhombicosidodecahedron.png Truncated dodecadodecahedron convex hull.png Icositruncated dodecadodecahedron convex hull.png Truncated icosahedron.png
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png

Truncated icosidodecahedral graph[edit]

Truncated icosidodecahedral graph
Truncated icosidodecahedral graph.png
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.[5]

Schlegel diagram graphs
Truncated icosidodecahedral graph-hexcenter.png
3-fold symmetry
Truncated icosidodecahedral graph-squarecenter.png
2-fold symmetry

Related polyhedra and tilings[edit]

Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532) [5,3]+, (532)
Uniform polyhedron-53-t0.png Uniform polyhedron-53-t01.png Uniform polyhedron-53-t1.png Uniform polyhedron-53-t12.png Uniform polyhedron-53-t2.png Uniform polyhedron-53-t02.png Uniform polyhedron-53-t012.png Uniform polyhedron-53-s012.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
{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
Icosahedron.svg Triakisicosahedron.jpg Rhombictriacontahedron.svg Pentakisdodecahedron.jpg Dodecahedron.svg Deltoidalhexecontahedron.jpg Disdyakistriacontahedron.jpg Pentagonalhexecontahedronccw.jpg
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 CDel node 1.pngCDel p.pngCDel node 1.pngCDel 3.pngCDel node 1.png. 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 Spherical truncated trigonal prism.png Uniform tiling 332-t012.png Uniform tiling 432-t012.png Uniform tiling 532-t012.png Uniform polyhedron-63-t012.png H2 tiling 237-7.png H2 tiling 238-7.png H2 tiling 23i-7.png H2 tiling 23j12-7.png H2 tiling 23j9-7.png H2 tiling 23j6-7.png H2 tiling 23j3-7.png
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 Spherical hexagonal bipyramid.png Spherical tetrakis hexahedron.png Spherical disdyakis dodecahedron.png Spherical disdyakis triacontahedron.png Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg H2checkers 237.png H2checkers 238.png H2checkers 23i.png H2 checkers 23j12.png H2 checkers 23j9.png H2 checkers 23j6.png H2 checkers 23j3.png
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

Notes[edit]

  1. ^ Wenninger, (Model 16, p. 30)
  2. ^ Williamson (Section 3-9, p. 94)
  3. ^ Cromwell (p. 82)
  4. ^ Weisstein, Eric W., "Icosahedral group", MathWorld.
  5. ^ Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269 

References[edit]

External links[edit]