Truncated icosidodecahedron

Truncated icosidodecahedron 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 62 faces: 30 squares, 20 regular hexagons, and 12 regular decagons. It has more vertices (120) and edges (180) 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.

Names

 The name truncated icosidodecahedron, given originally by Johannes Kepler, is misleading. An actual truncation of a icosidodecahedron has rectangles instead of squares. This nonuniform polyhedron is topologically equivalent to the Archimedean solid. Alternate interchangeable names are: Truncated icosidodecahedron (Johannes Kepler) Rhombitruncated icosidodecahedron (Magnus Wenninger) Great rhombicosidodecahedron (Robert Williams, Peter Cromwell) Omnitruncated dodecahedron or icosahedron (Norman Johnson) .mw-parser-output .tmulti .thumbinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{text-align:left;background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .thumbcaption{text-align:center}}

The name great rhombicosidodecahedron refers to the relationship with the (small) rhombicosidodecahedron (compare section Dissection).
There is a nonconvex uniform polyhedron with a similar name, the nonconvex great rhombicosidodecahedron.

Area and volume

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

{\begin{aligned}A&=30\left(1+{\sqrt {3}}+{\sqrt {5+2{\sqrt {5}}}}\right)a^{2}&&\approx 174.292\,0303a^{2}.\\V&=\left(95+50{\sqrt {5}}\right)a^{3}&&\approx 206.803\,399a^{3}.\end{aligned}}

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)),
(±(2φ − 1), ±2, ±(2 + φ)) and
φ, ±3, ±2φ),

where φ = 1 + 5/2 is the golden ratio.

Dissection

The truncated icosidodecahedron is the convex hull of a rhombicosidodecahedron with cuboids above its 30 squares whose height to base ratio is the golden ratio. The rest of its space can be dissected into 12 nonuniform pentagonal cupolas below the decagons and 20 nonuniform triangular cupolas below the hexagons.

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
Vertices120
Edges180
Diameter15
Girth4
Automorphisms120 (A5×2)
Chromatic number2
PropertiesCubic, Hamiltonian, regular, zero-symmetric
Table of graphs and parameters

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.

Related polyhedra and tilings  Bowtie icosahedron and dodecahedron contain two trapezoidal faces in place of the square.

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.