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} 
Conway notation  bD or taD 
Schläfli symbols  tr{5,3} or 
t_{0,1,2}{5,3}  
Wythoff symbol  2 3 5  
Coxeter diagram  
Symmetry group  I_{h}, H_{3}, [5,3], (*532), order 120 
Rotation group  I, [5,3]^{+}, (532), order 60 
Dihedral angle  610: 142.62° 410: 148.28° 46: 159.095° 
References  U_{28}, C_{31}, W_{16} 
Properties  Semiregular convex zonohedron 
Colored faces 
4.6.10 (Vertex figure) 
Disdyakis triacontahedron (dual polyhedron) 
Net 
In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron,^{[1]} great rhombicosidodecahedron,^{[2]}^{[3]} omnitruncated dodecahedron or omnitruncated icosahedron^{[4]} 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 the most edges and vertices of all Platonic and Archimedean solids, though the snub dodecahedron has more faces. Of all vertextransitive polyhedra, it occupies the largest percentage (89.80%) of the volume of a sphere in which it is inscribed, very narrowly beating the snub dodecahedron (89.63%) and small rhombicosidodecahedron (89.23%), and less narrowly beating the truncated icosahedron (86.74%); it also has by far the greatest volume (206.8 cubic units) when its edge length equals 1. Of all vertextransitive polyhedra that are not prisms or antiprisms, it has the largest sum of angles (90 + 120 + 144 = 354 degrees) at each vertex; only a prism or antiprism with more than 60 sides would have a larger sum. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated icosidodecahedron is a 15zonohedron.
The name truncated icosidodecahedron, given originally by Johannes Kepler, is misleading. An actual truncation of an icosidodecahedron has rectangles instead of squares. This nonuniform polyhedron is topologically equivalent to the Archimedean solid. Alternate interchangeable names are:


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.
The surface area A and the volume V of the truncated icosidodecahedron of edge length a are:^{[citation needed]}
If a set of all 13 Archimedean solids were constructed with all edge lengths equal, the truncated icosidodecahedron would be the largest.
Cartesian coordinates for the vertices of a truncated icosidodecahedron with edge length 2φ − 2, centered at the origin, are all the even permutations of:^{[5]}
where φ = 1 + √5/2 is the golden ratio.
The truncated icosidodecahedron is the convex hull of a rhombicosidodecahedron with cuboids above its 30 squares, whose height to base ratio is φ. The rest of its space can be dissected into nonuniform cupolas, namely 12 between inner pentagons and outer decagons and 20 between inner triangles and outer hexagons.
An alternative dissection also has a rhombicosidodecahedral core. It has 12 pentagonal rotundae between inner pentagons and outer decagons. The remaining part is a toroidal polyhedron.
dissection images 


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 A_{2} and H_{2} Coxeter planes.
Centered by  Vertex  Edge 46 
Edge 410 
Edge 610 
Face square 
Face hexagon 
Face decagon 

Solid  
Wireframe  
Projective symmetry 
[2]^{+}  [2]  [2]  [2]  [2]  [6]  [10] 
Dual image 
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  

Decagoncentered  Hexagoncentered  Squarecentered  
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  

Vertices  120 
Edges  180 
Radius  15 
Diameter  15 
Girth  4 
Automorphisms  120 (A_{5}×2) 
Chromatic number  2 
Properties  Cubic, Hamiltonian, regular, zerosymmetric 
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 zerosymmetric and cubic Archimedean graph.^{[6]}
3fold symmetry 
2fold symmetry 
Bowtie icosahedron and dodecahedron contain two trapezoidal faces in place of the square.^{[7]} 
Family of uniform icosahedral polyhedra  

Symmetry: [5,3], (*532)  [5,3]^{+}, (532)  
{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  
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 CoxeterDynkin 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.
*n32 symmetry mutation 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  
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  
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 