Truncated dodecahedron  

(Click here for rotating model) 

Type  Archimedean solid Uniform polyhedron 
Elements  F = 32, E = 90, V = 60 (χ = 2) 
Faces by sides  20{3}+12{10} 
Conway notation  tD 
Schläfli symbols  t{5,3} 
t_{0,1}{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  1010: 116.57° 310: 142.62° 
References  U_{26}, C_{29}, W_{10} 
Properties  Semiregular convex 
Colored faces 
3.10.10 (Vertex figure) 
Triakis icosahedron (dual polyhedron) 
Net 
In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.
This polyhedron can be formed from a dodecahedron by truncating (cutting off) the corners so the pentagon faces become decagons and the corners become triangles.
It is used in the celltransitive hyperbolic spacefilling tessellation, the bitruncated icosahedral honeycomb.
The area A and the volume V of a truncated dodecahedron of edge length a are:
Cartesian coordinates for the vertices of a truncated dodecahedron with edge length 2φ − 2, centered at the origin,^{[1]} are all even permutations of:
where φ = 1 + √5/2 is the golden ratio.
The truncated dodecahedron has five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces: hexagonal and pentagonal. The last two correspond to the A_{2} and H_{2} Coxeter planes.
Centered by  Vertex  Edge 310 
Edge 1010 
Face Triangle 
Face Decagon 

Image  
Projective symmetry 
[2]  [2]  [2]  [6]  [10] 
Dual image 
The truncated dodecahedron 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 
Trianglecentered 

It shares its vertex arrangement with three nonconvex uniform polyhedra:
Truncated dodecahedron 
Great icosicosidodecahedron 
Great ditrigonal dodecicosidodecahedron 
Great dodecicosahedron 
It is part of a truncation process between a dodecahedron and icosahedron:
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 is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry.
*n32 symmetry mutation of truncated spherical tilings: t{n,3}  

Symmetry *n32 [n,3] 
Spherical  Euclid.  Compact hyperb.  Paraco.  
*232 [2,3] 
*332 [3,3] 
*432 [4,3] 
*532 [5,3] 
*632 [6,3] 
*732 [7,3] 
*832 [8,3]... 
*∞32 [∞,3] 

Truncated figures 

Symbol  t{2,3}  t{3,3}  t{4,3}  t{5,3}  t{6,3}  t{7,3}  t{8,3}  t{∞,3}  
Triakis figures 

Config.  V3.4.4  V3.6.6  V3.8.8  V3.10.10  V3.12.12  V3.14.14  V3.16.16  V3.∞.∞ 
Truncated dodecahedral graph  

5fold symmetry schlegel diagram


Vertices  60 
Edges  90 
Automorphisms  120 
Chromatic number  2 
Properties  Cubic, Hamiltonian, regular, zerosymmetric 
In the mathematical field of graph theory, a truncated dodecahedral graph is the graph of vertices and edges of the truncated dodecahedron, one of the Archimedean solids. It has 60 vertices and 90 edges, and is a cubic Archimedean graph.^{[2]}
Circular 