Disdyakis triacontahedron
Disdyakis triacontahedron  

animated 

Type  Catalan 
Conway notation  mD or dbD 
Coxeter diagram  
Face polygon 
scalene triangle 
Faces  120 
Edges  180 
Vertices  62 = 12 + 20 + 30 
Face configuration  V4.6.10 
Symmetry group  I_{h}, H_{3}, [5,3], (*532) 
Rotation group  I, [5,3]^{+}, (532) 
Dihedral angle  164° 53' 17" 
Dual polyhedron  truncated icosidodecahedron 
Properties  convex, facetransitive 
Net 
In geometry, a disdyakis triacontahedron, hexakis icosahedron, decakis dodecahedron or kisrhombic triacontahedron^{[1]} is a Catalan solid with 120 faces and the dual to the Archimedean truncated icosidodecahedron. As such it is face uniform but with irregular face polygons. It looks a wee bit like an inflated rhombic triacontahedron—if one replaces each face of the rhombic triacontahedron with a single vertex and four triangles in a regular fashion one ends up with a disdyakis triacontahedron. That is, the disdyakis triacontahedron is the Kleetope of the rhombic triacontahedron. It also has the most faces among the Archimedean and Catalan solids, with the snub dodecahedron, with 92 faces, in second place.
If the bipyramids and the trapezohedra are excluded, the disdyakis triacontahedron has the most faces of any other strictly convex polyhedron where every face of the polyhedron has the same shape.
Projected into a sphere, the edges of a disdyakis triacontahedron define 15 great circles. Buckminster Fuller used these 15 great circles, along with 10 and 6 others in two other polyhedra to define his 31 great circles of the spherical icosahedron.
Contents
Symmetry
The edges of the polyhedron projected onto a sphere form 15 great circles, and represent all 15 mirror planes of reflective I_{h} icosahedral symmetry, as shown in this image. Combining pairs of light and dark triangles define the fundamental domains of the nonreflective (I) icosahedral symmetry. The edges of a compound of five octahedra also represent the 10 mirror planes of icosahedral symmetry.
Disdyakis triacontahedron 
Dodecahedron 
Icosahedron 
Rhombic triacontahedron 
Alternately colored 
Edges as great circles 
compound of five octahedra 
Orthogonal projections
The disdyakis triacontahedron has three types of vertices which can be centered in orthogonally projection:
Projective symmetry 
[2]  [6]  [10] 

Image  
Dual image 
Uses
The disdyakis triacontahedron, as a regular dodecahedron with pentagons divided into 10 triangles each, is considered the "holy grail" for combination puzzles like the Rubik's cube. This unsolved problem, often called the "big chop" problem, currently has no satisfactory mechanism. It is the most significant unsolved problem in mechanical puzzles.^{[2]}
This shape was used to create d120 dice using 3D printing.^{[3]} More recently, the Dice Lab has used the Disdyakis triacontahedron to mass market an injection moulded 120 sided die.^{[4]} It is claimed that the d120 is the largest number of possible faces on a fair dice, aside from infinite families (such as right regular prisms, bipyramids, and trapezohedra) that would be impractical in reality due to the tendency to roll for a long time.^{[5]}
Related polyhedra
Related polyhedra and tilings
Polyhedra similar to the disdyakis triacontahedron are duals to the Bowtie icosahedron and dodecahedron, containing extra pairs of triangular faces.^{[6]} 
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 
It is topologically related to a polyhedra sequence defined by the face configuration V4.6.2n. This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and continuing into the hyperbolic plane for any n ≥ 7.
With an even number of faces at every vertex, these polyhedra and tilings can be shown by alternating two colors so all adjacent faces have different colors.
Each face on these domains also corresponds to the fundamental domain of a symmetry group with order 2,3,n mirrors at each triangle face vertex. This is *n32 in orbifold notation, and [n,3] in Coxeter notation.
*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  
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 
References
 ^ Conway, Symmetries of things, p.284
 ^ Big Chop
 ^ Kevin Cook's Dice Collector website: d120 3D printed from Shapeways artist SirisC
 ^ The Dice Lab
 ^ http://nerdist.com/thisd120isthelargestmathematicallyfairdiepossible/
 ^ Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons Craig S. Kaplan
 Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 048623729X. (Section 39)
 Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 9780521543255, MR 0730208 (The thirteen semiregular convex polyhedra and their duals, Page 25, Disdyakistriacontahedron )
 The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim GoodmanStrass, ISBN 9781568812205 [1] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 285, kisRhombic triacontahedron )
External links
 Eric Wolfgang Weisstein, Disdyakis triacontahedron (Catalan solid) at MathWorld.
 Disdyakis triacontahedron (Hexakis Icosahedron) – Interactive Polyhedron Model