Great truncated icosidodecahedron
Polyhedron with 62 faces
From Wikipedia, the free encyclopedia
In geometry, the great truncated icosidodecahedron (or great quasitruncated icosidodecahedron or stellatruncated icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U68. It has 62 faces (30 squares, 20 hexagons, and 12 decagrams), 180 edges, and 120 vertices.[1] It is given a Schläfli symbol t0,1,2{5/3,3}, and Coxeter-Dynkin diagram, 
.
| Great truncated icosidodecahedron | |
|---|---|
 | |
| Type | Uniform star polyhedron |
| Elements | F = 62, E = 180 V = 120 (χ = 2) |
| Faces by sides | 30{4}+20{6}+12{10/3} |
| Coxeter diagram |  |
| Wythoff symbol | 2 3 5/3 | |
| Symmetry group | Ih, [5,3], *532 |
| Index references | U68, C87, W108 |
| Dual polyhedron | Great disdyakis triacontahedron |
| Vertex figure |  4.6.10/3 |
| Bowers acronym | Gaquatid |
Cartesian coordinates
Cartesian coordinates for the vertices of a great truncated icosidodecahedron centered at the origin are all the even permutations of
![{\displaystyle {\begin{array}{ccclc}{\Bigl (}&\pm \,\varphi ,&\pm \,\varphi ,&\pm {\bigl [}3-{\frac {1}{\varphi }}{\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm \,2\varphi ,&\pm \,{\frac {1}{\varphi }},&\pm \,{\frac {1}{\varphi ^{3}}}&{\Bigl )},\\{\Bigl (}&\pm \,\varphi ,&\pm \,{\frac {1}{\varphi ^{2}}},&\pm {\bigl [}1+{\frac {3}{\varphi }}{\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm \,{\sqrt {5}},&\pm \,2,&\pm \,{\frac {\sqrt {5}}{\varphi }}&{\Bigr )},\\{\Bigl (}&\pm \,{\frac {1}{\varphi }},&\pm \,3,&\pm \,{\frac {2}{\varphi }}&{\Bigr )},\end{array}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/87c7bcee4bc8882f8a020eac8c37db39c7ae393e)
where is the golden ratio.
Related polyhedra
Great disdyakis triacontahedron
| Great disdyakis triacontahedron | |
|---|---|
 | |
| Type | Star polyhedron |
| Face |  |
| Elements | F = 120, E = 180 V = 62 (χ = 2) |
| Symmetry group | Ih, [5,3], *532 |
| Index references | DU68 |
| dual polyhedron | Great truncated icosidodecahedron |
The great disdyakis triacontahedron (or trisdyakis icosahedron) is a nonconvex isohedral polyhedron. It is the dual of the great truncated icosidodecahedron. Its faces are triangles.
Proportions
The triangles have one angle of , one of and one of The dihedral angle equals Part of each triangle lies within the solid, hence is invisible in solid models.
