Great truncated icosidodecahedron

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, .

Cartesian coordinates for the vertices of a great truncated icosidodecahedron centered at the origin are all the even permutations of

φ ,

1 φ

2 φ ,

1 φ

φ ,

3 φ

{\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}}}

is the golden ratio.

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.

The triangles have one angle of

arccos ⁡

The dihedral angle equals

arccos ⁡

Part of each triangle lies within the solid, hence is invisible in solid models.

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