Great stellated truncated dodecahedron

In geometry, the great stellated truncated dodecahedron (or quasitruncated great stellated dodecahedron or great stellatruncated dodecahedron) is a nonconvex uniform polyhedron, indexed as U₆₆. It has 32 faces (20 triangles and 12 decagrams), 90 edges, and 60 vertices.^[1] It is given a Schläfli symbol t_0,1{5/3,3}.

Great stellated truncated dodecahedron
Type	Uniform star polyhedron
Elements	F = 32, E = 90 V = 60 (χ = 2)
Faces by sides	20{3}+12{10/3}
Coxeter diagram
Wythoff symbol	2 3 | 5/3
Symmetry group	Ih, [5,3], *532
Index references	U66, C83, W104
Dual polyhedron	Great triakis icosahedron
Vertex figure	3.10/3.10/3
Bowers acronym	Quit Gissid

3D model of a great stellated truncated dodecahedron

Related polyhedra

It shares its vertex arrangement with three other uniform polyhedra: the small icosicosidodecahedron, the small ditrigonal dodecicosidodecahedron, and the small dodecicosahedron:

Great stellated truncated dodecahedron

Small icosicosidodecahedron

Small ditrigonal dodecicosidodecahedron

Small dodecicosahedron

Cartesian coordinates

Cartesian coordinates for the vertices of a great stellated truncated dodecahedron are all the even permutations of $${\displaystyle {\begin{array}{crclc}{\Bigl (}&0,&\pm \,\varphi ,&\pm {\bigl [}2-{\frac {1}{\varphi }}{\bigr ]}&{\Bigr )}\\{\Bigl (}&\pm \,\varphi ,&\pm \,{\frac {1}{\varphi }},&\pm \,{\frac {2}{\varphi }}&{\Bigr )}\\{\Bigl (}&\pm \,{\frac {1}{\varphi ^{2}}},&\pm \,{\frac {1}{\varphi }},&\pm \,2&{\Bigr )}\end{array}}}$$

where ${\displaystyle \varphi ={\tfrac {1+{\sqrt {5}}}{2}}}$ is the golden ratio.

See also

• List of uniform polyhedra