Icositruncated dodecadodecahedron

In geometry, the icositruncated dodecadodecahedron or icosidodecatruncated icosidodecahedron is a nonconvex uniform polyhedron, indexed as U45.

Its convex hull is a nonuniform truncated icosidodecahedron.

Cartesian coordinates for the vertices of an icositruncated dodecadodecahedron are all the even permutations of

2 + φ

φ

3 φ − 1

2 φ

2 φ

φ

φ

{\displaystyle {\begin{array}{crrlc}{\Bigl (}&\pm {\bigl [}2-{\frac {1}{\varphi }}{\bigr ]},&\pm \,1,&\pm {\bigl [}2+\varphi {\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm \,1,&\pm \,{\frac {1}{\varphi ^{2}}},&\pm {\bigl [}3\varphi -1{\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm \,2,&\pm \,{\frac {2}{\varphi }},&\pm \,2\varphi &{\Bigr )},\\{\Bigl (}&\pm \,3,&\pm \,{\frac {1}{\varphi ^{2}}},&\pm \,\varphi ^{2}&{\Bigr )},\\{\Bigl (}&\pm \,\varphi ^{2},&\pm \,1,&\pm {\bigl [}3\varphi -2{\bigr ]}&{\Bigr )},\end{array}}}

is the golden ratio.

The tridyakis icosahedron is the dual polyhedron of the icositruncated dodecadodecahedron.

It has 44 vertices, 180 edges, and 120 scalene triangular faces.

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