Antitruncated dodecahedron

Antitruncated dodecahedron
Rank	3
Type	Semi-uniform
Elements
Faces	20 triangles, 12 pentapods
Edges	60+30
Vertices	60
Measures (edge length 1)
Central density	1
Related polytopes
Army	Srid
Dual	Antiapiculated icosahedron
Convex core	Dodecahedron
Abstract & topological properties
Orientable	Yes
Properties
Convex	No
Nature	Tame

The antitruncated dodecahedron or inflected truncated dodecahedron is a semi-uniform polyhedron. It consists of 20 triangles and 12 pentapods, and has 1 triangle and 2 pentapods at a vertex. It has an isosceles triangle as a vertex figure. It is the antitruncation of the dodecahedron and also a faceting of the small rhombicosidodecahedron.

Vertex coordinates

The antitruncated dodecahedron shares its vertices with the small rhombicosidodecahedron, being all permutations of

• $\left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),$

along with all even permutations of

• $\left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right).$

Variations

The antitruncated dodecahedron is part of a teepee where you continue to antitruncate the dodecahedron without the vertices crossing.