Antitruncated icosahedron

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

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

Vertex coordinates

The antitruncated icosahedron 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

Continuing to antitruncate the icosahedron without letting vertices cross will form a teepee.