# Compound of ten truncated tetrahedra

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Compound of ten truncated tetrahedra
Rank3
TypeUniform
Notation
Bowers style acronymTe
Elements
Components10 truncated tetrahedra
Faces40 triangles as 20 hexagrams, 40 hexagons as 20 stellated dodecagons
Edges60+120
Vertices120
Vertex figureIsosceles triangle, edge lengths 1. 3, 3
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {22}}{4}}\approx 1.17260}$
Volume${\displaystyle {\frac {115{\sqrt {2}}}{6}}\approx 27.10576}$
Dihedral angles3–6: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
6–6: ${\displaystyle \arccos \left({\frac {1}{3}}\right)\approx 70.52878^{\circ }}$
Central density10
Number of external pieces380
Level of complexity26
Related polytopes
ArmySemi-uniform Grid, edge lengths ${\displaystyle {\frac {3{\sqrt {2}}-{\sqrt {10}}}{4}}}$ (decagons), ${\displaystyle {\frac {{\sqrt {10}}-{\sqrt {2}}}{4}}}$ (ditrigon-rectangle)
RegimentTe
DualCompound of ten triakis tetrahedra
ConjugateCompound of ten truncated tetrahedra
Convex coreIcosahedron
Abstract & topological properties
Flag count720
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The truncated icosicosahedron, te, or compound of ten truncated tetrahedra is a uniform polyhedron compound. It consists of 40 triangles (which form coplanar pairs combining into 20 hexagrams) and 40 hexagons, with one triangle and two hexagons joining at each vertex. As the name suggests, it can be derived as the truncation of the icosicosahedron, the compound of ten tetrahedra. It can alternatively be constructed as the compound of the two chiral forms of the truncated chiricosahedron.

Its quotient prismatic equivalent is the truncated tetrahedral decayottoorthowedge, which is twelve-dimensional.

## Vertex coordinates

The vertices of a truncated icosicosahedron of edge length 1 can be given by all even permutations of:

• ${\displaystyle \left(\pm {\frac {3{\sqrt {2}}}{4}},\,\pm {\frac {\sqrt {2}}{4}},\,\pm {\frac {\sqrt {2}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{8}},\,\pm {\frac {3{\sqrt {2}}-{\sqrt {10}}}{8}},\,\pm {\frac {{\sqrt {2}}+{\sqrt {10}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{4}},\,\pm {\frac {3{\sqrt {2}}+{\sqrt {10}}}{8}}\right),}$
• ${\displaystyle \left(\pm {\frac {3{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {3{\sqrt {2}}-{\sqrt {10}}}{8}},\,\pm {\frac {\sqrt {2}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {10}}{4}},\,\pm {\frac {\sqrt {2}}{4}},\,\pm {\frac {\sqrt {10}}{4}}\right).}$