# Compound of five truncated cubes

Compound of five truncated cubes
Rank3
TypeUniform
Notation
Bowers style acronymTar
Elements
Components5 truncated cubes
Faces40 triangles as 20 hexagrams, 30 octagons
Edges60+120
Vertices120
Vertex figureIsosceles triangle, edge lengths 1. 2+2, 2+2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {7+4{\sqrt {2}}}}{2}}\approx 1.77882}$
Volume${\displaystyle 35{\frac {3+2{\sqrt {2}}}{3}}\approx 67.99832}$
Dihedral angles8–3: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
8–8: 90°
Central density5
Number of external pieces380
Level of complexity26
Related polytopes
ArmySemi-uniform Grid, edge lengths ${\displaystyle {\frac {{\sqrt {10}}-{\sqrt {2}}}{4}}}$ (dipentagon-ditrigon), ${\displaystyle {\frac {-2-3{\sqrt {2}}+2{\sqrt {5}}+{\sqrt {10}}}{4}}}$ (dipentagon-rectangle), ${\displaystyle {\frac {\sqrt {2}}{2}}}$ (ditrigon-rectangle)
RegimentTar
DualCompound of five triakis octahedra
ConjugateCompound of five quasitruncated hexahedra
Convex coreRhombic triacontahedron
Abstract & topological properties
Flag count720
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The truncated rhombihedron, hyperhombicosicosahedron, tar, or compound of five truncated cubes is a uniform polyhedron compound. It consists of 40 triangles (which form coplanar pairs combining into 20 hexagrams) and 30 octagons, with one triangle and two octagons joining at each vertex. As the name suggests, it can be derived as the truncation of the rhombihedron, the compound of five cubes.

Its quotient prismatic equivalent is the truncated cubic pentachoroorthowedge, which is seven-dimensional.

## Vertex coordinates

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

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

along with all even permutations of:

• ${\displaystyle \left(\pm {\frac {{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {-2-{\sqrt {2}}+2{\sqrt {5}}+{\sqrt {10}}}{8}},\,\pm {\frac {1+{\sqrt {5}}+{\sqrt {10}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {2}}{4}},\,\pm {\frac {-2-3{\sqrt {2}}+2{\sqrt {5}}+{\sqrt {10}}}{8}},\,\pm {\frac {2+3{\sqrt {2}}+2{\sqrt {5}}+{\sqrt {10}}}{8}}\right),}$
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {2}}}{4}},\,\pm {\frac {4+3{\sqrt {2}}-{\sqrt {10}}}{8}},\,\pm {\frac {4+3{\sqrt {2}}+{\sqrt {10}}}{8}}\right),}$
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {2}}+2{\sqrt {5}}+{\sqrt {10}}}{8}},\,\pm {\frac {{\sqrt {2}}-{\sqrt {10}}}{8}},\,\pm {\frac {-1+{\sqrt {5}}+{\sqrt {10}}}{4}}\right).}$