# Compound of five quasitruncated hexahedra

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Compound of five quasitruncated hexahedra
Rank3
TypeUniform
Notation
Bowers style acronymQuitar
Elements
Components5 quasitruncated hexahedra
Faces40 triangles as 20 hexagrams, 30 octagrams
Edges60+120
Vertices120
Vertex figureIsosceles triangle, edge lengths 1. 2–2, 2–2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {7-4{\sqrt {2}}}}{2}}\approx 0.57947}$
Volume${\displaystyle 35{\frac {3-2{\sqrt {2}}}{3}}\approx 2.00168}$
Dihedral angles8/3–8/3: 90°
8/3–3: ${\displaystyle \arccos \left({\frac {\sqrt {3}}{3}}\right)\approx 54.73561^{\circ }}$
Central density35
Number of external pieces1320
Level of complexity86
Related polytopes
ArmySemi-uniform Grid, edge lengths ${\displaystyle {\frac {-2+3{\sqrt {2}}-2{\sqrt {5}}+{\sqrt {10}}}{4}}}$ (dipentagon-ditrigon), ${\displaystyle {\frac {1+{\sqrt {5}}-{\sqrt {10}}}{2}}}$ (dipentagon-rectangle), ${\displaystyle {\frac {-2+{\sqrt {2}}+2{\sqrt {5}}-{\sqrt {10}}}{4}}}$ (ditrigon-rectangle)
RegimentQuitar
DualCompound of five great triakis octahedra
ConjugateCompound of five truncated cubes
Convex coreIcosahedron
Abstract & topological properties
Flag count720
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

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

## Vertex coordinates

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

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

along with all even permutations of:

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