# Compound of five great cubicuboctahedra

Rhombiquasihyperhomb-icosicosahedron
Rank3
TypeUniform
Notation
Bowers style acronymRaquahri
Elements
Components5 great cubicuboctahedra
Faces40 triangles as 20 hexagrams, 30 squares, 30 octagrams
Edges120+120
Vertices120
Vertex figureIsosceles trapezoid, edge lengths 1, 2–2, 2, 2–2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {5-2{\sqrt {2}}}}{2}}\approx 0.73681}$
Volume${\displaystyle 10{\frac {4{\sqrt {2}}-3}{3}}\approx 8.85618}$
Dihedral angles8/3–3: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
8/3–4: 90°
Central density20
Number of external pieces840
Level of complexity56
Related polytopes
ArmySemi-uniform Grid, edge lengths ${\displaystyle {\frac {-2+{\sqrt {2}}+2{\sqrt {5}}-{\sqrt {10}}}{4}}}$ (dipentagon-ditrigon), ${\displaystyle {\frac {-4+{\sqrt {2}}+{\sqrt {10}}}{4}}}$ (dipentagon-rectangle), ${\displaystyle {\frac {2-{\sqrt {2}}}{2}}}$ (ditrigon-rectangle)
RegimentRaquahri
DualCompound of five great hexacronic icositetrahedra
ConjugateCompound of five small cubicuboctahedra
Convex coreRhombic triacontahedron
Abstract & topological properties
Flag count960
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The rhombiquasihyperhombicosicosahedron, raquahri, or compound of five great cubicuboctahedra is a uniform polyhedron compound. It consists of 40 triangles (which form coplanar pairs combining into 20 hexagrams), 30 squares, and 30 octagrams, with one triangle, one square, and two octagrams joining at each vertex.

## Vertex coordinates

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

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