# Compound of five small rhombicuboctahedra

Compound of five small rhombicuboctahedra
Rank3
TypeUniform
Notation
Bowers style acronymRasseri
Elements
Components5 small rhombicuboctahedra
Faces40 triangles as 20 hexagrams, 30+60 squares
Edges120+120
Vertices120
Vertex figureIsosceles trapezoid, edge lengths 1, 2, 2, 2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {5+2{\sqrt {2}}}}{2}}\approx 1.39897}$
Volume${\displaystyle 10{\frac {6+5{\sqrt {2}}}{3}}\approx 43.57023}$
Dihedral angles4–3: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
4–4: 135°
Central density5
Number of external pieces1080
Level of complexity68
Related polytopes
ArmySemi-uniform Grid, edge lengths ${\displaystyle {\frac {4+{\sqrt {2}}-{\sqrt {10}}}{4}}}$ (dipentagon-ditrigon), ${\displaystyle {\frac {1+{\sqrt {2}}-{\sqrt {5}}}{2}}}$ (dipentagon-rectangle), ${\displaystyle {\frac {{\sqrt {10}}-{\sqrt {2}}}{4}}}$ (ditrigon-rectangle)
RegimentRasseri
DualCompound of five deltoidal icositetrahedra
ConjugateCompound of five quasirhombicuboctahedra
Abstract & topological properties
Flag count960
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The rhombisnub rhombicosicosahedron, rasseri, or compound of five small rhombicuboctahedra is a uniform polyhedron compound. It consists of 40 triangles (which form coplanar pairs combining into 20 hexagrams) and 30+60 squares, with one triangle and three squares joining at each vertex. It can be seen as the cantellation of the rhombihedron.

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

## Vertex coordinates

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

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