# Compound of five cuboctahedra

(Redirected from Antirhombicosicosahedron)
Compound of five cuboctahedra
Rank3
TypeUniform
Notation
Bowers style acronymArie
Elements
Components5 cuboctahedra
Faces40 triangles as 20 hexagrams, 30 squares
Edges120
Vertices60
Vertex figureRectangle, edge lengths 1 and 2
Measures (edge length 1)
Volume${\displaystyle {\frac {25{\sqrt {2}}}{3}}\approx 11.78511}$
Dihedral angle${\displaystyle \arccos \left({\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Central density5
Number of external pieces260
Level of complexity14
Related polytopes
ArmySemi-uniform Srid, edge lengths ${\displaystyle {\frac {3{\sqrt {2}}-{\sqrt {10}}}{4}}}$ (pentagons), ${\displaystyle {\frac {\sqrt {2}}{2}}}$ (triangles)
RegimentArie
DualCompound of five rhombic dodecahedra
ConjugateCompound of five cuboctahedra
Convex coreRhombic triacontahedron
Abstract & topological properties
Flag count480
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The antirhombicosicosahedron, arie, or compound of five cuboctahedra is a uniform polyhedron compound. It consists of 40 triangles (which form coplanar pairs combining into 20 hexagrams) and 30 squares, with two of each joining at a vertex.

It can be thought of as a rectification of either the small icosicosahedron or the rhombihedron, or the cantellation of the chiricosahedron. Each individual component has pyritohedral symmetry.

Its quotient prismatic equivalent is the cuboctahedral pentagyroprism, which is seven-dimensional.

## Vertex coordinates

The vertices of an antirhombicosicosahedron of edge length 1 can be given by all even permutations of:

• ${\displaystyle \left(\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,0\right),}$
• ${\displaystyle \left(\pm {\frac {{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{8}},\,\pm {\frac {\sqrt {10}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {2}}{4}},\,\pm {\frac {3{\sqrt {2}}-{\sqrt {10}}}{8}},\,\pm {\frac {3{\sqrt {2}}+{\sqrt {10}}}{8}}\right).}$