# Antirhombicosicosahedron

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Antirhombicosicosahedron
Rank3
TypeUniform
SpaceSpherical
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)
Circumradius1
Volume${\displaystyle \frac{25\sqrt2}{3} \approx 11.78511}$
Dihedral angle${\displaystyle \arccos\left(\frac{\sqrt3}{3}\right) \approx 125.26439^\circ}$
Central density5
Number of external pieces260
Level of complexity14
Related polytopes
ArmySemi-uniform Srid
RegimentArie
DualCompound of five rhombic dodecahedra
ConjugateAntirhombicosicosahedron
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.

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

## Vertex coordinates

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

• ${\displaystyle \left(\pm\frac{\sqrt2}{2},\,\pm\frac{\sqrt2}{2},\,0\right),}$
• ${\displaystyle \left(\pm\frac{\sqrt2+\sqrt{10}}{8},\,\pm\frac{\sqrt{10}-\sqrt2}{8},\,\pm\frac{\sqrt{10}}{4}\right),}$
• ${\displaystyle \left(\pm\frac{\sqrt2}{4},\,\pm\frac{3\sqrt2-\sqrt{10}}{8},\,\pm\frac{3\sqrt2+\sqrt{10}}{8}\right).}$