Antirhombicosicosahedron

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.

Antirhombicosicosahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Arie
Elements
Components	5 cuboctahedra
Faces	40 triangles as 20 hexagrams, 30 squares
Edges	120
Vertices	60
Vertex figure	Rectangle, edge lengths 1 and √2
Measures (edge length 1)
Circumradius	1
Volume	$\frac{25\sqrt2}{3} \approx 11.78511$
Dihedral angle	$\arccos\left(\frac{\sqrt3}{3}\right) \approx 125.26439^\circ$
Central density	5
Number of external pieces	260
Level of complexity	14
Related polytopes
Army	Semi-uniform Srid
Regiment	Arie
Dual	Compound of five rhombic dodecahedra
Conjugate	Antirhombicosicosahedron
Convex core	Rhombic triacontahedron
Abstract & topological properties
Flag count	480
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

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.

Gallery

Vertex coordinates

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

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

External links

• Bowers, Jonathan. "Polyhedron Category C3: Fivers" (#12).

• Klitzing, Richard. "arie".

• Wikipedia Contributors. "Compound of five cuboctahedra".