Compound of five cuboctahedra

Compound of five cuboctahedra
(3D model) (OFF file)
Rank	3
Type	Uniform
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	${\frac {25{\sqrt {2}}}{3}}\approx 11.78511$
Dihedral angle	$\arccos \left({\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$
Central density	5
Number of external pieces	260
Level of complexity	14
Related polytopes
Army	Semi-uniform Srid, edge lengths ${\frac {3{\sqrt {2}}-{\sqrt {10}}}{4}}$ (pentagons), ${\frac {\sqrt {2}}{2}}$ (triangles)
Regiment	Arie
Dual	Compound of five rhombic dodecahedra
Conjugate	Compound of five cuboctahedra
Convex core	Rhombic triacontahedron
Abstract & topological properties
Flag count	480
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

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 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:

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

External links

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

• Klitzing, Richard. "arie".
• Wikipedia contributors. "Compound of five cuboctahedra".