Rhombiquasihyperhombicosicosahedron

Rhombiquasihyperhomb-icosicosahedron
	(3D model); (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Raquahri
Elements
Components	5 great cubicuboctahedra
Faces	40 triangles as 20 hexagrams, 30 squares, 30 octagrams
Edges	120+120
Vertices	120
Vertex figure	Isosceles trapezoid, edge lengths 1, √2–√2, √2, √2–√2
Measures (edge length 1)
Circumradius
Volume
Dihedral angles	8/3–3:
	8/3–4: 90°
Central density	20
Number of external pieces	840
Level of complexity	56
Related polytopes
Army	Semi-uniform Grid
Regiment	Raquahri
Dual	Compound of five great hexacronic icositetrahedra
Conjugate	Rhombihyperhombicosicosahedron
Convex core	Rhombic triacontahedron
Abstract & topological properties
Flag count	960
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The rhombiquasihyperhombicosicosahedron, raquahri, or compound of five great cubicuboctahedra is a uniform polyhedron compound. It consists of 40 triangles (which form coplanar pairs combining into 20 hexagrams), 30 squares, and 30 octagrams, with one triangle, one square, and two octagrams joining at each vertex.

Gallery[edit | edit source]

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

$\left(\pm\frac{\sqrt2-1}{2},\,\pm\frac12,\,\pm\frac12\right),$
$\left(\pm\frac{\sqrt{10}-\sqrt2}{8},\,\pm\frac{2-\sqrt2+2\sqrt5–\sqrt{10}}{8},\,\pm\frac{1-\sqrt2-\sqrt5}{4}\right),$
$\left(\pm\frac{2-\sqrt2}{4},\,\pm\frac{4-\sqrt2-\sqrt{10}}{8},\,\pm\frac{4-\sqrt2+\sqrt{10}}{8}\right),$
$\left(\pm\frac{\sqrt2}{4},\,\pm\frac{-2-\sqrt2+2\sqrt5-\sqrt{10}}{8},\,\pm\frac{2+\sqrt2+\sqrt5-\sqrt{10}}{8}\right),$
$\left(\pm\frac{\sqrt2+\sqrt{10}}{8},\,\pm\frac{-2+\sqrt2+2\sqrt5-\sqrt{10}}{8},\,\pm\frac{1-\sqrt2+\sqrt5}{4}\right).$