Compound of five small rhombicuboctahedra

Compound of five small rhombicuboctahedra
(3D model) (OFF file)
Rank	3
Type	Uniform
Notation
Bowers style acronym	Rasseri
Elements
Components	5 small rhombicuboctahedra
Faces	40 triangles as 20 hexagrams, 30+60 squares
Edges	120+120
Vertices	120
Vertex figure	Isosceles trapezoid, edge lengths 1, √2, √2, √2
Measures (edge length 1)
Circumradius	${\frac {\sqrt {5+2{\sqrt {2}}}}{2}}\approx 1.39897$
Volume	$10{\frac {6+5{\sqrt {2}}}{3}}\approx 43.57023$
Dihedral angles	4–3: $\arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }$
	4–4: 135°
Central density	5
Number of external pieces	1080
Level of complexity	68
Related polytopes
Army	Semi-uniform Grid, edge lengths ${\frac {4+{\sqrt {2}}-{\sqrt {10}}}{4}}$ (dipentagon-ditrigon), ${\frac {1+{\sqrt {2}}-{\sqrt {5}}}{2}}$ (dipentagon-rectangle), ${\frac {{\sqrt {10}}-{\sqrt {2}}}{4}}$ (ditrigon-rectangle)
Regiment	Rasseri
Dual	Compound of five deltoidal icositetrahedra
Conjugate	Compound of five quasirhombicuboctahedra
Abstract & topological properties
Flag count	960
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The rhombisnub rhombicosicosahedron, rasseri, or compound of five small rhombicuboctahedra is a uniform polyhedron compound. It consists of 40 triangles (which form coplanar pairs combining into 20 hexagrams) and 30+60 squares, with one triangle and three squares joining at each vertex. It can be seen as the cantellation of the rhombihedron.

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

Gallery

Vertex coordinates

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

• $\left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{8}},\,\pm {\frac {2+{\sqrt {2}}+2{\sqrt {5}}+{\sqrt {10}}}{8}},\,\pm {\frac {1+{\sqrt {2}}-{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {2+{\sqrt {2}}}{4}},\,\pm {\frac {4+{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {4+{\sqrt {2}}-{\sqrt {10}}}{8}}\right),$
• $\left(\pm {\frac {\sqrt {2}}{4}},\,\pm {\frac {-2+{\sqrt {2}}+2{\sqrt {5}}+{\sqrt {10}}}{8}},\,\pm {\frac {2-{\sqrt {2}}+{\sqrt {5}}+{\sqrt {10}}}{8}}\right),$
• $\left(\pm {\frac {{\sqrt {2}}+{\sqrt {10}}}{8}},\,\pm {\frac {-2-{\sqrt {2}}+2{\sqrt {5}}+{\sqrt {10}}}{8}},\,\pm {\frac {1+{\sqrt {2}}+{\sqrt {5}}}{4}}\right).$

External links

• Bowers, Jonathan. "Polyhedron Category C3: Fivers" (#15).

• Klitzing, Richard. "rasseri".
• Wikipedia contributors. "Compound of five rhombicuboctahedra".