Compound of ten octahedra (small rhombicosidodecahedron hull)

Compound of ten octahedra (small rhombicosidodecahedron hull)
(3D model) (OFF file)
Rank	3
Type	Uniform
Notation
Bowers style acronym	Gissi
Elements
Components	10 octahedra
Faces	20+60 triangles
Edges	60+60
Vertices	60
Vertex figure	Square, edge length 1
Measures (edge length 1)
Circumradius	${\frac {\sqrt {2}}{2}}\approx 0.70711$
Inradius	${\frac {\sqrt {6}}{6}}\approx 0.40825$
Volume	${\frac {10{\sqrt {2}}}{3}}\approx 4.71405$
Dihedral angle	$\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$
Central density	10
Number of external pieces	600
Level of complexity	40
Related polytopes
Army	Semi-uniform Srid, edge lengths ${\frac {-3+2{\sqrt {2}}+{\sqrt {5}}}{6}}$ (pentagons), ${\frac {{\sqrt {5}}-{\sqrt {2}}}{3}}$ (triangles)
Regiment	Gissi
Dual	Second compound of ten cubes
Conjugate	Compound of ten octahedra (truncated icosahedron hull)
Convex core	Order-6-truncated pentakis dodecahedron
Abstract & topological properties
Flag count	480
Schläfli type	{3,4}
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The great snub icosahedron, gissi, or second compound of ten octahedra is a uniform polyhedron compound. It consists of 20+60 triangles, with 4 triangles joining at each vertex.

Each octahedral component has triangular antiprismatic symmetry. If each component is rotated by 60°, the snub icosahedron, the other uniform compound of ten octahedra, is produced. This compound's convex hull is a semi-uniform variant of the small rhombicosidodecahedron, while the snub icosahedron has a semi-uniform truncated icosahedron as its convex hull instead.

Its quotient prismatic equivalent is the great triangular antiprismatic decayottoorthowedge, which is twelve-dimensional.

Vertex coordinates

The vertices of a great snub icosahedron of edge length 1 are given by all even permutations of:

• $\left(0,\,\pm {\frac {-2-{\sqrt {2}}-2{\sqrt {5}}+{\sqrt {10}}}{12}},\,\pm {\frac {-2+{\sqrt {2}}+2{\sqrt {5}}+{\sqrt {10}}}{12}}\right),$
• $\left(\pm {\frac {6+4{\sqrt {2}}+2{\sqrt {5}}}{24}},\,\pm {\frac {{\sqrt {2}}-{\sqrt {5}}}{6}},\,\pm {\frac {-3+2{\sqrt {2}}+{\sqrt {5}}}{12}}\right),$
• $\left(\pm {\frac {1-{\sqrt {2}}+{\sqrt {5}}+{\sqrt {10}}}{6}},\,\pm {\frac {-1+{\sqrt {2}}+{\sqrt {5}}+{\sqrt {10}}}{12}},\,\pm {\frac {1}{2}}\right).$

External links

• Bowers, Jonathan. "Polyhedron Category C9: Octahedral Continuums" (#62).

• Klitzing, Richard. "gissi".
• Wikipedia contributors. "Compound of ten octahedra".