Small rhombicuboctahedral prism

Small rhombicuboctahedral prism
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Sircope
Coxeter diagram	x x4o3x ()
Elements
Cells	8 triangular prisms, 6+12 cubes, 2 small rhombicuboctahedra
Faces	16 triangles, 12+24+24+24 squares
Edges	24+48+48
Vertices	48
Vertex figure	Isosceles trapezoidal pyramid, edge lengths 1, √2, √2, √2 (base), √2 (legs)
Measures (edge length 1)
Circumradius	$\sqrt{\frac{3+\sqrt2}{2}} ≈ 1.48563$
Hypervolume	$2\frac{6+5\sqrt2}{3} ≈ 8.71404$
Dichoral angles	Trip–4–cube: $\arccos\left(-\frac{\sqrt6}{3}\right) ≈ 144.73561°$
	Cube–4–cube: 135°
	Sirco–4–cube: 90°
	Sirco–3–trip: 90°
Height	1
Central density	1
Number of external pieces	28
Level of complexity	16
Related polytopes
Army	Sircope
Regiment	Sircope
Dual	Deltoidal icositetrahedral tegum
Conjugate	Quasirhombicuboctahedral prism
Abstract & topological properties
Flag count	1536
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B3×A1, order 96
Convex	Yes
Nature	Tame

The small rhombicuboctahedral prism or sircope is a prismatic uniform polychoron that consists of 2 small rhombicuboctahedra, 6+12 cubes, and 8 triangular prisms. Each vertex joins 1 small rhombicuboctahedron, 1 triangular prism, and 3 cubes. As the name suggests, it is a prism based on the small rhombicuboctahedron. As such it is also a convex segmentochoron (designated K-4.66 on Richard Klitzing's list).

The small rhombicuboctahedral prism can be obtained from the small disprismatotesseractihexadecachoron by removing 2 cube atop small rhombicuboctahedron segmentochora.

Gallery

• 

  Card with cell counts, verf, and cross-sections

• 

  Segmentochoron display, sirco atop sirco

• 

  Net

Vertex coordinates

The vertices of a small rhombicuboctahedral prism of edge length 1 are given by all permutations of the first three coordinates of:

• $\left(±\frac{1+\sqrt2}{2},\,±\frac12,\,±\frac12,\,±\frac12\right).$

Representations

A small rhombicuboctahedral prism has the following Coxeter diagrams:

• x x4o3x (full symmetry)
• x2x4s3s () (bases as snubs)
• xx4oo3xx&#x (bases considered separately)
• xxxx xxxx4oxxo&#xt (BC₂×A₁ symmetry, cube-first)
• xxx wxx xwx xxw&#zx (A₁×A₁×A₁×A₁ symmetry)

Related polychora

The small rhombicuboctahedral prism can be constructed by attaching squarre cupolic prisms to 2 opposite octagonal prisms of the square-octagonal duoprism.

The regiment of the small rhombicuboctahedral prism also includes the small cubicuboctahedral prism and the small rhombihexahedral prism.

External links

• Bowers, Jonathan. "category 19: Prisms" (#923).

• Klitzing, Richard. "Sircope".

• Wikipedia Contributors. "Rhombicuboctahedral prism".