Small rhombicuboctahedral prism

Small rhombicuboctahedral prism
Rank4
TypeUniform
Notation
Bowers style acronymSircope
Coxeter diagramx x4o3x ()
Elements
Cells8 triangular prisms, 6+12 cubes, 2 small rhombicuboctahedra
Faces16 triangles, 12+24+24+24 squares
Edges24+48+48
Vertices48
Vertex figureIsosceles trapezoidal pyramid, edge lengths 1, 2, 2, 2 (base), 2 (legs)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {3+{\sqrt {2}}}{2}}}\approx 1.48563}$
Hypervolume${\displaystyle 2{\frac {6+5{\sqrt {2}}}{3}}\approx 8.71404}$
Dichoral anglesTrip–4–cube: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
Cube–4–cube: 135°
Sirco–4–cube: 90°
Sirco–3–trip: 90°
Height1
Central density1
Number of external pieces28
Level of complexity16
Related polytopes
ArmySircope
RegimentSircope
DualDeltoidal icositetrahedral tegum
ConjugateQuasirhombicuboctahedral prism
Abstract & topological properties
Flag count1536
Euler characteristic0
OrientableYes
Properties
SymmetryB3×A1, order 96
ConvexYes
NatureTame

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.

Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right).}$

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.