# Great rhombicuboctahedral prism

Great rhombicuboctahedral prism Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymGircope
Coxeter diagramx x4x3x (       )
Elements
Cells12 cubes, 8 hexagonal prisms, 6 octagonal prisms, 2 great rhombicuboctahedra
Faces24+24+24+24 squares, 16 hexagons, 12 octagons
Edges48+48+48+48
Vertices96
Vertex figureIrregular tetrahedron, edge lengths 2, 3, 2+2 (base), 2 (legs)
Measures (edge length 1)
Circumradius$\sqrt{\frac{7+3\sqrt2}{2}} ≈ 2.37093$ Hypervolume$2(11+7\sqrt2) ≈ 41.79899$ Dichoral anglesCube–4–hip: $\arccos\left(-\frac{\sqrt6}{3}\right) ≈ 144.73561°$ Cube–4–op: 135°
Hip–4–op: $\arccos\left(-\frac{\sqrt3}{3}\right) ≈ 125.26439°$ Girco–8–op: 90°
Girco–6–hip: 90°
Girco–4–cube: 90°
Height1
Central density1
Number of external pieces28
Level of complexity24
Related polytopes
ArmyGircope
RegimentGircope
DualDisdyakis dodecahedral tegum
ConjugateQuasitruncated cuboctahedral prism
Abstract & topological properties
Flag count2304
Euler characteristic0
OrientableYes
Properties
SymmetryB3×A1, order 96
ConvexYes
NatureTame

The great rhombicuboctahedral prism, or gircope, is a prismatic uniform polychoron that consists of 2 great rhombicuboctahedra, 6 octagonal prisms, 8 hexagonal prisms, and 12 cubes. Each vertex joins one of each type of cell. As the name suggests, it is a prism based on the great rhombicuboctahedron. As such it is also a convex segmentochoron (designated K-4.125 on Richard Klitzing's list).

The great rhombicuboctahedral prism can be obtained as the central segment of the prismatorhombated tesseract in rhombicuboctahedron-first orientation.

This polychoron can be alternated into a snub cubic antiprism, although it cannot be made uniform. The octagons can also be alternated into long rectangles to create a pyritosnub alterprism, which is also nonuniform.

## Vertex coordinates

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

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

The great rhombicuboctahedral prism has the following Coxeter diagrams:

• x x4x3x (full symmetry)
• xx4xx3xx&#x (bases considered separately)
• xxxxxx xuxxux4xxwwxx&#xt (BC2×A1 symmetry, octagonal prism-first)