# Great rhombicuboctahedral prism

Great rhombicuboctahedral prism
Rank4
TypeUniform
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${\displaystyle {\sqrt {\frac {7+3{\sqrt {2}}}{2}}}\approx 2.37093}$
Hypervolume${\displaystyle 2(11+7{\sqrt {2}})\approx 41.79899}$
Dichoral anglesCube–4–hip: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
Cube–4–op: 135°
Hip–4–op: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
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 (OBSA: 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:

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

## 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)