# Cuboctahedral prism

The cuboctahedral prism or cope is a prismatic uniform polychoron that consists of 2 cuboctahedra, 6 cubes and 8 triangular prisms. Each vertex joins 1 cuboctahedron, 2 cubes, and 2 triangular prisms. As the name suggests, it is a prism based on the cuboctahedron. As such it is also a convex segmentochoron (designated K-4.43 on Richard Klitzing's list).

Cuboctahedral prism
Rank4
TypeUniform
Notation
Bowers style acronymCope
Coxeter diagramx o4x3o ()
Elements
Cells8 triangular prisms, 6 cubes, 2 cuboctahedra
Faces16 triangles, 12+24 squares
Edges12+48
Vertices24
Vertex figureRectangular pyramid, edge lengths 1, 2 (base), 2 (legs)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {5}}{2}}\approx 1.11803}$
Hypervolume${\displaystyle {\frac {5{\sqrt {2}}}{3}}\approx 2.35702}$
Dichoral anglesCube–4–trip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Co–4–cube: 90°
co–3–trip: 90°
Height1
Central density1
Number of external pieces16
Level of complexity8
Related polytopes
ArmyCope
RegimentCope
DualRhombic dodecahedral tegum
ConjugateNone
Abstract & topological properties
Flag count768
Euler characteristic0
OrientableYes
Properties
SymmetryB3×A1, order 96
Flag orbits8
ConvexYes
NatureTame

## Vertex coordinates

The vertices of a cuboctahedral prism of edge length 1 are given by all permutations and sign changes of the first three coordinates of:

• ${\displaystyle \left(\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,\pm {\frac {1}{2}}\right).}$

## Representations

A cuboctahedral prism has the following Coxeter diagrams:

• x o4x3o (full symmetry)
• x x3o3x (       ) (rhombitetratetrahedral prism)
• s2s4x3o (       ) (bases as rhombitetratetrahedra)
• oo4xx3oo&#x (bases considered separately)
• xx3oo3xx&#x (rhombitetratetrahedral bases considered separately)
• xxx xox4oqo&#xt (BC2×A1 axial, cube-first)
• xxx xxo3oxx&#xt (A2×A1 axial, triangular prism-first)
• xxx qqo qoq oqq&#zx (A1×A1×A1×A1 symmetry)

## Related polychora

A cuboctahedral prism can be cut in half to produce two triangular cupolic prisms with the base triangular prisms in rotated orientations.

The regiment of the cuboctahedral prism also includes the octahemioctahedral prism and the cubohemioctahedral prism.