Cuboctahedral prism

Cuboctahedral prism
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Cope
Coxeter diagram	x o4x3o ()
Elements
Cells	8 triangular prisms, 6 cubes, 2 cuboctahedra
Faces	16 triangles, 12+24 squares
Edges	12+48
Vertices	24
Vertex figure	Rectangular pyramid, edge lengths 1, √2 (base), √2 (legs)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {5}}{2}}\approx 1.11803$
Hypervolume	${\frac {5{\sqrt {2}}}{3}}\approx 2.35702$
Dichoral angles	Cube–4–trip: $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$
	Co–4–cube: 90°
	co–3–trip: 90°
Height	1
Central density	1
Number of external pieces	16
Level of complexity	8
Related polytopes
Army	Cope
Regiment	Cope
Dual	Rhombic dodecahedral tegum
Conjugate	None
Abstract & topological properties
Flag count	768
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B3×A1, order 96
Flag orbits	8
Convex	Yes
Nature	Tame

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).

Gallery

• 
  Card with cell counts, verf, and cross-sections
• 
  Segmentochoron display, co atop co
• 
  Net

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:

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

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.

External links

• Bowers, Jonathan. "Category 19: Prisms" (#908).

• Klitzing, Richard. "Cope".
• Wikipedia contributors. "Cuboctahedral prism".