# Octahedral prism

Octahedral prism
Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymOpe
Coxeter diagramx o4o3x ()
Bracket notation[<III>I]
Elements
Cells8 triangular prisms, 2 octahedra
Faces16 triangles, 12 squares
Edges6+24
Vertices12
Vertex figureSquare pyramid, edge lengths 1 (base), 2 (legs)
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt3}{2} ≈ 0.86603}$
Hypervolume${\displaystyle \frac{\sqrt2}{3} ≈ 0.47141}$
Dichoral anglesTrip–4–trip: ${\displaystyle \arccos\left(-\frac13\right) ≈ 109.47122°}$
Oct–3–trip: 90°
HeightsOct atop oct: 1
Trip atop gyro trip: ${\displaystyle \frac{\sqrt6}{3} ≈ 0.81650}$
Central density1
Number of pieces10
Level of complexity4
Related polytopes
ArmyOpe
RegimentOpe
DualCubic tegum
ConjugateNone
Abstract properties
Flag count384
Euler characteristic0
Topological properties
OrientableYes
Properties
SymmetryB3×A1, order 96
ConvexYes
NatureTame
Discovered by{{{discoverer}}}

The octahedral prism or ope is a prismatic uniform polychoron that consists of 2 octahedra and 8 triangular prisms. Each vertex joins 1 octahedron and 4 triangular prisms. It is a prism based on the octahedron. As such it is also a convex segmentochoron (designated K-4.11 on Richard Klitzing's list).

## Vertex coordinates

Coordinates for the vertices of an octahedral prism of edge length 1 are given by all permutations of the first three coordinates of:

• ${\displaystyle \left(0,\,0,\,±\frac{\sqrt2}{2},\,±\frac12\right).}$

## Representations

An octahedral prism has the following Coxeter diagrams:

• x o4o3x (full symmetry)
• x o3x3o () (base has A3 symmetry, tetratetrahedral prism)
• x2s2s3s () (triangular antiprismatic prism)
• x2s2s6o () (base has G2×A1+ symmetry)
• oo4oo3xx&#x (bases considered separately)
• oo3xx3oo&#x (bases considered in tet symmetry)
• xx xo3ox&#x (A2×A1 axial. trip atop gyro trip)
• xxx oxo4ooo&#xt (BC2×A1 symmetry, as square bipyramidal prism)
• xxx oxo oxo&#xt (A1×A1×A1 symmetry, as rectangular bipyramidal prism)
• xxx xox oqo&#xt (A1×A1×A1 axial, bases are edge-first)
• xxx qoo oqo ooq&#zx (A1×A1×A1×A1 symmetry, as rhombic bipyramidal prism)
• xx qo ox4oo&#zx (BC2×A1×A1 symmetry)

## Related polychora

An octahedral prism can be cut into 2 square pyramidal prisms joining at a common cubic cell. If one half is rotated the result is instead a dyadic gyrotegmipucofastegium, which is also a segmentochoron.

The regiment of the octahedral prism also includes the tetrahemihexahedral prism.