Octahedral prism

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. As the name suggests, it is a prism based on the octahedron. As such it is also a 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:


 * $$\left(0,\,0,\,±\frac{\sqrt3}{3},\,±\frac12\right).$$

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.