Octahedral prism

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Octahedral prism
Octahedral prism.png
Bowers style acronymOpe
Coxeter diagramx o4o3x (CDel node 1.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png)
Bracket notation[<III>I]
Cells8 triangular prisms, 2 octahedra
Faces16 triangles, 12 squares
Vertex figureSquare pyramid, edge lengths 1 (base), 2 (legs)
Measures (edge length 1)
Dichoral anglesTrip–4–trip:
 Oct–3–trip: 90°
HeightsOct atop oct: 1
 Trip atop gyro trip:
Central density1
Number of external pieces10
Level of complexity4
Related polytopes
DualCubic tegum
Abstract & topological properties
Flag count384
Euler characteristic0
SymmetryB3×A1, order 96

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).

Gallery[edit | edit source]

Vertex coordinates[edit | edit source]

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

Representations[edit | edit source]

An octahedral prism has the following Coxeter diagrams:

  • x o4o3x (full symmetry)
  • x o3x3o (CDel node 1.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png) (base has A3 symmetry, tetratetrahedral prism)
  • x2s2s3s (CDel node 1.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 3.pngCDel node h.png) (triangular antiprismatic prism)
  • x2s2s6o (CDel node 1.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 6.pngCDel node.png) (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[edit | edit source]

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.

External links[edit | edit source]

  • Klitzing, Richard. "Ope".