Octahedral prism

Octahedral prism
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Ope
Coxeter diagram	x o4o3x ()
Bracket notation	[<III>I]
Elements
Cells	8 triangular prisms, 2 octahedra
Faces	16 triangles, 12 squares
Edges	6+24
Vertices	12
Vertex figure	Square pyramid, edge lengths 1 (base), √2 (legs)
Measures (edge length 1)
Circumradius	$\frac{\sqrt3}{2} ≈ 0.86603$
Hypervolume	$\frac{\sqrt2}{3} ≈ 0.47141$
Dichoral angles	Trip–4–trip: $\arccos\left(-\frac13\right) ≈ 109.47122°$
	Oct–3–trip: 90°
Heights	Oct atop oct: 1
	Trip atop gyro trip: $\frac{\sqrt6}{3} ≈ 0.81650$
Central density	1
Number of pieces	10
Level of complexity	4
Related polytopes
Army	Ope
Regiment	Ope
Dual	Cubic tegum
Conjugate	None
Abstract properties
Flag count	384
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	B3×A1, order 96
Convex	Yes
Nature	Tame
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).

Gallery

• 

  Card with cell counts, verf, and cross-sections

• 

  Segmentochoron display, oct atop oct

• 

  Segmentochoron display, trip atop gyro trip

• 

  Net

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:

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

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.

External links

• Bowers, Jonathan. "Category 19: Prisms" (#890).

• Klitzing, Richard. "Ope".

• Wikipedia Contributors. "Octahedral prism".
• Hi.gher.Space Wiki Contributors. "Octahedral prism".