Octahedral prism
Octahedral prism  

Rank  4 
Type  Uniform 
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  
Hypervolume  
Dichoral angles  Trip–4–trip: 
Oct–3–trip: 90°  
Heights  Oct atop oct: 1 
Trip atop gyro trip:  
Central density  1 
Number of external pieces  10 
Level of complexity  4 
Related polytopes  
Army  Ope 
Regiment  Ope 
Dual  Cubic tegum 
Conjugate  None 
Abstract & topological properties  
Flag count  384 
Euler characteristic  0 
Orientable  Yes 
Properties  
Symmetry  B_{3}×A_{1}, order 96 
Flag orbits  4 
Convex  Yes 
Nature  Tame 
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 K4.11 on Richard Klitzing's list).
Gallery[edit  edit source]

Segmentochoron display, oct atop oct

Segmentochoron display, trip atop gyro trip

Net
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 () (base has A_{3} symmetry, tetratetrahedral prism)
 x2s2s3s () (triangular antiprismatic prism)
 x2s2s6o () (base has G_{2}×A_{1}+ symmetry)
 oo4oo3xx&#x (bases considered separately)
 oo3xx3oo&#x (bases considered in tet symmetry)
 xx xo3ox&#x (A_{2}×A_{1} axial. trip atop gyro trip)
 xxx oxo4ooo&#xt (BC_{2}×A_{1} symmetry, as square bipyramidal prism)
 xxx oxo oxo&#xt (A_{1}×A_{1}×A_{1} symmetry, as rectangular bipyramidal prism)
 xxx xox oqo&#xt (A_{1}×A_{1}×A_{1} axial, bases are edgefirst)
 xxx qoo oqo ooq&#zx (A_{1}×A_{1}×A_{1}×A_{1} 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]
 Bowers, Jonathan. "Category 19: Prisms" (#890).
 Klitzing, Richard. "Ope".
 Wikipedia contributors. "Octahedral prism".
 Hi.gher.Space Wiki Contributors. "Octahedral prism".