Octahedral prism

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Octahedral prism
Octahedral prism.png
Schlegel diagram
Type Prismatic uniform 4-polytope
Uniform index 51
Schläfli symbol t0,3{3,4,2} or {3,4}×{}
t1,3{3,3,2} or r{3,3}×{}
s{2,6}×{}
sr{3,2}×{}
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
Cells 2 (3.3.3.3)Octahedron.png
8 (3.4.4)Triangular prism.png
Faces 16 {3}, 12 {4}
Edges 30
Vertices 12
Vertex figure Tetratetrahedral prism verf.png
Square pyramid
Symmetry [3,4,2], order 96
[3,3,2], order 48
[6,2+,2], order 24
[(3,2)+,2], order 12
Properties convex
Octahedral prism net.png
Net

In geometry, a octahedral prism is a convex uniform 4-polytope. This 4-polytope has 10 polyhedral cells: 2 octahedra connected by 8 triangular prisms.

Octahedral hyperprism Schlegel.png
Transparent Schlegel diagram

Alternative names[edit]

  • Octahedral dyadic prism (Norman W. Johnson)
  • Ope (Jonathan Bowers, for octahedral prism)
  • Triangular antiprismatic prism
  • Triangular antiprismatic hyperprism

Structure[edit]

The octahedral prism consists of two octahedra connected to each other via 8 triangular prisms. The triangular prisms are joined to each other via their square faces.

Projections[edit]

The octahedron-first orthographic projection of the octahedral prism into 3D space has an octahedral envelope. The two octahedral cells project onto the entire volume of this envelope, while the 8 triangular prismic cells project onto its 8 triangular faces.

The triangular-prism-first orthographic projection of the octahedral prism into 3D space has a hexagonal prismic envelope. The two octahedral cells project onto the two hexagonal faces. One triangular prismic cell projects onto a triangular prism at the center of the envelope, surrounded by the images of 3 other triangular prismic cells to cover the entire volume of the envelope. The remaining four triangular prismic cells are projected onto the entire volume of the envelope as well, in the same arrangement, except with opposite orientation.

Related polytopes[edit]

It is the second in an infinite series of uniform antiprismatic prisms.

Convex p-gonal antiprismatic prisms
Name s{2,2}×{} s{2,3}×{} s{2,4}×{} s{2,5}×{} s{2,6}×{} s{2,7}×{} s{2,8}×{} s{2,p}×{}
Coxeter
diagram
CDel node.pngCDel 4.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 8.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel 4.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 10.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel 5.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 12.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 14.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel 7.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 16.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel 8.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 2x.pngCDel p.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel p.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
Image Digonal antiprismatic prism.png Triangular antiprismatic prism.png Square antiprismatic prism.png Pentagonal antiprismatic prism.png Hexagonal antiprismatic prism.png Heptagonal antiprismatic prism.png Octagonal antiprismatic prism.png 15-gonal antiprismatic prism.png
Vertex
figure
Tetrahedral prism verf.png Tetratetrahedral prism verf.png Square antiprismatic prism verf2.png Pentagonal antiprismatic prism verf.png Hexagonal antiprismatic prism verf.png Heptagonal antiprismatic prism verf.png Octagonal antiprismatic prism verf.png Uniform antiprismatic prism verf.png
Cells 2 s{2,2}
(2) {2}×{}={4}
4 {3}×{}
2 s{2,3}
2 {3}×{}
6 {3}×{}
2 s{2,4}
2 {4}×{}
8 {3}×{}
2 s{2,5}
2 {5}×{}
10 {3}×{}
2 s{2,6}
2 {6}×{}
12 {3}×{}
2 s{2,7}
2 {7}×{}
14 {3}×{}
2 s{2,8}
2 {8}×{}
16 {3}×{}
2 s{2,p}
2 {p}×{}
2p {3}×{}
Net Tetrahedron prism net.png Octahedron prism net.png 4-antiprismatic prism net.png 5-antiprismatic prism net.png 6-antiprismatic prism net.png 7-antiprismatic prism net.png 8-antiprismatic prism net.png 15-gonal antiprismatic prism verf.png

It is one of 18 uniform polyhedral prisms created by using uniform prisms to connect pairs of parallel Platonic solids and Archimedean solids.

It is one of four four-dimensional Hanner polytopes; the other three are the tesseract, the 16-cell, and the dual of the octahedral prism (a double pyramid with a cubical base).

References[edit]

  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)
  • Norman Johnson Uniform Polytopes, Manuscript (1991)

External links[edit]