5-orthoplex
Regular 5-orthoplex (pentacross) | |
---|---|
Orthogonal projection inside Petrie polygon | |
Type | Regular 5-polytope |
Family | orthoplex |
Schläfli symbol | {3,3,3,4} {3,3,3^{1,1}} |
Coxeter-Dynkin diagrams | |
4-faces | 32 {3^{3}} |
Cells | 80 {3,3} |
Faces | 80 {3} |
Edges | 40 |
Vertices | 10 |
Vertex figure | 16-cell |
Petrie polygon | decagon |
Coxeter groups | BC_{5}, [3,3,3,4] D_{5}, [3^{2,1,1}] |
Dual | 5-cube |
Properties | convex, Hanner polytope |
In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.
It has two constructed forms, the first being regular with Schläfli symbol {3^{3},4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3^{1,1}} or Coxeter symbol 2_{11}.
It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 5-hypercube or 5-cube.
Alternate names
- pentacross, derived from combining the family name cross polytope with pente for five (dimensions) in Greek.
- Triacontaditeron (or triacontakaiditeron) - as a 32-facetted 5-polytope (polyteron).
As a configuration
This configuration matrix represents the 5-orthoplex. The rows and columns correspond to vertices, edges, faces, cells and 4-faces. The diagonal numbers say how many of each element occur in the whole 5-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^{[1]}^{[2]}
[math]\displaystyle{ \begin{bmatrix}\begin{matrix} 10 & 8 & 24 & 32 & 16 \\ 2 & 40 & 6 & 12 & 8 \\ 3 & 3 & 80 & 4 & 4 \\ 4 & 6 & 4 & 80 & 2 \\ 5 & 10 & 10 & 5 & 32 \end{matrix}\end{bmatrix} }[/math]
Cartesian coordinates
Cartesian coordinates for the vertices of a 5-orthoplex, centered at the origin are
- (±1,0,0,0,0), (0,±1,0,0,0), (0,0,±1,0,0), (0,0,0,±1,0), (0,0,0,0,±1)
Construction
There are three Coxeter groups associated with the 5-orthoplex, one regular, dual of the penteract with the C_{5} or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of 5-cell facets, alternating, with the D_{5} or [3^{2,1,1}] Coxeter group, and the final one as a dual 5-orthotope, called a 5-fusil which can have a variety of subsymmetries.
Name | Coxeter diagram | Schläfli symbol | Symmetry | Order | Vertex figure(s) |
---|---|---|---|---|---|
regular 5-orthoplex | {3,3,3,4} | [3,3,3,4] | 3840 | ||
Quasiregular 5-orthoplex | {3,3,3^{1,1}} | [3,3,3^{1,1}] | 1920 | ||
5-fusil | |||||
{3,3,3,4} | [4,3,3,3] | 3840 | |||
{3,3,4}+{} | [4,3,3,2] | 768 | |||
{3,4}+{4} | [4,3,2,4] | 384 | |||
{3,4}+2{} | [4,3,2,2] | 192 | |||
2{4}+{} | [4,2,4,2] | 128 | |||
{4}+3{} | [4,2,2,2] | 64 | |||
5{} | [2,2,2,2] | 32 |
Other images
Coxeter plane | B_{5} | B_{4} / D_{5} | B_{3} / D_{4} / A_{2} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [10] | [8] | [6] |
Coxeter plane | B_{2} | A_{3} | |
Graph | |||
Dihedral symmetry | [4] | [4] |
The perspective projection (3D to 2D) of a stereographic projection (4D to 3D) of the Schlegel diagram (5D to 4D) of the 5-orthoplex. 10 sets of 4 edges form 10 circles in the 4D Schlegel diagram: two of these circles are straight lines in the stereographic projection because they contain the center of projection. |
Related polytopes and honeycombs
This polytope is one of 31 uniform 5-polytopes generated from the B_{5} Coxeter plane, including the regular 5-cube and 5-orthoplex.
References
- H.S.M. Coxeter:
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN:978-0-471-01003-6 [1]
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
- Klitzing, Richard. "5D uniform polytopes (polytera) x3o3o3o4o - tac". https://bendwavy.org/klitzing/dimensions/polytera.htm.
External links
- Olshevsky, George. "Cross polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007. https://web.archive.org/web/20070204075028/members.aol.com/Polycell/glossary.html#Cross.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
Original source: https://en.wikipedia.org/wiki/5-orthoplex.
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