- Alternate names
- As a configuration
- Cartesian coordinates
- Construction
- Other images
- Related polytopes and honeycombs
- References
- External links
| Regular 5-orthoplex (pentacross) | |
|---|---|
Orthogonal projection inside Petrie polygon | |
| Type | Regular 5-polytope |
| Family | orthoplex |
| Schläfli symbol | {3,3,3,4} {3,3,31,1} |
| Coxeter-Dynkin diagrams | node_1|3|node|3|node|3|node|4|node}} {{CDD|node_1|3|node|3|node|split1|nodes}} |
| 4-faces | 32 {33} |
| Cells | 80 {3,3} |
| Faces | 80 {3} |
| Edges | 40 |
| Vertices | 10 |
| Vertex figure | 16-cell |
| Petrie polygon | decagon |
| Coxeter groups | BC5, [3,3,3,4] D5, [32,1,1] |
| Dual | 5-cube |
| Properties | convex |
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 {33,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,31,1} or Coxeter symbol 211.
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]
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 C5 or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of 5-cell facets, alternating, with the D5 or [32,1,1] 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 | node_1|3|node|3|node|3|node|4|node}} | {3,3,3,4} | [3,3,3,4] | 3840 | node_1|3|node|3|node|4|node}} |
| Quasiregular 5-orthoplex | node_1|3|node|3|node|split1|nodes}} | {3,3,31,1} | [3,3,31,1] | 1920 | node_1|3|node|split1|nodes}} |
| 5-fusil | |||||
| node_f1|4|node|3|node|3|node|3|node}} | {3,3,3,4} | [4,3,3,3] | 3840 | node_f1|4|node|3|node|3|node}} | |
| node_f1|4|node|3|node|3|node|2|node_f1}} | {3,3,4}+{} | [4,3,3,2] | 768 | node_f1|4|node|3|node|2|node_f1}} | |
| node_f1|4|node|3|node|2|node_f1|4|node}} | {3,4}+{4} | [4,3,2,4] | 384 | node_f1|4|node|3|node|2|node_f1}} {{CDD|node_f1|4|node|2|node_f1|4|node}} | |
| node_f1|4|node|3|node|2|node_f1|2|node_f1}} | {3,4}+2{} | [4,3,2,2] | 192 | node_f1|4|node|3|node|2|node_f1}} {{CDD|node_f1|4|node|2|node_f1|2|node_f1}} | |
| node_f1|4|node|2|node_f1|4|node|2|node_f1}} | 2{4}+{} | [4,2,4,2] | 128 | node_f1|4|node|2|node_f1|4|node}} | |
| node_f1|4|node|2|node_f1|2|node_f1|2|node_f1}} | {4}+3{} | [4,2,2,2] | 64 | node_f1|4|node|2|node_f1|2|node_f1}} {{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1}} | |
| node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}} | 5{} | [2,2,2,2] | 32 | node_f1|2|node_f1|2|node_f1|2|node_f1}} |
Other images
{{5-cube Coxeter plane graphs|t4|150}}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
{{2 k1 polytopes}}This polytope is one of 31 uniform 5-polytopes generated from the B5 Coxeter plane, including the regular 5-cube and 5-orthoplex.
{{Penteract family}}References
2. ^Coxeter, Complex Regular Polytopes, p.117
- 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}}
- (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)
- {{KlitzingPolytopes|polytera.htm|5D uniform polytopes (polytera)|x3o3o3o4o - tac}}
External links
- {{GlossaryForHyperspace | anchor=Cross | title=Cross polytope }}
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
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