7-orthoplex
Regular 7-orthoplex (heptacross) | |
---|---|
![]() Orthogonal projection inside Petrie polygon | |
Type | Regular 7-polytope |
Family | orthoplex |
Schläfli symbol | {35,4} {3,3,3,3,31,1} |
Coxeter-Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6-faces | 128 {35}![]() |
5-faces | 448 {34}![]() |
4-faces | 672 {33}![]() |
Cells | 560 {3,3}![]() |
Faces | 280 {3}![]() |
Edges | 84 |
Vertices | 14 |
Vertex figure | 6-orthoplex |
Petrie polygon | tetradecagon |
Coxeter groups | C7, [3,3,3,3,3,4] D7, [34,1,1] |
Dual | 7-cube |
Properties | convex, Hanner polytope |
In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cells 4-faces, 448 5-faces, and 128 6-faces.
It has two constructed forms, the first being regular with Schläfli symbol {35,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,3,31,1} or Coxeter symbol 411.
It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 7-hypercube, or hepteract.
Alternate names
- Heptacross, derived from combining the family name cross polytope with hept for seven (dimensions) in Greek.
- Hecatonicosoctaexon as a 128-facetted 7-polytope (polyexon).
As a configuration
This configuration matrix represents the 7-orthoplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-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} 14 & 12 & 60 & 160 & 240 & 192 & 64 \\ 2 & 84 & 10 & 40 & 80 & 80 & 32 \\ 3 & 3 & 280 & 8 & 24 & 32 & 16 \\ 4 & 6 & 4 & 560 & 6 & 12 & 8 \\ 5 & 10 & 10 & 5 & 672 & 4 & 4 \\ 6 & 15 & 20 & 15 & 6 & 448 & 2 \\ 7 & 21 & 35 & 35 & 21 & 7 & 128 \end{matrix}\end{bmatrix} }[/math]
Images
Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | ![]() |
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Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | ![]() |
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Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | ![]() |
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Dihedral symmetry | [6] | [4] |
Construction
There are two Coxeter groups associated with the 7-orthoplex, one regular, dual of the hepteract with the C7 or [4,3,3,3,3,3] symmetry group, and a half symmetry with two copies of 6-simplex facets, alternating, with the D7 or [34,1,1] symmetry group. A lowest symmetry construction is based on a dual of a 7-orthotope, called a 7-fusil.
Name | Coxeter diagram | Schläfli symbol | Symmetry | Order | Vertex figure |
---|---|---|---|---|---|
regular 7-orthoplex | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{3,3,3,3,3,4} | [3,3,3,3,3,4] | 645120 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Quasiregular 7-orthoplex | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{3,3,3,3,31,1} | [3,3,3,3,31,1] | 322560 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7-fusil | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7{} | [26] | 128 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cartesian coordinates
Cartesian coordinates for the vertices of a 7-orthoplex, centered at the origin are
- (±1,0,0,0,0,0,0), (0,±1,0,0,0,0,0), (0,0,±1,0,0,0,0), (0,0,0,±1,0,0,0), (0,0,0,0,±1,0,0), (0,0,0,0,0,±1,0), (0,0,0,0,0,0,±1)
Every vertex pair is connected by an edge, except opposites.
See also
- Rectified 7-orthoplex
- Truncated 7-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. "7D uniform polytopes (polyexa) x3o3o3o3o3o4o - zee". https://bendwavy.org/klitzing/dimensions/polyexa.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
Fundamental convex regular and uniform polytopes in dimensions 2–10
| ||||||||||||
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Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn | |||||||
Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon | |||||||
Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
Uniform 4-polytope | 5-cell | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell | |||||||
Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube | |||||||||
Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 | ||||||||
Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 | ||||||||
Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 | ||||||||
Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube | |||||||||
Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube | |||||||||
Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope | |||||||
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds |
![]() | Original source: https://en.wikipedia.org/wiki/7-orthoplex.
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