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 |
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
The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.[1][2]
Images
Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | |||
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | |||
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
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
- (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".
External links
- Olshevsky, George. "Cross polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
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 |