In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a 7-orthoplex, or 7-
cross polytope
In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahe ...
, is a regular
7-polytope
In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets.
A uniform 7-polytope is one whose symmetry group is transitive on vertices and whose f ...
with 14
vertices, 84
edge
Edge or EDGE may refer to:
Technology Computing
* Edge computing, a network load-balancing system
* Edge device, an entry point to a computer network
* Adobe Edge, a graphical development application
* Microsoft Edge, a web browser developed by ...
s, 280 triangle
faces
The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...
, 560 tetrahedron
cells
Cell most often refers to:
* Cell (biology), the functional basic unit of life
Cell may also refer to:
Locations
* Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...
, 672 5-cells ''4-faces'', 448 ''5-faces'', and 128 ''6-faces''.
It has two constructed forms, the first being regular with Schläfli symbol , and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol or Coxeter symbol 4
11.
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 language, Greek.
* Hecatonicosoctaexon as a 128-Facet (geometry), facetted
7-polytope
In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets.
A uniform 7-polytope is one whose symmetry group is transitive on vertices and whose f ...
(polyexon).
As a configuration
This Regular 4-polytope#As configurations, 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.
[Coxeter, Complex Regular Polytopes, p.117]
Images
Construction
There are two Coxeter groups associated with the 7-orthoplex, one regular polytope, regular, Dual polytope, dual of the hepteract with the C
7 or [4,3,3,3,3,3] symmetry group, and a half symmetry with two copies of 6-simplex facets, alternating, with the D
7 or [3
4,1,1] symmetry group. A lowest symmetry construction is based on a dual of a 7-orthotope, called a 7-fusil.
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 (geometry), vertex pair is connected by an
edge
Edge or EDGE may refer to:
Technology Computing
* Edge computing, a network load-balancing system
* Edge device, an entry point to a computer network
* Adobe Edge, a graphical development application
* Microsoft Edge, a web browser developed by ...
, except opposites.
See also
* Rectified 7-orthoplex
* Truncated 7-orthoplex
References
* Harold Scott MacDonald Coxeter, 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,
*** (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 (mathematician), Norman Johnson ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966)
*
External links
*
Polytopes of Various Dimensions
{{Polytopes
7-polytopes