In geometry, a 7-orthoplex, or 7- cross polytope, 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 edges, 280 triangle faces, 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-cell In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It i ...

s ''4-faces'', 448 ''5-faces'', and 128 ''6-faces''. It has two constructed forms, the first being regular with

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol or Coxeter symbol 4₁₁. It is a part of an infinite family of polytopes, called

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 octahed ...

s or ''orthoplexes''. The dual polytope is the 7-

hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, ...

, or

hepteract In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces. It can be named by its Schläfli symbol , bein ...

Alternate names

* Heptacross, derived from combining the family name ''cross polytope'' with ''hept'' for seven (dimensions) in Greek. * Hecatonicosoctaexon as a 128- facetted

(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.Coxeter, Complex Regular Polytopes, p.117

\begin\begin
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\end

Images

Construction

There are two Coxeter groups associated with the 7-orthoplex, one

regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrum ...

dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...

of the

with the C₇ or ,3,3,3,3,3symmetry group, and a half symmetry with two copies of 6-simplex facets, alternating, with the D₇ or ^4,1,1symmetry group. A lowest symmetry construction is based on a dual of a 7- orthotope, called a 7-fusil.

Cartesian coordinates

Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...

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.

References

H.S.M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

: ** 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'', ath. Zeit. 46 (1940) 380-407, MR 2,10*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', ath. Zeit. 188 (1985) 559-591*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3-45* 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