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 ...

, an 8-orthoplex or 8-

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 ...

is a regular

8-polytope In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets. A uniform 8-polytope is one which is vertex-transitive, ...

with 16 vertices, 112

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 ...

s, 448 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 ...

, 1120 tetrahedron cells, 1792

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'', 1792 ''5-faces'', 1024 ''6-faces'', and 256 ''7-faces''. It has two constructive forms, the first being regular with Schläfli symbol , and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol or

Coxeter symbol 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 ...

5₁₁. It is a part of an infinite family of polytopes, called cross-polytopes or ''orthoplexes''. The

dual polytope In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other ...

is an 8- hypercube, or

octeract In geometry, an 8-cube is an eight-dimensional hypercube. It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 5-cube 5-faces, 112 6-cube 6-faces, and 16 7-cube 7-faces. It is represented by Schlä ...

Alternate names

* Octacross, derived from combining the family name ''cross polytope'' with ''oct'' for eight (dimensions) in

Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...

* Diacosipentacontahexazetton as a 256- facetted

(polyzetton)

As a configuration

This configuration matrix represents the 8-orthoplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces and 7-faces. The diagonal numbers say how many of each element occur in the whole 8-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.

\begin\begin
 16 & 14 & 84 & 280 & 560 & 672 & 448 & 128
\\ 2 & 112 & 12 & 60 & 160 & 240 & 192 & 64
\\ 3 & 3 & 448 & 10 & 40 & 80 & 80 & 32
\\ 4 & 6 & 4 & 1120 & 8 & 24 & 32 & 16
\\ 5 & 10 & 10 & 5 & 1792 & 6 & 12 & 8
\\ 6 & 15 & 20 & 15 & 6 & 1792 & 4 & 4
\\ 7 & 21 & 35 & 35 & 21 & 7 & 1024 & 2
\\ 8 & 28 & 56 & 70 & 56 & 28 & 8 & 256
\end\end

The diagonal

f-vector Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral co ...

numbers are derived through the

Wythoff construction In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process ...

, dividing the full group order of a subgroup order by removing individual mirrors.

Construction

There are two

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...

s associated with the 8-cube, one regular, dual of the

with the C₈ or ,3,3,3,3,3,3symmetry group, and a half symmetry with two copies of 7-simplex facets, alternating, with the D₈ or ^5,1,1symmetry group. A lowest symmetry construction is based on a dual of an 8-

orthotope In geometry, an orthotopeCoxeter, 1973 (also called a hyperrectangle or a box) is the generalization of a rectangle to higher dimensions. A necessary and sufficient condition is that it is congruent to the Cartesian product of intervals. If all o ...

, called an 8-fusil.

Cartesian coordinates

Cartesian coordinates for the vertices of an 8-cube, centered at the origin are : (±1,0,0,0,0,0,0,0), (0,±1,0,0,0,0,0,0), (0,0,±1,0,0,0,0,0), (0,0,0,±1,0,0,0,0), : (0,0,0,0,±1,0,0,0), (0,0,0,0,0,±1,0,0), (0,0,0,0,0,0,0,±1), (0,0,0,0,0,0,0,±1) Every

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet * Vertex (computer graphics), a data structure that describes the positio ...

pair is connected by an

, except opposites.

Images

It is used in its alternated form 5₁₁ with the

8-simplex In geometry, an 8- simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 126 5-cell 4-faces, 84 5-simplex 5-faces, 36 6-simplex 6-faces, and 9 7-simplex 7-faces. Its dihedral angle is ...

to form the 5₂₁ honeycomb.

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,

*** (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. *

External links

*
Polytopes of Various Dimensions

{{Polytopes 8-polytopes