In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, an 8-cube is an eight-
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
al
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...
. It has 256
vertices, 1024
edges, 1792 square
faces, 1792 cubic
cells, 1120
tesseract
In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. Just as the perimeter of the square consists of four edges and the surface of the cube consists of six ...
4-faces, 448
5-cube 5-faces, 112
6-cube 6-faces, and 16
7-cube 7-faces.
It is represented by
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
, being composed of 3
7-cubes around each 6-face. It is called an octeract, a
portmanteau
In linguistics, a blend—also known as a blend word, lexical blend, or portmanteau—is a word formed by combining the meanings, and parts of the sounds, of two or more words together. of
tesseract
In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. Just as the perimeter of the square consists of four edges and the surface of the cube consists of six ...
(the ''4-cube'') and ''oct'' for eight (dimensions) in
Greek. It can also be called a regular hexdeca-8-tope or hexadecazetton, being an
8-dimensional polytope constructed from 16 regular
facet
Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cu ...
s.
It is a part of an infinite family of polytopes, called
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...
s. The
dual of an 8-cube can be called an
8-orthoplex and is a part of the infinite family of
cross-polytopes.
Cartesian coordinates
Cartesian coordinates
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
for the vertices of an 8-cube centered at the origin and edge length 2 are
: (±1,±1,±1,±1,±1,±1,±1,±1)
while the interior of the same consists of all points (x
0, x
1, x
2, x
3, x
4, x
5, x
6, x
7) with -1 < x
i < 1.
As a configuration
This
configuration matrix represents the 8-cube. 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-cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.
The diagonal
f-vector numbers are derived through the
Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.
Projections
Derived polytopes
Applying an ''
alternation'' operation, deleting alternating vertices of the octeract, creates another
uniform polytope, called a ''
8-demicube'', (part of an infinite family called
demihypercubes), which has 16
demihepteractic and 128 8-simplex facets.
Related polytopes
The ''8-cube'' is 8th in an infinite series of
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...
:
References
*
H.S.M. Coxeter:
** Coxeter, ''
Regular Polytopes'', (3rd edition, 1973), Dover edition, , p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
** 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
*
*
Multi-dimensional Glossary: hypercubeGarrett Jones
{{Polytopes
8-polytopes