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 ...
, a 7-cube is a
seven-dimensional hypercube with 128
vertices, 448
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, 672 square
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 cubic
cells, 280
tesseract
In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of e ...
4-faces, 84
penteract
In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.
It is represented by Schläfli symbol or , constructed as 3 tesseract ...
5-faces, and 14
hexeract 6-faces.
It can be named by its
Schläfli symbol , being composed of 3
6-cube
In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces.
It has Schläfli symbol , being composed of 3 5-cubes around each 4-face. It can ...
s around each 5-face. It can be called a hepteract, a
portmanteau
A portmanteau word, or portmanteau (, ) is a blend of words[tesseract
In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of e ...](_blank)
(the ''4-cube'') and ''hepta'' for seven (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 ...
. It can also be called a regular tetradeca-7-tope or tetradecaexon, being a
7 dimensional polytope constructed from 14 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 cut ...
s.
Related polytopes
The ''7-cube'' is 7th in a series of
hypercube:
The
dual of a 7-cube is called a
7-orthoplex
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 fi ...
, and is a part of the infinite family of
cross-polytopes.
Applying an ''
alternation'' operation, deleting alternating vertices of the hepteract, creates another
uniform polytope
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vert ...
, called a
demihepteract
In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube ( hepteract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.
E. L. El ...
, (part of an infinite family called
demihypercube
In geometry, demihypercubes (also called ''n-demicubes'', ''n-hemicubes'', and ''half measure polytopes'') are a class of ''n''- polytopes constructed from alternation of an ''n''- hypercube, labeled as ''hγn'' for being ''half'' of the hy ...
s), which has 14
demihexeract
In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a ''6-cube'' (hexeract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.
E. L. Elte ...
ic and 64
6-simplex
In geometry, a 6-simplex is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5-cell 4-faces, and 7 5-simplex 5-faces. Its dihedral angle is cos−1(1/6), or approximately 80.41°.
Alte ...
6-faces.
As a configuration
This
configuration matrix represents the 7-cube. 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-cube. 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]
Cartesian coordinates
Cartesian coordinates for the vertices of a hepteract centered at the origin and edge length 2 are
: (±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) with -1 < x
i < 1.
Projections
References
*
H.S.M. Coxeter:
** Coxeter, ''
Regular Polytopes
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
'', (3rd edition, 1973), Dover edition, , p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973, 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
Peter McMullen (born 11 May 1942) is a British mathematician, a professor emeritus of mathematics at University College London.
Education and career
McMullen earned bachelor's and master's degrees from Trinity College, Cambridge, and studied at ...
, 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
www.4d-screen.de
{{Polytopes
7-polytopes
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