In
geometry, a 6-cube is a six-
dimensional
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, ...
with 64
vertices, 192
edges, 240 square
faces, 160 cubic
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 ...
, 60
tesseract 4-face
In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a ''polyhedron''.
In more technical treatments of the geometry of polyhedra ...
s, and 12
5-cube
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 tesseracts, ...
5-face
In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a ''polyhedron''.
In more technical treatments of the geometry of polyhedra ...
s.
It has
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 ...
, being composed of 3
5-cube
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 tesseracts, ...
s around each 4-face. It can be called a hexeract, a
portmanteau of
tesseract (the ''4-cube'') with ''hex'' for six (dimensions) in
Greek. It can also be called a regular dodeca-6-tope or dodecapeton, being a
6-dimensional polytope constructed from 12 regular
facets.
Related polytopes
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 (). 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, ...
s. The
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 a 6-cube can be called a
6-orthoplex, and is a part of the infinite family of
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.
Applying an ''
alternation'' operation, deleting alternating vertices of the 6-cube, creates another
uniform polytope, called a
6-demicube
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 i ...
, (part of an infinite family called
demihypercubes), which has 12
5-demicube
In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a ''5-hypercube'' (penteract) with alternated vertices removed.
It was discovered by Thorold Gosset. Since it was the only semiregular 5- ...
and 32
5-simplex
In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(), or approximately 78.46°.
The 5-s ...
facets.
As a configuration
This
configuration matrix represents the 6-cube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-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
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 6-cube centered at the origin and edge length 2 are
: (±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) with −1 < x
i < 1.
Construction
There are three
Coxeter groups associated with the 6-cube, one
regular, with the C
6 or
,3,3,3,3 Coxeter group, and a half symmetry (D
6) or
3,1,1Coxeter group. The lowest symmetry construction is based on
hyperrectangles or
proprisms,
cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ti ...
s of lower dimensional hypercubes.
Projections
Related polytopes
The ''6-cube'' is 6th in a series of
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, ...
:
This polytope is one of 63
uniform 6-polytopes generated from the B
6 Coxeter plane, including the regular 6-cube or
6-orthoplex.
References
*
Coxeter, H.S.M. ''
Regular Polytopes'', (3rd edition, 1973), Dover edition, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n>=5)
*
External links
*
*
Multi-dimensional Glossary: hypercubeGarrett Jones
{{Polytopes
6-polytopes
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