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 ...
, a 6-cube is a six-
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 ...
with 64
vertices, 192
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 by ...
s, 240 square
faces
The face is the front of the 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 affect the ...
, 160 cubic
cells
Cell most often refers to:
* Cell (biology), the functional basic unit of life
* Cellphone, a phone connected to a cellular network
* Clandestine cell, a penetration-resistant form of a secret or outlawed organization
* Electrochemical cell, a d ...
, 60
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-face
In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object. For example, a cube has six faces in this sense.
In more modern treatments of the geometry of polyhedra and higher-dimensional polyto ...
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. For example, a cube has six faces in this sense.
In more modern treatments of the geometry of polyhedra and higher-dimensional polyto ...
s.
It has
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
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
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'') with ''hex'' for six (dimensions) in
Greek
Greek may refer to:
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 of all kno ...
. It can also be called a regular dodeca-6-tope or dodecapeton, being a
6-dimensional polytope constructed from 12 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.
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 ( ); 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 a 6-cube can be called a
6-orthoplex
In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 Vertex (geometry), vertices, 60 Edge (geometry), edges, 160 triangle Face (geometry), faces, 240 tetrahedron Cell (mathematics), cells, 192 5-cell ''4-faces'', and 64 ...
, and is a part of the infinite family of
cross-polytope
In geometry, a cross-polytope, hyperoctahedron, orthoplex, staurotope, 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 reg ...
s. It is composed of various
5-cubes, at perpendicular angles on the u-axis, forming coordinates (x,y,z,w,v,u).
Applying an ''
alternation'' operation, deleting alternating vertices of the 6-cube, creates another
uniform polytope
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform Facet (mathematics), facets. Here, "vertex-transitive" means that it has symmetries taking every vertex to every other vertex; the sam ...
, called a
6-demicube, (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 hype ...
s), which has 12
5-demicube
In Five-dimensional space, five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a ''5-hypercube'' (penteract) with Alternation (geometry), alternated vertices removed.
It was discovered by Thorold ...
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 ...
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
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 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 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 ref ...
s associated with the 6-cube, one
regular, with the C
6 or
,3,3,3,3Coxeter 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 ref ...
, and a half symmetry (D
6) or
3,1,1Coxeter group. The lowest symmetry construction is based on
hyperrectangle
In geometry, a hyperrectangle (also called a box, hyperbox, k-cell or orthotopeCoxeter, 1973), is the generalization of a rectangle (a plane figure) and the rectangular cuboid (a solid figure) to higher dimensions. A necessary and sufficient cond ...
s or
proprism
In geometry of 4 dimensions or higher, a proprism is a polytope resulting from the Cartesian product of two or more polytopes, each of two dimensions or higher. The term was coined by John Horton Conway for ''product prism''. The dimension of the s ...
s,
cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is
A\times B = \.
A table c ...
s of lower dimensional hypercubes.
Projections
Related polytopes
The 64 vertices of a 6-cube also represent a regular skew 4-polytope . Its net can be seen as a 4×4×4 matrix of 64 cubes, a periodic subset of the
cubic honeycomb
The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb (geometry), honeycomb) in Euclidean 3-space made up of cube, cubic cells. It has 4 cubes around every edge, and 8 cubes around each verte ...
, , in 3-dimensions. It has 192 edges, and 192 square faces. Opposite faces fold together into a 4-cycle. Each fold direction adds 1 dimension, raising it into 6-space.
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 ( ); 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 ...
:
This polytope is one of 63
uniform 6-polytopes generated from the B
6 Coxeter plane
In mathematics, a Coxeter element is an element of an irreducible Coxeter group which is a product of all simple reflections. The product depends on the order in which they are taken, but different orderings produce conjugate elements, which hav ...
, including the regular 6-cube or
6-orthoplex
In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 Vertex (geometry), vertices, 60 Edge (geometry), edges, 160 triangle Face (geometry), faces, 240 tetrahedron Cell (mathematics), cells, 192 5-cell ''4-faces'', and 64 ...
.
References
*
Coxeter, H.S.M. ''
Regular Polytopes
''Regular Polytopes'' is a geometry book on regular polytopes written by Harold Scott MacDonald Coxeter. It was originally published by Methuen in 1947 and by Pitman Publishing in 1948, with a second edition published by Macmillan in 1963 and a th ...
'', (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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