6-cube

Related polytopes

It is a part of an infinite family of polytopes, called hypercubes. The dual of a 6-cube can be called a 6-orthoplex, and is a part of the infinite family of cross-polytopes. 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, called a 6-demicube, (part of an infinite family called demihypercubes), which has 12 5-demicube and 32 5-simplex 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.

\begin{bmatrix}\begin{matrix}64 & 6 & 15 & 20 & 15 & 6 \ 2 & 192 & 5 & 10 & 10 & 5 \ 4 & 4 & 240 & 4 & 6 & 4 \ 8 & 12 & 6 & 160 & 3 & 3 \ 16 & 32 & 24 & 8 & 60 & 2 \ 32 & 80 & 80 & 40 & 10 & 12 \end{matrix}\end{bmatrix}

Cartesian coordinates

Cartesian coordinates 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 (x0, x1, x2, x3, x4, x5) with −1 i

Construction

There are three Coxeter groups associated with the 6-cube, one regular, with the C6 or [4,3,3,3,3] Coxeter group, and a half symmetry (D6) or [33,1,1] Coxeter group. The lowest symmetry construction is based on hyperrectangles or proprisms, cartesian products of lower dimensional hypercubes.

Projections

Related polytopes

The 64 vertices of a 6-cube also represent a regular skew 4-polytope {4,3,4 | 4}. Its net can be seen as a 4×4×4 matrix of 64 cubes, a periodic subset of the cubic honeycomb, {4,3,4}, 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:

This polytope is one of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.

References

• H.S.M. Coxeter, Regular Polytopes, 1973, 3rd edition, Dover, New York, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n dimensions (n ≥ 5),

References

1. (2019). "2019 International Engineering Conference (IEC)".
2. (February 1988). "An improved projection operation for cylindrical algebraic decomposition of three-dimensional space". Journal of Symbolic Computation.
3. Coxeter, Regular Polytopes, sec 1.8 Configurations
4. Coxeter, Complex Regular Polytopes, p.117