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 be called a hexeract, a portmanteau of tesseract with hex for six 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 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.
Applying an alternation operation, deleting alternating vertices of the 6-cube, creates another uniform polytope, called a 6-demicube,, 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.

Cartesian coordinates

for the vertices of a 6-cube centered at the origin and edge length 2 are
while the interior of the same consists of all points with −1 < x_i < 1.

Construction

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

Name	Coxeter	Schläfli	Symmetry	Order
Regular 6-cube				46080
Quasiregular 6-cube				23040
hyperrectangle		×		7680
hyperrectangle		×		3072
hyperrectangle		²		2304
hyperrectangle		×²		1536
hyperrectangle		××		768
hyperrectangle		³		512
hyperrectangle		×³		384
hyperrectangle		²×²		256
hyperrectangle		×⁴		128
hyperrectangle		⁶		64

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,, 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.