6-cube Hexeract | |
---|---|
Orthogonal projection inside Petrie polygon Orange vertices are doubled, and the center yellow has 4 vertices | |
Type | Regular 6-polytope |
Family | hypercube |
Schläfli symbol | {4,3^{4}} |
Coxeter diagram | |
5-faces | 12 {4,3,3,3} |
4-faces | 60 {4,3,3} |
Cells | 160 {4,3} |
Faces | 240 {4} |
Edges | 192 |
Vertices | 64 |
Vertex figure | 5-simplex |
Petrie polygon | dodecagon |
Coxeter group | B_{6}, [3^{4},4] |
Dual | 6-orthoplex |
Properties | convex, Hanner polytope |
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 {4,3^{4}}, being composed of 3 5-cubes 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.
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).^{[1]}^{[2]}
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.
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.^{[3]}^{[4]}
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 (x_{0}, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) with −1 < x_{i} < 1.
There are three Coxeter groups associated with the 6-cube, one regular, with the C_{6} or [4,3,3,3,3] Coxeter group, and a half symmetry (D_{6}) or [3^{3,1,1}] 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 | {4,3,3,3,3} | [4,3,3,3,3] | 46080 | |
Quasiregular 6-cube | [3,3,3,3^{1,1}] | 23040 | ||
hyperrectangle | {4,3,3,3}×{} | [4,3,3,3,2] | 7680 | |
{4,3,3}×{4} | [4,3,3,2,4] | 3072 | ||
{4,3}^{2} | [4,3,2,4,3] | 2304 | ||
{4,3,3}×{}^{2} | [4,3,3,2,2] | 1536 | ||
{4,3}×{4}×{} | [4,3,2,4,2] | 768 | ||
{4}^{3} | [4,2,4,2,4] | 512 | ||
{4,3}×{}^{3} | [4,3,2,2,2] | 384 | ||
{4}^{2}×{}^{2} | [4,2,4,2,2] | 256 | ||
{4}×{}^{4} | [4,2,2,2,2] | 128 | ||
{}^{6} | [2,2,2,2,2] | 64 |
Coxeter plane | B_{6} | B_{5} | B_{4} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [12] | [10] | [8] |
Coxeter plane | Other | B_{3} | B_{2} |
Graph | |||
Dihedral symmetry | [2] | [6] | [4] |
Coxeter plane | A_{5} | A_{3} | |
Graph | |||
Dihedral symmetry | [6] | [4] |
3D Projections | |
6-cube 6D simple rotation through 2Pi with 6D perspective projection to 3D. |
6-cube quasicrystal structure orthographically projected to 3D using the golden ratio. |
A 3D perspective projection of an hexeract undergoing a triple rotation about the X-W1, Y-W2 and Z-W3 orthogonal planes. |
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:
Line segment | Square | Cube | 4-cube | 5-cube | 6-cube | 7-cube | 8-cube |
This polytope is one of 63 uniform 6-polytopes generated from the B_{6} Coxeter plane, including the regular 6-cube or 6-orthoplex.