6-cube

6-cube
Hexeract

Orthogonal projection
inside Petrie polygon
Orange vertices are doubled, and the center yellow has 4 vertices
TypeRegular 6-polytope
Familyhypercube
Schläfli symbol {4,34}
Coxeter-Dynkin diagram
5-faces12 {4,3,3,3}
4-faces60 {4,3,3}
Cells160 {4,3}
Faces240 {4}
Edges192
Vertices64
Vertex figure5-simplex
Petrie polygondodecagon
Coxeter groupB6, [34,4]
Dual6-orthoplex
Propertiesconvex

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 {4,34}, 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.

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.

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.

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 < xi < 1.

Images

orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane Other B3 B2
Graph
Dihedral symmetry [2] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]
3D Projections

6-cube 6D simple rotation through 2Pi with 6D perspective projection to 3D.

Hexeract Quasicrystal structure orthographically projected
to 3D using the Golden Ratio.

Related polytopes

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

References

External links

This article is issued from Wikipedia - version of the Tuesday, April 21, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.