6-cubeHexeractOrthogonal projectioninside Petrie polygon.Orange vertices are doubled, and the center yellow has 4 vertices.TypeRegular 6-polytopeFamilyhypercubeSchläfli symbol {4,34}Coxeter diagram5-faces12 {4,3,3,3} 4-faces60 {4,3,3} Cells160 {4,3} Faces240 {4} Edges192Vertices64Vertex figure5-simplexPetrie polygondodecagonCoxeter groupB6, [34,4]Dual6-orthoplex Propertiesconvex, 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 {4,34}, 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. Acronym: ax
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.
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.
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.
NameCoxeterSchläfliSymmetryOrderRegular 6-cube{4,3,3,3,3}[4,3,3,3,3]46080Quasiregular 6-cube[3,3,3,31,1]23040hyperrectangle{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
Projections + Orthographic projectionsCoxeter planeB6B5B4GraphDihedral symmetryCoxeter planeOtherB3B2GraphDihedral symmetryCoxeter planeA5A3GraphDihedral symmetry
3D Projections6-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 a hexeract undergoing a triple rotation about the X-W1, Y-W2 and Z-W3 orthogonal planes.
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 hypercubes:
This polytope is one of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including also the regular 6-orthoplex.
References
o3o3o3o3o4x - ax
External links
Multi-dimensional Glossary: hypercube Garrett Jones