6-cube
Name Schläfli Symmetry Order
Regular 6-cube { 4,3,3,3,3 } [ 4,3,3,3,3 ] 46080
Quasiregular 6-cube [ 3,3,3,3 ] 23040
hyperrectangle { 4,3,3,3 } × { } [ 4,3,3,3,2 ] 7680
hyperrectangle { 4,3,3 } × { 4 } [ 4,3,3,2,4 ] 3072
hyperrectangle { 4,3 } [ 4,3,2,4,3 ] 2304
hyperrectangle { 4,3,3 } × { } [ 4,3,3,2,2 ] 1536
hyperrectangle { 4,3 } × { 4 } × { } [ 4,3,2,4,2 ] 768
hyperrectangle { 4 } [ 4,2,4,2,4 ] 512
hyperrectangle { 4,3 } × { } [ 4,3,2,2,2 ] 384
hyperrectangle { 4 } × { } [ 4,2,4,2,2 ] 256
hyperrectangle { 4 } × { } [ 4,2,2,2,2 ] 128
hyperrectangle { } [ 2,2,2,2,2 ] 64
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 .
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}, 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.