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 can be called a hexeract, a portmanteau of tesseract (the 4-cube) with hex for six (dimensions) in Greek.
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.
The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces.
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 (x0, x1, x2, x3, x4, x5) with −1 < xi < 1.
The lowest symmetry construction is based on hyperrectangles or proprisms, cartesian products of lower dimensional hypercubes.
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.
Each fold direction adds 1 dimension, raising it into 6-space.
This polytope is one of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.