Hexeract

The hexeract, or ax, also called the 6-cube or dodecapeton, is one of the 3 regular polypeta. It has 12 penteracts as facets, joining 6 to a vertex. It is the 6-dimensional hypercube.

It is also a penteractic prism, cubic duoprism, square trioprism, dodecahedron-small stellated dodecahedral gyropeton, and 12-3-5 gyropeton.

It can be alternated into a demihexeract, which is uniform.

Vertex coordinates
The vertices of a hexeract of edge length 1, centered at the origin, are given by:
 * $$\left(\pm\frac12,\,\pm\frac12,\,\pm\frac12,\,\pm\frac12,\,\pm\frac12,\,\pm\frac12\right).$$

Representations
A hexeract has the following Coxeter diagrams:


 * x4o3o3o3o3o (full symmetry)
 * x4o3o x4o3o (B3×B3 symmetry, cubic duoprism)
 * x4o x4o3o3o (B4×B2 symmetry, square-tesseractic duoprism)
 * x x4o3o3o3o (B5×B1 symmetry, penteractic prism)
 * x4o x4o x4o (B2×B2×B2 symmetry, square trioprism)
 * x x4o x4o3o (B3×B3×A1 symmetry, square-cubic duoprismatic prism)
 * x x x4o3o3o (B4×A1×A1 symmetry, tesseract prismatic prism)
 * x x x4o x4o (B2×B2×A1×A1 symmetry, square duoprismatic prismatic prism)
 * x x x x4o3o (B3×A1×A1×A1 symmetry, cubic prism prism prism)
 * x x x x x4o (five dimensions different)
 * x x x x x x (all six dimensions different)
 * xx4oo3oo3oo3oo&#x (B5 axial, penteract prism)
 * qo3oo3oq *b3oo3oo3oo&#zx (D4 symmetry, hull of two demihexeracts)
 * oqooooo3ooqoooo3oooqooo3ooooqoo3oooooqo&#xt (A5 axial, vertex-first)