Penteract

The penteract, or pent, also called the 5-cube or decateron, is one of the 3 regular polytera. It has 10 tesseracts as facets, joining 5 to a vertex. It is the 5-dimensional hypercube. As such, it is also a tesseractic prism and square-cube duoprism.

Like the hypercubes of every other dimension, the penteract can tile 5D Euclidean space in the penteractic pentacomb.

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

Vertex coordinates
The vertices of a penteract of edge length 1, centered at the origin, are given by:


 * $$\left(±\frac12,\,±\frac12,\,±\frac12,\,±\frac12,\,±\frac12\right).$$

Representations
A penteract has the following Coxeter diagrams:


 * x4o3o3o3o (full ysmmetry)
 * x x4o3o3o (BC4×A1 symmetry, tesseractic prism)
 * x4o x4o3o (BC3×BC2 symmetry, square-cubic duoprism)
 * x x x4o3o (BC3×A1×A1 symmetry, cubic prismatic prism)
 * x x4o x4o (BC2×BC2×A1 symmetry, square duoprismatic prism)
 * x x x x4o (BC2×A1×A1×A1 symmetry, square prismatic prismatic prism)
 * x x x x x (A1×A1×A1×A1×A1 symmetry, all five dimensions separate)
 * xx4oo3oo3oo&#x (BC4 axial, tesseract atop tesseract)
 * xx xx4oo3oo&#x (BC3×A1 axial, cubic prism bases)
 * xx4oo xx4oo&#x (BC2×BC2 axial, square duoprismatic bases)
 * xx xx xx4oo&#x (BC2×A1×A1 axial, square prismatic prism bases)
 * xx xx xx xx&#x (A1×A1×A1×A1 symetry, bases have four separate dimensions)
 * oqo xxx4ooo3ooo&#xt (BC3×A1 axial, cell-first)
 * xxxx4oooo oqoo3ooqo&#xt (BC2×A1 axial, face-first)
 * xxxxx oqooo3ooqoo3oooqo&#xt (A3×A1 axial, edge-first)
 * oqoooo3ooqooo3oooqoo3ooooqo&#xt (A4 axial, vertex-first)
 * qo3oo3oq *b3oo3oo&#zx (D5 symmetry)