Hexadecachoron

The hexadecachoron, or hex, also commonly called the 16-cell, is one of the 6 convex regular polychora. It has 16 regular tetrahedra as cells, joining 4 to an edge and 8 to a vertex in an octahedral arrangment. It is the 4-dimensional orthoplex.

Vertex coordinates
The vertices of a regular hexadecachoron of dege length 1, centered at the origin, are given by all permutations and sign changes of:
 * (±$\sqrt{2}$/2, 0, 0, 0).

It can also be given as the even changes of sign of:


 * ($\sqrt{2}$/4, $\sqrt{2}$/4, $\sqrt{2}$/4, $\sqrt{2}$/4).

These are formed by alternating the vertices of a tesseract.

Demitesseract
The hexadecachoron can also be constructed as the alternation of the tesseract. In this variation, called a demitesseract and having D4 symmetry, the tetrahedral cells come in 2 groups of 8, with all cells in one group sharing faces only with those of the other group. This makes it the 4-dimensional demihypercube.