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 and also the 2-2 duoantiprism and the 8-3 step prism.

It is one of the three regular polychora that can tile 4D space.

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

They 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. It can be represented as x3o3o *b3o.