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 square duotegum, the digonal duoantiprism, the digonal diswirlprism and the 8-3 step prism. Together with its dual, the hexadecachoron is the first in a series of tetrahedral and digonal antiprismatic swirlchora and the first in a series of square dihedral swirlchora.

It and the great duoantiprism are the only uniform duoantiprisms, and it is the only one that is convex and regular.

It is one of the three regular polychora that can tile 4D space, the others being the tesseract and the icositetrachoron.

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.

Isogonal derivatives
Substitution by vertices of these following elements will produce these convex isogonal polychora:
 * Tetrahedron (16): Tesseract
 * Triangle (32): Rectified tesseract
 * Edge (24): Icositetrachoron

Related polychora
The hexadecachoron is the colonel of a two-member regiment that includes the tesseractihemioctachoron