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 arrangement. It is the 4-dimensional orthoplex.

It is also the square duotegum, the digonal duoantiprism, the digonal diswirlprism, and the 8-3 step prism. It is the first in an infinite family of isogonal tetrahedral swirlchora, the first in an infinite family of isogonal square hosohedral swirlchora and also the first in an infinite family of isochoric square dihedral swirlchora. It can also be seen as a tetrahedral antiprism in two senses, being both a segmentochoron of a tetrahedron atop dual tetrahedron (being designated K-4.2 in Richard Klitzing's list of convex segmentochora) and the alternated cubic prism (that is a tesseract). It is also a regular-faced octahedral bipyramid.

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. Tiling it results in the hexadecachoric tetracomb.

The hexadecachoral army also contains the tesseractihemioctachoron.

Vertex coordinates
The vertices of a regular hexadecachoron of edge length 1, centered at the origin, are given by all permutations of:
 * $$\left(±\frac{\sqrt2}{2},\,0,\,0,\,0\right).$$

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


 * $$\left(\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4}\right).$$

These are formed by alternating the vertices of a tesseract.

Surtope Angles
The surtope angle represents the portion of solid space occupied by the polytope at that surtope.


 * A2  0:40.00.00   120° = 1/3   Dichoral or margin angle
 * A3: 0:20.00.00   120°E = 1/6  Edge angle.
 * A4  0:05.00.00     1/24

These are derived from the regular tiling x3o3o4o3o.

Representations
A hexadecachoron has the following Coxeter diagrams:


 * o4o3o3x (full symmetry)
 * x3o3o *b3o (D4 symmetry, as demitesseract)
 * s4o3o3o (as alternated tesseract)
 * s2s4o3o (as tetrahedral antiprism/alternated cubic prism)
 * s4o2s4o (as digonal duoantiprism)
 * s2s2s4o (as disphenoidal antiprism)
 * s2s2s2s (as alternated 4D block)
 * xo3oo3ox&#x (A3 axial, tetrahedron atop dual tetrahedron)
 * ooo4ooo3oxo&#xt (BC3 axial, as octahedral bipyramid)
 * ooo3oxo3ooo&#xt (A3 axial, as tetratetrahedral bipyramid)
 * o(qo)o o(ox)o4o(oo)o&#xt (as square bipyramidal bipyramid)
 * o(qoo)o o(oqo)o o(ooq)o&#xt (as rhombic bipyramidal bipyramid)
 * xox oxo4ooo&#xt (BC2×A1 axial, edge-first)
 * xox oxo oxo&#xt (A1×A1 axial, edge-first)
 * xoo3oox oqo&#xt (A2×A1 axial, face-first)
 * oxoo3ooox&#xr (A2 axial)
 * xo4oo ox4oo&#zx (BC2×BC2 symmetry, as square duotegum)
 * xo xo ox4oo&#zx (as square-rectangular duotegum)
 * xo xo ox ox&#zx (as rectangular duotegum)
 * xoxo oxox&#xr (A1×A1 axial)
 * qo oo4oo3ox&#zx (BC2×A1 symmetry)
 * qo oo3ox3oo&#zx (A3×A1 symmetry)
 * qo os2os3os&#zx (as triangular antiprismatic bipyramid)
 * qooo oqoo ooqo oooq&#zx (A1×A1×A1×A1 symmetry)
 * qoo oqo oox4ooo&#zx (BC2×A1×A1 symmetry)

Variations
Besides the regular hexadecachoron, other types of polychora with 16 tetrahedral cells exist:


 * Demitesseract - tetrahedra as two types, but no metrical variations - alternated tesseract
 * Rectangular duotegum - 4 sets of 4 phyllic disphenoids, isogonal
 * Tetrahedral antiprism - 2 tetrahedra, 8 triangular pyramids, and 6 tetragonal disphenoids, isogonal
 * Tetrahedral antipodium - 2 different sized base tetrahedra, 2 sets of 4 triangular pyramid sides, and 6 digonal disphenoids
 * Octahedral bipyramid - 16 identical triangular pyramids
 * Square duotegum - 16 identical digonal disphenoids
 * Square-rhombic duotegum - 16 identical sphenoids
 * Rhombic duotegum - 16 identical phyllic disphenoids
 * Rhombic-rhombic duotegum - 16 identical irregular tetrahedra
 * 8-3 step prism - isogonal with rhombic and phyllic dispenoids
 * Digonal duoantiprism - isogonal, 2 sets of 8 tetragonal disphenoids
 * Tetragonal disphenoidal antiprism, alternated square-rectangle duoprism, 4 rhombic and 4 digonal disphenoids, 8 sphenoids
 * Rhombic disphenoidal antiprism - 4 pairs of opposite rhombic disphenoids connected by irregular tetrahedra
 * Difold tetraswirlchoron - tetrahedra as digonal antiprisms, no metrical variations

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

A hexadecachoron can be cut in half to produce 2 octahedral pyramids. Each of these can further be cut in half to produce 2 square scalenes, so a hexadecachoron can be consructed by joining four square scalenes together.

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