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, the digonal duotransitionaltertegum, 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 tegum.

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 hexadecachoron company 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.

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 (B3 axial, as octahedral tegum)
 * ooo3oxo3ooo&#xt (A3 axial, as tetratetrahedral tegum)
 * o(qo)o o(ox)o4o(oo)o&#xt (as square tegmatic tegum)
 * o(qoo)o o(oqo)o o(ooq)o&#xt (as rhombic tegmatic tegum)
 * xox oxo4ooo&#xt (B2×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 (B2×B2 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 (B2×A1 symmetry)
 * qo oo3ox3oo&#zx (A3×A1 symmetry)
 * qo os2os3os&#zx (as triangular antiprismatic tegum)
 * qooo oqoo ooqo oooq&#zx (A1×A1×A1×A1 symmetry)
 * qoo oqo oox4ooo&#zx (B2×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 tegum - 16 identical triangular pyramids
 * Square duotegum - 16 identical tetragonal disphenoids, noble
 * Square-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.

Uniform polychoron compounds composed of hexadecachora include:


 * Demidistesseract (2)
 * Stellated icositetrachoron (3)
 * Chiricosahedral antiprism (5)
 * Small stellated tetracontoctachoron (6)
 * Snubahedral antiprism (6)
 * Small snubahedral antiprism (6)
 * Chirocrossifissal hexacontatetrachoron (8)
 * Chirotrigonal antiprismatic enneacontahexachoron (12)
 * Disnubahedral antiprism (12)
 * Stellated chirohecatonicosachoron (15)
 * Crossifissal dishexacontatetrachoron (16)
 * Compound of 18 hexadecachora (18)
 * Bitrigonal antiprismatic enneacontahexachoron (24)
 * Stellated diacositetracontachoron (30)
 * Stellated chirodishecatonicosachoron (30)
 * Stellated triacosihexacontachoron (45)
 * Stellated chirotrishecatonicosachoron (45)
 * Compound of 50 hexadecachora (50)
 * Stellated tetracosioctacontachoron (60)
 * Stellated chirotetrishecatonicosachoron (60)
 * Stellated dodecahedronary hexacosichoron (75)
 * Stellated hexacosichoron (75)
 * Stellated snub dodecahedronary pentishecatonicosachoron (375)
 * Dodecahedronary disnubachoron (600)
 * Compound of 675 hexadecachora (675)
 * Several infinite families based on treating the hexadecachoron as a square duotegum, digonal duoantiprism, and 8-3 step prism

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