(Redirected from 16 cell)
Rank4
TypeRegular
SpaceSpherical
Bowers style acronymHex
Info
Coxeter diagramo4o3o3x
Schläfli symbol{3,3,4}
Bracket notation<IIII>
SymmetryBC4, order 384
ArmyHex
RegimentHex
Elements
Vertex figureOctahedron, edge length 1
Cells16 tetrahedra
Faces32 triangles
Edges24
Vertices8
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt2}{2} ≈ 0.70711}$
Edge radius${\displaystyle \frac12 = 0.5}$
Face radius${\displaystyle \frac{\sqrt6}{6} ≈ 0.40825}$
Inradius${\displaystyle \frac{\sqrt2}{4} ≈ 0.35355}$
Hypervolume${\displaystyle \frac16 ≈ 0.1667}$
Dichoral angle120°
Height${\displaystyle \frac{\sqrt2}{2} ≈ 0.70711}$
Central density1
Euler characteristic0
Number of pieces16
Level of complexity1
Related polytopes
DualTesseract
Properties
ConvexYes
OrientableYes
NatureTame

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 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 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:

• ${\displaystyle \left(±\frac{\sqrt2}{2},\,0,\,0,\,0\right).}$

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

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

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 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 (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 tegum)
• 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:

## 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.

Two of the seven regular polychoron compounds are composed of hexadecachora:

o4o3o3o truncations
Name OBSA Schläfli symbol CD diagram Picture
Tesseract tes {4,3,3}
Truncated tesseract tat t{4,3,3}
Rectified tesseract rit r{4,3,3}
Rectified hexadecachoron = Icositetrachoron ico r{3,3,4}