Hexadecachoron

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Hexadecachoron
Rank4
TypeRegular
Notation
Bowers style acronymHex
Coxeter diagramo4o3o3x ()
Schläfli symbol{3,3,4}
Bracket notation<IIII>
Elements
Cells16 tetrahedra
Faces32 triangles
Edges24
Vertices8
Vertex figureOctahedron, edge length 1
Edge figuretet 3 tet 3 tet 3 tet 3
Petrie polygonsOctagonal-octagrammic coil:
Measures (edge length 1)
Circumradius
Edge radius
Face radius
Inradius
Hypervolume
Dichoral angle120°
Interior anglesAt triangle:
At edge:
At vertex:
Height
Central density1
Number of external pieces16
Level of complexity1
Related polytopes
ArmyHex
RegimentHex
DualTesseract
ConjugateNone
Van Oss polygonSquare
Abstract & topological properties
Flag count384
Euler characteristic0
OrientableYes
Properties
SymmetryB4, order 384
ConvexYes
Net count261
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, 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.

If the vertex coordinates are mapped from to the complex coordinate space in the obvious way, the resulting set of points is precisely the vertices of the Möbius-Kantor polygon, which is a complex polygon. The containing planes of the hexadecachoron's triangular faces become the edges of the Möbius-Kantor polygon (which are not edges in the traditional sense, but instead complex 1-spaces that contain three points each).

Gallery[edit | edit source]

Vertex coordinates[edit | edit source]

The vertices of a regular hexadecachoron of edge length 1, centered at the origin, are given by all permutations of:

  • .

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

  • .

These are formed by alternating the vertices of a tesseract.

Representations[edit | edit source]

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 (K2 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 (K2 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 (K4 symmetry)
  • qoo oqo oox4ooo&#zx (B2×A1×A1 symmetry)

Variations[edit | edit source]

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

Related polychora[edit | edit source]

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:

Isogonal derivatives[edit | edit source]

Substitution by vertices of these following elements will produce these convex isogonal polychora:

External links[edit | edit source]