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Schlegel wireframe 16-cell.png
Bowers style acronymHex
Coxeter diagramo4o3o3x (CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png)
Schläfli symbol{3,3,4}
Bracket notation<IIII>
Cells16 tetrahedra
Faces32 triangles
Vertex figureOctahedron, edge length 1 16-cell verf.png
Edge figuretet 3 tet 3 tet 3 tet 3
Petrie polygonsOctagonal-octagrammic coil:
Measures (edge length 1)
Edge radius
Face radius
Dichoral angle120°
Interior anglesAt triangle:
At edge:
At vertex:
Central density1
Number of external pieces16
Level of complexity1
Related polytopes
Abstract & topological properties
Flag count384
Euler characteristic0
SymmetryB4, order 384
Net count261

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.

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 (CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png) (full symmetry)
  • x3o3o *b3o (CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png) (D4 symmetry, as demitesseract)
  • s4o3o3o (CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png) (as alternated tesseract)
  • s2s4o3o (CDel node h.pngCDel 2.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png) (as tetrahedral antiprism/alternated cubic prism)
  • s4o2s4o (CDel node h.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node h.pngCDel 4.pngCDel node.png) (as digonal duoantiprism)
  • s2s2s4o (CDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 4.pngCDel node.png) (as disphenoidal antiprism)
  • s2s2s2s (CDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.png) (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[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:

o4o3o3o truncations
Name OBSA CD diagram Picture
Tesseract tes x4o3o3o
Schlegel wireframe 8-cell.png
Truncated tesseract tat x4x3o3o
Schlegel half-solid truncated tesseract.png
Rectified tesseract rit o4x3o3o
Schlegel half-solid rectified 8-cell.png
Tesseractihexadecachoron tah o4x3x3o
Schlegel half-solid bitruncated 8-cell.png
Rectified hexadecachoron = Icositetrachoron ico o4o3x3o
Schlegel half-solid rectified 16-cell.png
Truncated hexadecachoron thex o4o3x3x
Schlegel half-solid truncated 16-cell.png
Hexadecachoron hex o4o3o3x
Schlegel wireframe 16-cell.png
Small rhombated tesseract srit x4o3x3o
Schlegel half-solid cantellated 8-cell.png
Great rhombated tesseract grit x4x3x3o
Schlegel half-solid cantitruncated 8-cell.png
Small rhombated hexadecachoron = Rectified icositetrachoron rico o4x3o3x
Schlegel half-solid cantellated 16-cell.png
Great rhombated hexadecachoron = Truncated icositetrachoron tico o4x3x3x
Schlegel half-solid cantitruncated 16-cell.png
Small disprismatotesseractihexadecachoron sidpith x4o3o3x
Schlegel half-solid runcinated 8-cell.png
Prismatorhombated hexadecachoron proh x4x3o3x
Schlegel half-solid runcitruncated 8-cell.png
Prismatorhombated tesseract prit x4o3x3x
Schlegel half-solid runcitruncated 16-cell.png
Great disprismatotesseractihexadecachoron gidpith x4x3x3x
Schlegel half-solid omnitruncated 8-cell.png

Isogonal derivatives[edit | edit source]

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

External links[edit | edit source]

  • Klitzing, Richard. "Hex".