Hexadecachoron

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Hexadecachoron
Schlegel wireframe 16-cell.png
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
Edge radius
Face radius
Inradius
Hypervolume
Dichoral angle120°
Height
Central density1
Euler characteristic0
Number of pieces16
Level of complexity1
Related polytopes
DualTesseract
ConjugateHexadecachoron
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.

Cross-sections[edit | edit source]

Hex sections Bowers.png

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.

Surtope Angles[edit | edit source]

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[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 (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)

Segmentochoron display[edit | edit source]

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.

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} CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Schlegel wireframe 8-cell.png
Truncated tesseract tat t{4,3,3} CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Schlegel half-solid truncated tesseract.png
Rectified tesseract rit r{4,3,3} CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Schlegel half-solid rectified 8-cell.png
Tesseractihexadecachoron tah 2t{4,3,3} CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Schlegel half-solid bitruncated 8-cell.png
Rectified hexadecachoron = Icositetrachoron ico r{3,3,4} CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Schlegel half-solid rectified 16-cell.png
Truncated hexadecachoron thex t{3,3,4} CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schlegel half-solid truncated 16-cell.png
Hexadecachoron hex {3,3,4} CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Schlegel wireframe 16-cell.png
Small rhombated tesseract srit rr{4,3,3} CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Schlegel half-solid cantellated 8-cell.png
Great rhombated tesseract grit tr{4,3,3} CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Schlegel half-solid cantitruncated 8-cell.png
Small rhombated hexadecachoron = Rectified icositetrachoron rico rr{3,3,4} CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Schlegel half-solid cantellated 16-cell.png
Great rhombated hexadecachoron = Truncated icositetrachoron tico tr{3,3,4} CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schlegel half-solid cantitruncated 16-cell.png
Small disprismatotesseractihexadecachoron sidpith t0,3{4,3,3} CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Schlegel half-solid runcinated 8-cell.png
Prismatorhombated hexadecachoron proh t0,1,3{4,3,3} CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Schlegel half-solid runcitruncated 8-cell.png
Prismatorhombated tesseract prit t0,1,3{3,3,4} CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schlegel half-solid runcitruncated 16-cell.png
Great disprismatotesseractihexadecachoron gidpith t0,1,2,3{4,3,3} CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
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".