Truncated hexadecachoron

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Truncated hexadecachoron
Schlegel half-solid truncated 16-cell.png
Bowers style acronymThex
Coxeter diagramo4o3x3x (CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png)
Cells8 octahedra, 16 truncated tetrahedra
Faces64 triangles, 32 hexagons
Vertex figureSquare pyramid, edge lengths 1 (base) and 3 (sides) Truncated 16-cell verf.png
Measures (edge length 1)
Dichoral anglesTut–3–oct: 120°
 Tut–6–tut: 120°
Central density1
Number of external pieces24
Level of complexity4
Related polytopes
DualHexakis tesseract
Abstract & topological properties
Flag count1536
Euler characteristic0
SymmetryB4, order 384

The truncated hexadecachoron, or thex, also commonly called the truncated 16-cell, is a convex uniform polychoron that consists of 8 regular octahedra and 16 truncated tetrahedra. 1 octahedron and four truncated tetrahedra join at each vertex. As the name suggests, it can be obtained as the truncation of a hexadecachoron.

Gallery[edit | edit source]

Vertex coordinates[edit | edit source]

The vertices of a truncated hexadecachoron of edge length 1 are given by all permutations of:

Alternatively under D4 symmetry it can be given by all permutations and even sign changes of:

Representations[edit | edit source]

A truncated hexadecachoron has the following Coxeter diagrams:

  • CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png (full symmetry)
  • CDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png (D4 symmetry, as truncated demitesseract)
  • CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png (as cantic tesseract)
  • xuxo3xoox3oxux&#xt (A3 axial, truncated tetrahedron-first)
  • ooooo4ooxoo3xuxux&#xt (BC3 axial, octahedron-first)
  • ooxoo3xuxux3ooxoo&#xt (A3 axial, octahedron-first)
  • ooxxuuxd ooxxuudx QqQoqooo qQoQoqoo &#zx (A1×A1×A1×A1 symmetry)
  • oxux4oqoo xoxu4qooo&#zx (BC2×BC2 symmetry)
  • Qqo ooo4oxo3xux&#zx (BC3×A1 symmetry)

Semi-uniform variant[edit | edit source]

The truncated hexadecachoron has a semi-uniform variant of the form o4o3y3x that maintains its full symmetry. This variant uses 8 octahedra of size y and 16 semi-uniform truncated tetrahedra of form x3y3o as cells, with 2 edge lengths.

With edges of length a (surrounded by truncated tetrahedra only) and b (of octahedra), its circumradius is given by and its hypervolume is given by .

It has coordinates given by all permutations of:

Both the uniform and semi-uniform truncated hexadecachoron are also isogonal under D4 symmetry, where it can be called the truncated demitesseract or cantic tesseract, but this does not give rise to any additional variations. This can be seen by its alternative coordinates as all permutations and even sign changes of:

Related polychora[edit | edit source]

The truncated hexadecachoron is the colonel of a regiment that also includes the truncated tesseractihemioctachoron.

Uniform polychoron compounds composed of truncated 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

External links[edit | edit source]