Truncated hexadecachoron

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.

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


Vertex coordinatesEdit

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:



A truncated hexadecachoron has the following Coxeter diagrams:

  •         (full symmetry)
  •       (D4 symmetry, as truncated demitesseract)
  •         (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 variantEdit

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 polychoraEdit

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
Truncated tesseract tat x4x3o3o
Rectified tesseract rit o4x3o3o
Tesseractihexadecachoron tah o4x3x3o
Rectified hexadecachoron = Icositetrachoron ico o4o3x3o
Truncated hexadecachoron thex o4o3x3x
Hexadecachoron hex o4o3o3x
Small rhombated tesseract srit x4o3x3o
Great rhombated tesseract grit x4x3x3o
Small rhombated hexadecachoron = Rectified icositetrachoron rico o4x3o3x
Great rhombated hexadecachoron = Truncated icositetrachoron tico o4x3x3x
Small disprismatotesseractihexadecachoron sidpith x4o3o3x
Prismatorhombated hexadecachoron proh x4x3o3x
Prismatorhombated tesseract prit x4o3x3x
Great disprismatotesseractihexadecachoron gidpith x4x3x3x

External linksEdit