Rank4
TypeUniform
Notation
Bowers style acronymThex
Coxeter diagramo4o3x3x ()
Elements
Cells8 octahedra, 16 truncated tetrahedra
Faces64 triangles, 32 hexagons
Edges24+96
Vertices48
Vertex figureSquare pyramid, edge lengths 1 (base) and 3 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {10}}{2}}\approx 1.58114}$
Hypervolume${\displaystyle {\frac {77}{6}}\approx 12.83333}$
Dichoral anglesTut–3–oct: 120°
Tut–6–tut: 120°
Central density1
Number of external pieces24
Level of complexity4
Related polytopes
ArmyThex
RegimentThex
DualHexakis tesseract
ConjugateNone
Abstract & topological properties
Flag count1536
Euler characteristic0
OrientableYes
Properties
SymmetryB4, order 384
ConvexYes
NatureTame

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.

## Vertex coordinates

The vertices of a truncated hexadecachoron of edge length 1 can be given as all permutations of:

• ${\displaystyle \left(\pm {\sqrt {2}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,0\right)}$.

Multiplying these coordinates by ${\displaystyle {\sqrt {2}}}$ gives integral coordinates.

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

• ${\displaystyle \left({\frac {3{\sqrt {2}}}{4}},\,{\frac {3{\sqrt {2}}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}}\right)}$.

## Representations

A truncated hexadecachoron has the following Coxeter diagrams:

• x3x3o4o () (full symmetry)
• x3x3o *b3o () (D4 symmetry, as truncated demitesseract)
• s4o3x3o () (as cantic tesseract)
• xuxo3xoox3oxux&#xt (A3 axial, truncated tetrahedron-first)
• ooooo4ooxoo3xuxux&#xt (B3 axial, octahedron-first)
• ooxoo3xuxux3ooxoo&#xt (A3 axial, octahedron-first)
• ooxxuuxd ooxxuudx QqQoqooo qQoQoqoo &#zx (K4 symmetry)
• oxux4oqoo xoxu4qooo&#zx (B2×B2 symmetry)
• Qqo ooo4oxo3xux&#zx (B3×A1 symmetry)

## Semi-uniform variant

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 ${\displaystyle {\sqrt {\frac {a^{2}+2b^{2}+2ab}{2}}}}$ and its hypervolume is given by ${\displaystyle {\frac {a^{4}+8a^{3}b+24a^{2}b^{2}+32ab^{3}+12b^{4}}{6}}}$.

It has coordinates given by all permutations of:

• ${\displaystyle \left(\pm (a+b){\frac {\sqrt {2}}{2}},\,\pm {\frac {b{\sqrt {2}}}{2}},\,0,\,0\right)}$.

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:

• ${\displaystyle \left((a+2b){\frac {\sqrt {2}}{4}},\,(a+2b){\frac {\sqrt {2}}{4}},\,a{\frac {\sqrt {2}}{4}},\,a{\frac {\sqrt {2}}{4}}\right)}$.

## Related polychora

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

Uniform polychoron compounds composed of truncated hexadecachora include: