Truncated hexadecachoron
Truncated hexadecachoron  

Rank  4 
Type  Uniform 
Notation  
Bowers style acronym  Thex 
Coxeter diagram  o4o3x3x () 
Elements  
Cells  8 octahedra, 16 truncated tetrahedra 
Faces  64 triangles, 32 hexagons 
Edges  24+96 
Vertices  48 
Vertex figure  Square pyramid, edge lengths 1 (base) and √3 (sides) 
Measures (edge length 1)  
Circumradius  
Hypervolume  
Dichoral angles  Tut–3–oct: 120° 
Tut–6–tut: 120°  
Central density  1 
Number of external pieces  24 
Level of complexity  4 
Related polytopes  
Army  Thex 
Regiment  Thex 
Dual  Hexakis tesseract 
Conjugate  None 
Abstract & topological properties  
Flag count  1536 
Euler characteristic  0 
Orientable  Yes 
Properties  
Symmetry  B_{4}, order 384 
Convex  Yes 
Nature  Tame 
The truncated hexadecachoron, or thex, also commonly called the truncated 16cell, 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]

B4 Coxeter plane projection

Stereographic projection

Net

Crosssection animation
Vertex coordinates[edit  edit source]
The vertices of a truncated hexadecachoron of edge length 1 can be given as all permutations of:
 .
Multiplying these coordinates by gives integral coordinates.
Alternatively under D_{4} 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:
 x3x3o4o () (full symmetry)
 x3x3o *b3o () (D_{4} symmetry, as truncated demitesseract)
 s4o3x3o () (as cantic tesseract)
 xuxo3xoox3oxux&#xt (A_{3} axial, truncated tetrahedronfirst)
 ooooo4ooxoo3xuxux&#xt (B_{3} axial, octahedronfirst)
 ooxoo3xuxux3ooxoo&#xt (A_{3} axial, octahedronfirst)
 ooxxuuxd ooxxuudx QqQoqooo qQoQoqoo &#zx (K_{4} symmetry)
 oxux4oqoo xoxu4qooo&#zx (B_{2}×B_{2} symmetry)
 Qqo ooo4oxo3xux&#zx (B_{3}×A_{1} symmetry)
Semiuniform variant[edit  edit source]
The truncated hexadecachoron has a semiuniform variant of the form o4o3y3x that maintains its full symmetry. This variant uses 8 octahedra of size y and 16 semiuniform 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 semiuniform truncated hexadecachoron are also isogonal under D_{4} 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:
 Truncated demidistesseract (2)
 Truncated stellated icositetrachoron (3)
 Truncated small stellated tetracontoctachoron (6)
 Truncated stellated dodecahedronary hexacosichoron (75)
 Truncated stellated hexacosichoron (75)
External links[edit  edit source]
 Bowers, Jonathan. "Category 2: Truncates" (#20).
 Bowers, Jonathan. "Tessic Isogonals".
 Klitzing, Richard. "thex".
 Quickfur. "The Truncated 16cell".
 Wikipedia contributors. "Truncated 16cell".