Tesseractihexadecachoron
Tesseractihexadecachoron  

Rank  4 
Type  Uniform 
Notation  
Bowers style acronym  Tah 
Coxeter diagram  o4x3x3o () 
Elements  
Cells  16 truncated tetrahedra, 8 truncated octahedra 
Faces  32 triangles, 24 squares, 64 hexagons 
Edges  96+96 
Vertices  96 
Vertex figure  Digonal disphenoid, edge lengths 1 (base 1), √2 (base 2) and √3 (sides) 
Measures (edge length 1)  
Circumradius  
Hypervolume  
Dichoral angles  Toe–6–tut: 120° 
Tut–3–tut: 120°  
Toe–4–toe: 90°  
Central density  1 
Number of external pieces  24 
Level of complexity  6 
Related polytopes  
Army  Tah 
Regiment  Tah 
Dual  Disphenoidal enneacontahexachoron 
Conjugate  None 
Abstract & topological properties  
Flag count  2304 
Euler characteristic  0 
Orientable  Yes 
Properties  
Symmetry  B_{4}, order 384 
Convex  Yes 
Nature  Tame 
The tesseractihexadecachoron, or tah, also commonly called the bitruncated tesseract, is a convex uniform polychoron that consists of 16 truncated tetrahedra and 8 truncated octahedra. 2 truncated tetrahedra and 2 truncated octahedra join at each vertex. It is the medial stage of the truncation series between a tesseract and its dual hexadecachoron. As such is could also be called a bitruncated 16cell.
Gallery[edit  edit source]

B4 Coxeter plane projection

Alternate view, centered on truncated octahedron

Stereographic projection

Net

Card with vertex figure, cell counts, and crosssections
Vertex coordinates[edit  edit source]
The vertices of a tesseractihexadecachoron of edge length 1 are all permutations of:
 .
Alternatively it can be given under D4 symmetry as all even sign changes of:
 .
Representations[edit  edit source]
The tesseractihexadecachoron has the following Coxeter diagrams:
 o4x3x3o () (full symmetry)
 x3x3x *b3o () (D_{4} symmetry, great rhombated demitesseract)
 s4o3x3x () (as runcicantic tesseract)
 s4x3x3o () (also as snub)
 ooqoo4xuxux3xooox&#xt (B_{3} axial, truncated octahedronfirst)
 xxuxoo3xuxxux3ooxuxx&#xt (A_{3} axial, truncated tetrahedronfirst)
 Qqo ooq4xux3xoo&#zx (B_{3}×A_{1} symmetry)
Semiuniform variant[edit  edit source]
The tesseractihexadecachoron has a semiuniform variant of the form o4x3y3o that maintains its full symmetry. This variant uses 16 semiuniform truncated tetrahedra of form x3y3o and 8 semiuniform truncated octahedra of form o4x3y as cells, with 2 edge lengths.
With edges of length a (of square faces) and b (of triangular faces), its circumradius is given by and its hypervolume is given by .
It has coordinates given by all permutations of:
 .
There is also a semiuniform variant with D_{4} symmetry known as the great rhombated demitesseract.
Related polychora[edit  edit source]
Uniform polychoron compounds composed of tesseractihexadecachora include:
 Bitruncated demidistesseract (2)
 Bitruncated great icositetrachoron (3)
 Bitruncated great stellated tetracontoctachoron (6)
 Bitruncated dodecahedronary hexacosichoron (75)
External links[edit  edit source]
 Bowers, Jonathan. "Category 7: Bitruncates" (#299).
 Bowers, Jonathan. "Tessic Isogonals".
 Klitzing, Richard. "tah".
 Quickfur. "The Bitruncated Tesseract".
 Wikipedia contributors. "Bitruncated tesseract".