Rank4
TypeUniform
Notation
Bowers style acronymTah
Coxeter diagramo4x3x3o ()
Elements
Cells16 truncated tetrahedra, 8 truncated octahedra
Faces32 triangles, 24 squares, 64 hexagons
Edges96+96
Vertices96
Vertex figureDigonal disphenoid, edge lengths 1 (base 1), 2 (base 2) and 3 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {3{\sqrt {2}}}{2}}\approx 2.12132}$
Hypervolume${\displaystyle {\frac {307}{6}}\approx 51.16667}$
Dichoral anglesToe–6–tut: 120°
Tut–3–tut: 120°
Toe–4–toe: 90°
Central density1
Number of external pieces24
Level of complexity6
Related polytopes
ArmyTah
RegimentTah
DualDisphenoidal enneacontahexachoron
ConjugateNone
Abstract & topological properties
Flag count2304
Euler characteristic0
OrientableYes
Properties
SymmetryB4, order 384
ConvexYes
NatureTame

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 16-cell.

## Vertex coordinates

The vertices of a tesseractihexadecachoron of edge length 1 are all permutations of:

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

Alternatively it can be given under D4 symmetry as all even sign changes of:

• ${\displaystyle \left({\frac {5{\sqrt {2}}}{4}},\,{\frac {3{\sqrt {2}}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}}\right)}$.

## Representations

The tesseractihexadecachoron has the following Coxeter diagrams:

• o4x3x3o () (full symmetry)
• x3x3x *b3o () (D4 symmetry, great rhombated demitesseract)
• s4o3x3x () (as runcicantic tesseract)
• s4x3x3o () (also as snub)
• ooqoo4xuxux3xooox&#xt (B3 axial, truncated octahedron-first)
• xxuxoo3xuxxux3ooxuxx&#xt (A3 axial, truncated tetrahedron-first)
• Qqo ooq4xux3xoo&#zx (B3×A1 symmetry)

## Semi-uniform variant

The tesseractihexadecachoron has a semi-uniform variant of the form o4x3y3o that maintains its full symmetry. This variant uses 16 semi-uniform truncated tetrahedra of form x3y3o and 8 semi-uniform 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 ${\displaystyle {\sqrt {\frac {3a^{2}+2b^{2}+4ab}{2}}}}$ and its hypervolume is given by ${\displaystyle {\frac {23a^{4}+88a^{3}b+120a^{2}b^{2}+64ab^{3}+12b^{4}}{6}}}$.

It has coordinates given by all permutations of:

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

There is also a semi-uniform variant with D4 symmetry known as the great rhombated demitesseract.

## Related polychora

Uniform polychoron compounds composed of tesseractihexadecachora include: