 Rank4
TypeUniform
SpaceSpherical
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$\frac{3\sqrt2}{2} ≈ 2.12132$ Hypervolume$\frac{307}{6} ≈ 51.16667$ Dichoral anglesToe–6–tut: 120°
Tut–3–tut: 120°
Toe–4–toe: 90°
Central density1
Number of pieces24
Level of complexity6
Related polytopes
ArmyTah
RegimentTah
DualDisphenoidal enneacontahexachoron
ConjugateNone
Abstract properties
Euler characteristic0
Topological properties
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:

• $\left(±\sqrt2,\,±\sqrt2,\,±\frac{\sqrt2}{2},\,0\right).$ Alternatively it can be given under D4 symmetry as all even sign changes of:

• $\left(\frac{5\sqrt2}{4},\,\frac{3\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4}\right).$ ## 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 (BC3 axial, truncated octahedron-first)
• xxuxoo3xuxxux3ooxuxx&#xt (A3 axial, truncated tetrahedron-first)
• Qqo ooq4xux3xoo&#zx (BC3×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 $\sqrt{\frac{3a^2+2b^2+4ab}{2}}$ and its hypervolume is given by $\frac{23a^4+88a^3b+120a^2b^2+64ab^3+12b^4}{6}$ .

It has coordinates given by all permutations of:

• $\left(±(a+b)\frac{\sqrt2}{2},\,±(a+b)\frac{\sqrt2}{2},\,±a\frac{\sqrt2}{2},\,0\right).$ There is also a semi-uniform variant with D4 symmetry known as the great rhombated demitesseract.