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Schlegel half-solid bitruncated 16-cell.png
Bowers style acronymTah
Coxeter diagramo4x3x3o (CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png)
Cells16 truncated tetrahedra, 8 truncated octahedra
Faces32 triangles, 24 squares, 64 hexagons
Vertex figureDigonal disphenoid, edge lengths 1 (base 1), 2 (base 2) and 3 (sides) Bitruncated 8-cell verf.png
Measures (edge length 1)
Dichoral anglesToe–6–tut: 120°
 Tut–3–tut: 120°
 Toe–4–toe: 90°
Central density1
Number of pieces24
Level of complexity6
Related polytopes
DualDisphenoidal enneacontahexachoron
Abstract properties
Euler characteristic0
Topological properties
SymmetryB4, order 384

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.

Gallery[edit | edit source]

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 (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[edit | edit source]

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 and its hypervolume is given by .

It has coordinates given by all permutations of:

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

Related polychora[edit | edit source]

o4o3o3o truncations
Name OBSA CD diagram Picture
Tesseract tes x4o3o3o
Schlegel wireframe 8-cell.png
Truncated tesseract tat x4x3o3o
Schlegel half-solid truncated tesseract.png
Rectified tesseract rit o4x3o3o
Schlegel half-solid rectified 8-cell.png
Tesseractihexadecachoron tah o4x3x3o
Schlegel half-solid bitruncated 8-cell.png
Rectified hexadecachoron = Icositetrachoron ico o4o3x3o
Schlegel half-solid rectified 16-cell.png
Truncated hexadecachoron thex o4o3x3x
Schlegel half-solid truncated 16-cell.png
Hexadecachoron hex o4o3o3x
Schlegel wireframe 16-cell.png
Small rhombated tesseract srit x4o3x3o
Schlegel half-solid cantellated 8-cell.png
Great rhombated tesseract grit x4x3x3o
Schlegel half-solid cantitruncated 8-cell.png
Small rhombated hexadecachoron = Rectified icositetrachoron rico o4x3o3x
Schlegel half-solid cantellated 16-cell.png
Great rhombated hexadecachoron = Truncated icositetrachoron tico o4x3x3x
Schlegel half-solid cantitruncated 16-cell.png
Small disprismatotesseractihexadecachoron sidpith x4o3o3x
Schlegel half-solid runcinated 8-cell.png
Prismatorhombated hexadecachoron proh x4x3o3x
Schlegel half-solid runcitruncated 8-cell.png
Prismatorhombated tesseract prit x4o3x3x
Schlegel half-solid runcitruncated 16-cell.png
Great disprismatotesseractihexadecachoron gidpith x4x3x3x
Schlegel half-solid omnitruncated 8-cell.png

External links[edit | edit source]

  • Klitzing, Richard. "tah".