Tesseractihexadecachoron

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Tesseractihexadecachoron
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
Hypervolume
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.

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 (B3 axial, truncated octahedron-first)
  • xxuxoo3xuxxux3ooxuxx&#xt (A3 axial, truncated tetrahedron-first)
  • Qqo ooq4xux3xoo&#zx (B3×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]

Uniform polychoron compounds composed of tesseractihexadecachora include:

External links[edit | edit source]