Tesseractihexadecachoron

Tesseractihexadecachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
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	$\frac{3\sqrt2}{2} ≈ 2.12132$
Hypervolume	$\frac{307}{6} ≈ 51.16667$
Dichoral angles	Toe–6–tut: 120°
	Tut–3–tut: 120°
	Toe–4–toe: 90°
Central density	1
Number of pieces	24
Level of complexity	6
Related polytopes
Army	Tah
Regiment	Tah
Dual	Disphenoidal enneacontahexachoron
Conjugate	None
Abstract properties
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	B4, 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 16-cell.

Gallery

• 

  B4 Coxeter plane projection

• 

  Alternate view, centered on truncated octahedron

• 

  Stereographic projection

• 

  Net

• 

  Cross-sections

• 

  Card with vertex figure, cell counts, and cross-sections

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.

External links

• Bowers, Jonathan. "Category 7: Bitruncates" (#299).

• Bowers, Jonathan. "Tessic Isogonals".

• Klitzing, Richard. "tah".

• Quickfur. "The Bitruncated Tesseract".

• Wikipedia Contributors. "Bitruncated tesseract".