Truncated hexadecachoron

Truncated hexadecachoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Thex
Coxeter diagram	o4o3x3x ()
Elements
Cells	8 octahedra, 16 truncated tetrahedra
Faces	64 triangles, 32 hexagons
Edges	24+96
Vertices	48
Vertex figure	Square pyramid, edge lengths 1 (base) and √3 (sides)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {10}}{2}}\approx 1.58114$
Hypervolume	${\frac {77}{6}}\approx 12.83333$
Dichoral angles	Tut–3–oct: 120°
	Tut–6–tut: 120°
Central density	1
Number of external pieces	24
Level of complexity	4
Related polytopes
Army	Thex
Regiment	Thex
Dual	Hexakis tesseract
Conjugate	None
Abstract & topological properties
Flag count	1536
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B4, order 384
Convex	Yes
Nature	Tame

The truncated hexadecachoron, or thex, also commonly called the truncated 16-cell, is a convex uniform polychoron that consists of 8 regular octahedra and 16 truncated tetrahedra. 1 octahedron and four truncated tetrahedra join at each vertex. As the name suggests, it can be obtained as the truncation of a hexadecachoron.

Gallery

• 

  B4 Coxeter plane projection

• 

  Stereographic projection

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  Net

• 

  Cross-sections

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  Cross-section animation

Vertex coordinates

The vertices of a truncated hexadecachoron of edge length 1 can be given as all permutations of:

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

Multiplying these coordinates by ${\sqrt {2}}$ gives integral coordinates.

Alternatively under D₄ symmetry it can be given by all permutations and even sign changes of:

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

Representations

A truncated hexadecachoron has the following Coxeter diagrams:

• x3x3o4o () (full symmetry)
• x3x3o *b3o () (D₄ symmetry, as truncated demitesseract)
• s4o3x3o () (as cantic tesseract)
• xuxo3xoox3oxux&#xt (A₃ axial, truncated tetrahedron-first)
• ooooo4ooxoo3xuxux&#xt (B₃ axial, octahedron-first)
• ooxoo3xuxux3ooxoo&#xt (A₃ axial, octahedron-first)
• ooxxuuxd ooxxuudx QqQoqooo qQoQoqoo &#zx (K₄ symmetry)
• oxux4oqoo xoxu4qooo&#zx (B₂×B₂ symmetry)
• Qqo ooo4oxo3xux&#zx (B₃×A₁ symmetry)

Semi-uniform variant

The truncated hexadecachoron has a semi-uniform variant of the form o4o3y3x that maintains its full symmetry. This variant uses 8 octahedra of size y and 16 semi-uniform truncated tetrahedra of form x3y3o as cells, with 2 edge lengths.

With edges of length a (surrounded by truncated tetrahedra only) and b (of octahedra), its circumradius is given by ${\sqrt {\frac {a^{2}+2b^{2}+2ab}{2}}}$ and its hypervolume is given by ${\frac {a^{4}+8a^{3}b+24a^{2}b^{2}+32ab^{3}+12b^{4}}{6}}$.

It has coordinates given by all permutations of:

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

Both the uniform and semi-uniform truncated hexadecachoron are also isogonal under D₄ symmetry, where it can be called the truncated demitesseract or cantic tesseract, but this does not give rise to any additional variations. This can be seen by its alternative coordinates as all permutations and even sign changes of:

• $\left((a+2b){\frac {\sqrt {2}}{4}},\,(a+2b){\frac {\sqrt {2}}{4}},\,a{\frac {\sqrt {2}}{4}},\,a{\frac {\sqrt {2}}{4}}\right)$.

Related polychora

The truncated hexadecachoron is the colonel of a regiment that also includes the truncated tesseractihemioctachoron.

Uniform polychoron compounds composed of truncated hexadecachora include:

• Truncated demidistesseract (2)
• Truncated stellated icositetrachoron (3)
• Truncated small stellated tetracontoctachoron (6)
• Truncated stellated dodecahedronary hexacosichoron (75)
• Truncated stellated hexacosichoron (75)

External links

• Bowers, Jonathan. "Category 2: Truncates" (#20).

• Bowers, Jonathan. "Tessic Isogonals".

• Klitzing, Richard. "thex".
• Quickfur. "The Truncated 16-cell".

• Wikipedia contributors. "Truncated 16-cell".