# Truncated tesseract

(Redirected from Tat)
Truncated tesseract
Rank4
TypeUniform
Notation
Bowers style acronymTat
Coxeter diagramx4x3o3o ()
Elements
Cells16 tetrahedra, 8 truncated cubes
Faces64 triangles, 24 octagons
Edges32+96
Vertices64
Vertex figureTriangular pyramid, edge lengths 1 (base) and 2+2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5+3{\sqrt {2}}}{2}}}\approx 2.14973}$
Hypervolume${\displaystyle {\frac {101+72{\sqrt {2}}}{6}}\approx 33.80390}$
Dichoral anglesTic–3–tet: 120°
Tic–8–tic: 90°
Central density1
Number of external pieces24
Level of complexity4
Related polytopes
ArmyTat
RegimentTat
ConjugateQuasitruncated tesseract
Abstract & topological properties
Flag count1536
Euler characteristic0
OrientableYes
Properties
SymmetryB4, order 384
Flag orbits4
ConvexYes
NatureTame

The truncated tesseract, or tat, is a convex uniform polychoron that consists of 16 regular tetrahedra and 8 truncated cubes. 1 tetrahedron and three truncated cubes join at each vertex. As the name suggests, it can be obtained by truncating the tesseract.

As the truncated tesseract, it is the square member of an infinite family of isogonal truncated duoprisms, and could be called the truncated square duoprism.

## Vertex coordinates

The vertices of a truncated tesseract of edge length 1 are given by all permutations of:

• ${\displaystyle \left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right)}$.

## Representations

A truncated tesseract has the following Coxeter diagramms:

• x4x3o3o () (full symmetry)
• xwwx4xoox3oooo&#xt (B3 symmetry, truncated cube-first)
• xwwxoooo3ooxwwxoo3ooooxwwx&#xt (A3 axial, tetrahedron-first)
• wx3oo3xw *b3oo&#zx (D4 symmetry)
• wx xw4xo3oo&#zx (B3×A1 symmetry)
• ox4wx xo4xw&#zx (B2×B2 symmetry, truncated square duoprism)
• xwww wxww wwxw wwwx&#zx (K4 symmetry)

## Semi-uniform variant

The truncated tesseract has a semi-uniform variant of the form x4y3o3o that maintains its full symmetry. This variant uses 16 tetrahedra of size y and 8 semi-uniform truncated cubes of form x4y3o as cells, with 2 edge lengths.

With edges of length a (surrounded by truncated cubes only) and b (of tetrahedra), its circumradius is given by ${\displaystyle {\sqrt {\frac {2a^{2}+3b^{2}+3ab{\sqrt {2}}}{2}}}}$ and its hypervolume is given by ${\displaystyle {\frac {6a^{4}+72a^{2}b^{2}+23b^{4}+(24a^{3}b+48ab^{3}){\sqrt {2}}}{6}}}$.

It has coordinates given by all permutations of:

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

## Related polychora

Uniform polychoron compounds composed of truncated tesseracts include: