Truncated tesseract
Truncated tesseract  

Rank  4 
Type  Uniform 
Notation  
Bowers style acronym  Tat 
Coxeter diagram  x4x3o3o () 
Elements  
Cells  16 tetrahedra, 8 truncated cubes 
Faces  64 triangles, 24 octagons 
Edges  32+96 
Vertices  64 
Vertex figure  Triangular pyramid, edge lengths 1 (base) and √2+√2 (sides) 
Measures (edge length 1)  
Circumradius  
Hypervolume  
Dichoral angles  Tic–3–tet: 120° 
Tic–8–tic: 90°  
Central density  1 
Number of external pieces  24 
Level of complexity  4 
Related polytopes  
Army  Tat 
Regiment  Tat 
Dual  Tetrakis hexadecachoron 
Conjugate  Quasitruncated tesseract 
Abstract & topological properties  
Flag count  1536 
Euler characteristic  0 
Orientable  Yes 
Properties  
Symmetry  B_{4}, order 384 
Flag orbits  4 
Convex  Yes 
Nature  Tame 
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.
Gallery[edit  edit source]

B4 Coxeter plane projection

Stereographic projection

Net
Vertex coordinates[edit  edit source]
The vertices of a truncated tesseract of edge length 1 are given by all permutations of:
 .
Representations[edit  edit source]
A truncated tesseract has the following Coxeter diagramms:
 x4x3o3o () (full symmetry)
 xwwx4xoox3oooo&#xt (B_{3} symmetry, truncated cubefirst)
 xwwxoooo3ooxwwxoo3ooooxwwx&#xt (A_{3} axial, tetrahedronfirst)
 wx3oo3xw *b3oo&#zx (D_{4} symmetry)
 wx xw4xo3oo&#zx (B_{3}×A_{1} symmetry)
 ox4wx xo4xw&#zx (B_{2}×B_{2} symmetry, truncated square duoprism)
 xwww wxww wwxw wwwx&#zx (K_{4} symmetry)
Semiuniform variant[edit  edit source]
The truncated tesseract has a semiuniform variant of the form x4y3o3o that maintains its full symmetry. This variant uses 16 tetrahedra of size y and 8 semiuniform 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 and its hypervolume is given by .
It has coordinates given by all permutations of:
 .
Related polychora[edit  edit source]
Uniform polychoron compounds composed of truncated tesseracts include:
 Truncated great icositetrachoron (3)
 Truncated great stellated tetracontoctachoron (6)
 Truncated dodecahedronary cubichoron (75)
External links[edit  edit source]
 Bowers, Jonathan. "Category 2: Truncates" (#19).
 Bowers, Jonathan. "Tessic Isogonals".
 Klitzing, Richard. "tat".
 Quickfur. "The Truncated Tesseract".
 Wikipedia contributors. "Truncated tesseract".