Truncated tesseract

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Truncated tesseract
Schlegel half-solid truncated tesseract.png
Bowers style acronymTat
Coxeter diagramx4x3o3o (CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png)
Cells16 tetrahedra, 8 truncated cubes
Faces64 triangles, 24 octagons
Vertex figureTriangular pyramid, edge lengths 1 (base) and 2+2 (sides) Truncated 8-cell verf.png
Measures (edge length 1)
Dichoral anglesTic–3–tet: 120°
 Tic–8–tic: 90°
Central density1
Number of external pieces24
Level of complexity4
Related polytopes
DualTetrakis hexadecachoron
ConjugateQuasitruncated tesseract
Abstract & topological properties
Flag count1536
Euler characteristic0
SymmetryB4, order 384

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]

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 (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 (A1×A1×A1×A1 symmetry)

Semi-uniform variant[edit | edit source]

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 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:

o4o3o3o truncations
Name OBSA CD diagram Picture
Tesseract tes x4o3o3o
Schlegel wireframe 8-cell.png
Truncated tesseract tat x4x3o3o
Schlegel half-solid truncated tesseract.png
Rectified tesseract rit o4x3o3o
Schlegel half-solid rectified 8-cell.png
Tesseractihexadecachoron tah o4x3x3o
Schlegel half-solid bitruncated 8-cell.png
Rectified hexadecachoron = Icositetrachoron ico o4o3x3o
Schlegel half-solid rectified 16-cell.png
Truncated hexadecachoron thex o4o3x3x
Schlegel half-solid truncated 16-cell.png
Hexadecachoron hex o4o3o3x
Schlegel wireframe 16-cell.png
Small rhombated tesseract srit x4o3x3o
Schlegel half-solid cantellated 8-cell.png
Great rhombated tesseract grit x4x3x3o
Schlegel half-solid cantitruncated 8-cell.png
Small rhombated hexadecachoron = Rectified icositetrachoron rico o4x3o3x
Schlegel half-solid cantellated 16-cell.png
Great rhombated hexadecachoron = Truncated icositetrachoron tico o4x3x3x
Schlegel half-solid cantitruncated 16-cell.png
Small disprismatotesseractihexadecachoron sidpith x4o3o3x
Schlegel half-solid runcinated 8-cell.png
Prismatorhombated hexadecachoron proh x4x3o3x
Schlegel half-solid runcitruncated 8-cell.png
Prismatorhombated tesseract prit x4o3x3x
Schlegel half-solid runcitruncated 16-cell.png
Great disprismatotesseractihexadecachoron gidpith x4x3x3x
Schlegel half-solid omnitruncated 8-cell.png

External links[edit | edit source]

  • Klitzing, Richard. "tat".