Great rhombated tesseract

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Great rhombated tesseract
Bowers style acronymGrit
Coxeter diagramx4x3x3o ()
Cells32 triangular prisms, 16 truncated tetrahedra, 8 great rhombicuboctahedra
Faces64 triangles, 96 squares, 64 hexagons, 24 octagons
Vertex figureSphenoid edge lengths 1 (1), 2 (2), 3 (2), and 2+2 (1)
Measures (edge length 1)
Dichoral anglesTut–3–trip: 150°
 Girco–6–tut: 120°
 Girco–8–girco: 90°
Central density1
Number of external pieces56
Level of complexity12
Related polytopes
DualSphenoidal hecatonenneacontadichoron
ConjugateGreat quasirhombated tesseract
Abstract & topological properties
Flag count4608
Euler characteristic0
SymmetryB4, order 384

The great rhombated tesseract, or grit, also commonly called the cantitruncated tesseract, is a convex uniform polychoron that consists of 32 triangular prisms, 16 truncated tetrahedra, and 8 great rhombicuboctahedra. 1 triangular prism, 1 truncated tetrahedron, and 2 great rhombicuboctahedra join at each vertex. As one of its names suggests, it can be obtained by cantitruncating the tesseract.

Gallery[edit | edit source]

Vertex coordinates[edit | edit source]

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

  • .

Representations[edit | edit source]

A great rhombated tesseract has the following Coxeter diagrams:

  • x4x3x3o () (full symmetry)
  • xxwwxx4xuxxux3xoooox&#xt (B3 symmetry, great rhombicuboctahedron-first)
  • wx3xx3xw *b3oo&#zx (D4 symmetry)
  • Xwx xxw4xux3xoo&#zx (B3×A1 symmetry)

Semi-uniform variant[edit | edit source]

The great rhombated tesseract has a semi-uniform variant of the form x4y3z3o that maintains its full symmetry. This variant uses 16 truncated tetrahedra of form y3z3o, 8 great rhombicuboctahedra of form x4y3z, and 32 triangular prisms of form x z3o as cells, with 3 edge lengths.

With edges of length a, b, and c (such that it forms a4b3c3o), its circumradius is given by .

It has coordinates given by all permutations of:

  • .

Related polychora[edit | edit source]

Uniform polychoron compounds composed of great rhombated tesseracts include:

External links[edit | edit source]