Great rhombated tesseract

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Great rhombated tesseract
Schlegel half-solid cantitruncated 8-cell.png
Bowers style acronymGrit
Coxeter diagramCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
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) Cantitruncated 8-cell verf.png
Measures (edge length 1)
Dichoral anglesTut–3–trip: 150°
 Girco–6–tut: 120°
 Girco–8–girco: 90°
Central density1
Number of pieces56
Level of complexity12
Related polytopes
DualSphenoidal hecatonenneacontadichoron
ConjugateGreat quasirhombated tesseract
Abstract properties
Euler characteristic0
Topological properties
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 (BC3 symmetry, great rhombicuboctahedron-first)
  • wx3xx3xw *b3oo&#zx (D4 symmetry)
  • Xwx xxw4xux3xoo&#zx (BC3×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]

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]