Great rhombated tesseract

Great rhombated tesseract
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Grit
Coxeter diagram
Elements
Cells	32 triangular prisms, 16 truncated tetrahedra, 8 great rhombicuboctahedra
Faces	64 triangles, 96 squares, 64 hexagons, 24 octagons
Edges	96+96+192
Vertices	192
Vertex figure	Sphenoid edge lengths 1 (1), √2 (2), √3 (2), and √2+√2 (1)
Measures (edge length 1)
Circumradius	$\sqrt{\frac{11+5\sqrt2}{2}} ≈ 3.00592$
Hypervolume	$\frac{601+424\sqrt2}{6} ≈ 200.10443$
Dichoral angles	Tut–3–trip: 150°
	Girco–4–trip: $\arccos\left(-\frac{\sqrt3}{3}\right) ≈ 125.26439°$
	Girco–6–tut: 120°
	Girco–8–girco: 90°
Central density	1
Number of pieces	56
Level of complexity	12
Related polytopes
Army	Grit
Regiment	Grit
Dual	Sphenoidal hecatonenneacontadichoron
Conjugate	Great quasirhombated tesseract
Abstract properties
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	B4, order 384
Convex	Yes
Nature	Tame

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

• 

  B4 Coxeter plane projection

• 

  Stereographic projection

• 

  Net

• 

  Cross-sections

Vertex coordinates

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

• $\left(±\frac{1+2\sqrt2}{2},\,±\frac{1+2\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac12\right).$

Representations

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

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 $\sqrt{\frac{2a^2+3b^2+2c^2+4bc+(3ab+2ac)\sqrt2}{2} }$.

It has coordinates given by all permutations of:

• $\left(±\frac{a+(b+c)\sqrt2}{2},\,±\frac{a+(b+c)\sqrt2}{2},\,±\frac{a+b\sqrt2}{2},\,±\frac{a}{2}\right).$

External links

• Bowers, Jonathan. "Category 8: Great Rhombates" (#308).

• Bowers, Jonathan. "Tessic Isogonals".

• Klitzing, Richard. "grit".

• Quickfur. "The Cantitruncated Tesseract".

• Wikipedia Contributors. "Cantitruncated tesseract".