Great rhombated hexateron

Great rhombated hexateron
(OFF file)
Rank	5
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Garx
Coxeter diagram	x3x3x3o3o ()
Elements
Tera	15 tetrahedral prisms, 6 truncated pentachora, 6 great rhombated pentachora
Cells	30 tetrahedra, 60 triangular prisms, 30 truncated tetrahedra, 15 truncated octahedra
Faces	120 triangles, 90 squares, 20+60 hexagons
Edges	60+60+180
Vertices	120
Vertex figure	Triangular pyramidal pyramid, edge lengths 1 (base), √2 (sides of pyramid 1), and √3 (sides of pyramid 2 and lacing between apices)
Measures (edge length 1)
Circumradius	2
Hypervolume	$\frac{1043\sqrt3}{80} ≈ 22.58161$
Diteral angles	Tip–tet–tepe: $\arccos\left(-\frac{\sqrt{10}}{5}\right) ≈ 129.23152°$
	Grip–trip–tepe: $\arccos\left(-\frac{\sqrt{10}}{10}\right) ≈ 108.43495°$
	Grip–tut–tip: $\arccos\left(-\frac15\right) ≈ 101.53696°$
	Grip–toe–grip: $\arccos\left(\frac15\right) ≈ 78.46304°$
Central density	1
Number of external pieces	27
Level of complexity	20
Related polytopes
Army	Garx
Regiment	Garx
Dual	Triangular-pyramidal pyramidal hecatonicosateron
Conjugate	None
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A5, order 720
Convex	Yes
Nature	Tame

The great rhombated hexateron, or garx, also called the cantitruncated 5-simplex, is a convex uniform polyteron. It consists of 15 tetrahedral prisms, 6 truncated pentachora, and 6 great rhombated pentachora. One truncated pentachoron, 1 tetrahedral prism, and 3 great rhombated pentachora join at each vertex. As the name suggests, it is the cantitruncation of the hexateron.

Vertex coordinates

The vertices of a great rhombated hexateron of edge length 1 can be given in 6 dimensions as all permutations of:

• $\left(\frac{3\sqrt2}{2},\,\sqrt2,\,\frac{\sqrt2}{2},\,0,\,0,\,0\right).$

Representations

A great rhombated hexateron has the following Coxeter diagrams:

• x3x3x3o3o (full symmetry)
• xuxx3xxux3ooox3oooo&#xt (A₅ axial, truncated pentachoron-first)

Gallery

• 

  A₄ orthographic projection

• 

  A₃

• 

  A₂

External links

• Bowers, Jonathan. "Category 10: Greater Truncates" (#722).

• Klitzing, Richard. "garx".

• Wikipedia Contributors. "Cantitruncated 5-simplex".