Prismatorhombated hexateron

Prismatorhombated hexateron
(OFF file)
Rank	5
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Pirx
Coxeter diagram	x3o3x3x3o ()
Elements
Tera	20 triangular duoprisms, 15 truncated tetrahedral prisms, 6 decachora, 6 prismatorhombated pentachora
Cells	60+60 triangular prisms, 60 hexagonal prisms, 30+30 truncated tettrahedra, 15 cuboctahedra
Faces	60+60+60 triangles, 90+180 squares, 120 hexagons
Edges	180+180+180
Vertices	180
Vertex figure	Skewed rectangular scalene, base edge lengths 1 and √2, opposite edge length 1, lacing lengths √2 and √3
Measures (edge length 1)
Circumradius	$\frac{\sqrt{33}}{3} ≈ 1.91485$
Hypervolume	$\frac{241\sqrt3}{10} ≈ 41.74242$
Diteral angles	Tuttip–trip–triddip: 135°
	Deca–tut–tuttip: $\arccos\left(-\frac{\sqrt{10}}{5}\right) ≈ 129.23152°$
	Prip–trip–triddip: $\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505°$
	Prip–hip–tuttip: $\arccos\left(-\frac{\sqrt{10}}{10}\right) ≈ 108.43495°$
	Prip–tut–deca: $\arccos\left(-\frac15\right) ≈ 101.53696°$
	Prip–co–prip: $\arccos\left(\frac15\right) ≈ 78.46304°$
Central density	1
Number of external pieces	47
Level of complexity	40
Related polytopes
Army	Pirx
Regiment	Pirx
Dual	Rhombiscalenic hecatonicoctacontateron
Conjugate	None
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A5, order 720
Convex	Yes
Nature	Tame

The prismatorhombated hexateron, or pirx, also called the runcicantellated 5-simplex, is a convex uniform polyteron. It consists of 20 triangular duoprisms, 15 truncated tetrahedral prisms, 6 decachora, and 6 prismatorhombated pentachora. 2 prismatorhombated pentachora, 1 decachoron, 1 triangular duoprism, and 2 truncated tetrahedral prisms join at each vertex. As the name suggests, it is the runcicantellation of the hexateron.

Vertex coordinates

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

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

Representations

A prismatorhombated hexateron has the following Coxeter diagrams:

• x3o3x3x3o (full symmetry)
• oxxx3xxoo3xxux3ooox&#xt (A₄ axial, decachoron-first)

Gallery

• 

  A₄ orthographic projection

• 

  A₃

• 

  A₂

Related polytopes

The prismatorhombated hexateron is the colonel of a three-member regiment that also includes the retroprismatorhombated hexateron and lepidoprismatorhombated hexateron.

External links

• Bowers, Jonathan. "Category 13: Prismatorhombates and Kin" (#829).

• Klitzing, Richard. "pirx".

• Wikipedia Contributors. "Runcicantellated 5-simplex".