Prismatotruncated hexateron

Prismatotruncated hexateron
(OFF file)
Rank	5
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Pattix
Coxeter diagram	x3x3o3x3o ()
Elements
Tera	15 octahedral prisms, 20 triangular-hexagonal duoprisms, 6 small rhombated pentachora, 6 prismatorhombated pentachora
Cells	60+60+60 triangular prisms, 30 octahedra, 60 hexagonal prisms, 15 truncated tettrahedra, 30 cuboctahedra
Faces	60+120+120 triangles, 180+180 squares, 60 hexagons
Edges	90+180+360
Vertices	180
Vertex figure	Wedge pyramid, edge lengths 1 (base square and top edge of wedge), √2 (sides of wedge and lacings to square), and √3 (lacings to top edge)
Measures (edge length 1)
Circumradius	$\frac{\sqrt{123}}{6} ≈ 1.84842$
Hypervolume	$\frac{9023\sqrt3}{480} ≈ 32.55895$
Diteral angles	Ope–trip–thiddip: 135°
	Srip–oct–ope: $\arccos\left(-\frac{\sqrt{10}}{5}\right) ≈ 129.23152°$
	Srip–trip–thiddip: $\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505°$
	Prip–hip–thiddip: $\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505°$
	Prip–trip–ope: $\arccos\left(-\frac{\sqrt{10}}{10}\right) ≈ 108.43495°$
	Prip–co–srip: $\arccos\left(-\frac15\right) ≈ 101.53696°$
	Prip–tut–prip: $\arccos\left(\frac15\right) ≈ 78.46304°$
Central density	1
Number of external pieces	47
Level of complexity	45
Related polytopes
Army	Pattix
Regiment	Pattix
Dual	Notch-pyramidal hecatonoctacontateron
Conjugate	None
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A5, order 720
Convex	Yes
Nature	Tame

The prismatotruncated hexateron, or pattix, also called the runcitruncated 5-simplex, is a convex uniform polyteron. It consists of 15 octahedral prisms, 20 triangular-hexagonal duoprisms, 6 small rhombated pentachora, and 6 prismatorhombated pentachora. One small rhombated pentachoron, 1 octahedral prism, 2 prismatorhombated pentachora, and 2 triangular-hexagonal duoprisms join at each vertex. As the name suggests, it is the runcitruncation of the hexateron.

Vertex coordinates

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

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

Representations

A prismatotruncated hexateron has the following Coxeter diagrams:

• x3x3o3x3o (full symmetry)
• xuxx3ooxx3xxxo3ooox&#xt (A₄ axial, small rhombated pentachoron-first)

Gallery

• 

  A₄ orthographic projection

• 

  A₃

• 

  A₂

Related polytopes

The prismatotruncated hexateron is the colonel of a regiment of 7 members.

External links

• Klitzing, Richard. "pattix".

• Wikipedia Contributors. "Runcitruncated 5-simplex".