Runcitruncated 5-simplex

Runcitruncated 5-simplex
(OFF file)
Rank	5
Type	Uniform
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 tetrahedra, 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	${\frac {\sqrt {123}}{6}}\approx 1.84842$
Hypervolume	${\frac {9023{\sqrt {3}}}{480}}\approx 32.55895$
Diteral angles	Ope–trip–thiddip: 135°
	Srip–oct–ope: $\arccos \left(-{\frac {\sqrt {10}}{5}}\right)\approx 129.23152^{\circ }$
	Srip–trip–thiddip: $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$
	Prip–hip–thiddip: $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$
	Prip–trip–ope: $\arccos \left(-{\frac {\sqrt {10}}{10}}\right)\approx 108.43495^{\circ }$
	Prip–co–srip: $\arccos \left(-{\frac {1}{5}}\right)\approx 101.53696^{\circ }$
	Prip–tut–prip: $\arccos \left({\frac {1}{5}}\right)\approx 78.46304^{\circ }$
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
Flag count	32400
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A5, order 720
Convex	Yes
Nature	Tame

The runcitruncated 5-simplex, also called the prismatotruncated hexateron or pattix, is a convex uniform 5-polytope. 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 5-simplex.

Vertex coordinates

The vertices of a runcitruncated 5-simplex of edge length 1 can be given in 6 dimensions as all permutations of:

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

Representations

A runcitruncated 5-simplex 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 runcitruncated 5-simplex is the colonel of a regiment of 7 members.

External links

• Klitzing, Richard. "pattix".
• Wikipedia contributors. "Runcitruncated 5-simplex".