Omnitruncated 5-simplex

Omnitruncated 5-simplex
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Gocad
Coxeter diagram	x3x3x3x3x ()
Elements
Tera	20 hexagonal duoprisms, 30 truncated octahedral prisms, 12 great prismatodecachora
Cells	90 cubes, 120+120+120 hexagonal prisms, 30+60 truncated octahedra
Faces	180+180+360+360 squares, 240+240 hexagons
Edges	360+720+720
Vertices	720
Vertex figure	Phyllic disphenoidal pyramid, edge lengths √2 (6) and √3 (4)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {35}}{2}}\approx 2.95804$
Hypervolume	$324{\sqrt {3}}\approx 561.18446$
Diteral angles	Tope–hip–hiddip: 135°
	Gippid–toe–tope: $\arccos \left(-{\frac {\sqrt {10}}{5}}\right)\approx 129.23152^{\circ }$
	Tope–cube–tope: 120°
	Gippid–hip–hiddip: $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$
	Gippid–hip–tope: $\arccos \left(-{\frac {\sqrt {10}}{10}}\right)\approx 108.43495^{\circ }$
	Gippid–toe–gippid: $\arccos \left(-{\frac {1}{5}}\right)\approx 101.53696^{\circ }$
Central density	1
Number of external pieces	62
Level of complexity	60
Related polytopes
Army	Gocad
Regiment	Gocad
Dual	Disphenoidal pyramidal heptacosiicosateron
Conjugate	None
Abstract & topological properties
Flag count	86400
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A5×2, order 1440
Convex	Yes
Nature	Tame

The omnitruncated 5-simplex, also called the great cellated hexateron, great cellidodecateron, or gocad, is a convex uniform 5-polytope. It consists of 20 hexagonal duoprisms, 30 truncated octahedral prisms, and 12 great prismatodecachora. 2 great prismatodecachora, 2 truncated octahedral prisms, and 1 hexagonal duoprism join at each vertex. As the name suggests, it is the omnitruncate of the A₅ family.

This polyteron can be alternated into a snub dodecateron, but that cannot be made uniform.

It is the 6th-order permutohedron, and therefore fills 5D space.

Vertex coordinates

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

• $\left({\frac {5{\sqrt {2}}}{2}},\,2{\sqrt {2}},\,{\frac {3{\sqrt {2}}}{2}},\,{\sqrt {2}},\,{\frac {\sqrt {2}}{2}},\,0\right)$.

Gallery

• 

  A₄ orthographic projection

• 

  A₃

• 

  A₂

• 

  Stereographic projection

External links

• Bowers, Jonathan. "Category 10: Greater Truncates" (#742).

• Klitzing, Richard. "gocad".
• Wikipedia contributors. "Omnitruncated 5-simplex".