Great cellidodecateron

Great cellidodecateron
(OFF file)
Rank	5
Type	Uniform
Space	Spherical
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	$\frac{\sqrt{35}}{2} ≈ 2.95804$
Hypervolume	$324\sqrt3 ≈ 561.18446$
Diteral angles	Tope–hip–hiddip: 135°
	Gippid–toe–tope: $\arccos\left(-\frac{\sqrt{10}}{5}\right) ≈ 129.23152°$
	Tope–cube–tope: 120°
	Gippid–hip–hiddip: $\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505°$
	Gippid–hip–tope: $\arccos\left(-\frac{\sqrt{10}}{10}\right) ≈ 108.43495°$
	Gippid–toe–gippid: $\arccos\left(-\frac15\right) ≈ 101.53696°$
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
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A5×2, order 1440
Convex	Yes
Nature	Tame

The great cellidodecateron, or gocad, also called the great cellated hexateron or omnitruncated 5-simplex, is a convex uniform polyteron. 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 great cellidodecateron of edge length 1 can be given in 6 dimensions as all permutations of:

• $\left(\frac{5\sqrt2}{2},\,2\sqrt2,\,\frac{3\sqrt2}{2},\,\sqrt2,\,\frac{\sqrt2}{2},\,0\right).$

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".