# Great cellidodecateron

(Redirected from Great cellihexahexateron)
Great cellidodecateron
Rank5
TypeUniform
Notation
Coxeter diagramx3x3x3x3x ()
Elements
Tera20 hexagonal duoprisms, 30 truncated octahedral prisms, 12 great prismatodecachora
Cells90 cubes, 120+120+120 hexagonal prisms, 30+60 truncated octahedra
Faces180+180+360+360 squares, 240+240 hexagons
Edges360+720+720
Vertices720
Vertex figurePhyllic disphenoidal pyramid, edge lengths 2 (6) and 3 (4)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {35}}{2}}\approx 2.95804}$
Hypervolume${\displaystyle 324{\sqrt {3}}\approx 561.18446}$
Diteral anglesTope–hip–hiddip: 135°
Gippid–toe–tope: ${\displaystyle \arccos \left(-{\frac {\sqrt {10}}{5}}\right)\approx 129.23152^{\circ }}$
Tope–cube–tope: 120°
Gippid–hip–hiddip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Gippid–hip–tope: ${\displaystyle \arccos \left(-{\frac {\sqrt {10}}{10}}\right)\approx 108.43495^{\circ }}$
Gippid–toe–gippid: ${\displaystyle \arccos \left(-{\frac {1}{5}}\right)\approx 101.53696^{\circ }}$
Central density1
Number of external pieces62
Level of complexity60
Related polytopes
DualDisphenoidal pyramidal heptacosiicosateron
ConjugateNone
Abstract & topological properties
Flag count86400
Euler characteristic2
OrientableYes
Properties
SymmetryA5×2, order 1440
ConvexYes
NatureTame

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:

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