# Hexagonal-great rhombicosidodecahedral duoprism

Hexagonal-great rhombicosidodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHagrid
Coxeter diagramx6o x5x3x
Elements
Tera30 square-hexagonal duoprisms, 20 hexagonal duoprisms, 12 hexagonal-decagonal duoprisms, 6 great rhombicosidodecahedral prisms
Cells180 cubes, 60+60+60+120 hexagonal prisms, 72 decagonal prisms, 6 great rhombicosidodecahedra
Faces180+360+360+360 squares, 120+120 hexagons, 72 decagons
Edges360+360+360+720
Vertices720
Vertex figureMirror-symmetric pentachoron, edge lengths 2, 3, (5+5)/2 (base triangle), 3 (top edge), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {35+12{\sqrt {5}}}}{2}}\approx 3.93169}$
Hypervolume${\displaystyle 15{\frac {19{\sqrt {3}}+10{\sqrt {15}}}{2}}\approx 537.29099}$
Diteral anglesShiddip–hip–hiddip: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
Shiddip–hip–hadedip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
Hiddip–hip–hadedip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Griddip–grid–griddip: 120°
Shiddip–cube–griddip: 90°
Hiddip–hip–griddip: 90°
Central density1
Number of external pieces68
Level of complexity60
Related polytopes
ArmyHagrid
RegimentHagrid
DualHexagonal-disdyakis triacontahedral duotegum
ConjugateHexagonal-great quasitruncated icosidodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×G2, order 1440
ConvexYes
NatureTame

The hexagonal-great rhombicosidodecahedral duoprism or hagrid is a convex uniform duoprism that consists of 6 great rhombicosidodecahedral prisms, 12 hexagonal-decagonal duoprisms, 20 hexagonal duoprisms, and 30 square-hexagonal duoprisms. Each vertex joins 2 great rhombicosidodecahedral prisms, 1 square-hexagonal duoprism, 1 hexagonal duoprism, and 1 hexagonal-decagonal duoprism.

This polyteron can be alternated into a triangular-snub dodecahedral duoantiprism, although it cannot be made uniform.

## Vertex coordinates

The vertices of a hexagonal-great rhombicosidodecahedral duoprism of edge length 1 are given by all permutations of the last three coordinates of:

• ${\displaystyle \left(0,\,\pm 1,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right),}$

along with all even permutations of the last three coordinates of:

• ${\displaystyle \left(0,\,\pm 1,\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(0,\,\pm 1,\,\pm 1,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,\pm 1,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(0,\,\pm 1,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(0,\,\pm 1,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right).}$

## Representations

A hexagonal-great rhombicosidodecahedral duoprism has the following Coxeter diagrams:

• x6o x5x3x (full symmetry)
• x3x x5x3x (hexagons as ditrigons)