# Great rhombicosidodecahedral prism

Great rhombicosidodecahedral prism
Rank4
TypeUniform
Notation
Bowers style acronymGriddip
Coxeter diagramx x5x3x ()
Elements
Cells30 cubes, 20 hexagonal prisms, 12 decagonal prisms, 2 great rhombicosidodecahedra
Faces60+60+60+60 squares, 40 hexagons, 24 decagons
Edges120+120+120+120
Vertices240
Vertex figureIrregular tetrahedron, edge lengths 2, 3, (5+5)/2 (base), 2 (legs)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {8+3{\sqrt {5}}}}\approx 3.83513}$
Hypervolume${\displaystyle 5(19+10{\sqrt {5}})\approx 206.80340}$
Dichoral anglesCube–4–hip: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
Cube–4–dip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
Hip–4–dip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Grid–10–dip: 90°
Grid–6–hip: 90°
Grid–4–cube: 90°
Height1
Central density1
Number of external pieces64
Level of complexity24
Related polytopes
ArmyGriddip
RegimentGriddip
DualDisdyakis triacontahedral tegum
ConjugateGreat quasitruncated icosidodecahedral prism
Abstract & topological properties
Flag count5760
Euler characteristic0
OrientableYes
Properties
SymmetryH3×A1, order 240
ConvexYes
NatureTame

The great rhombicosidodecahedral prism (OBSA: griddip) is a prismatic uniform polychoron that consists of 2 great rhombicosidodecahedra, 12 decagonal prisms, 20 hexagonal prisms, and 30 cubes. Each vertex joins one of each type of cell. It is a prism based on the great rhombicosidodecahedron. As such it is also a convex segmentochoron (designated K-4.150 on Richard Klitzing's list).

This polychoron can be alternated into a snub dodecahedral antiprism, which cannot be made uniform.

The great rhombicosidodecahedral prism can be vertex-inscribed into the small tritrigonary prismatohecatonicosidishexacosichoron.

## Vertex coordinates

The vertices of a great rhombicosidodecahedral prism of edge length 1 are given by all permutations of the first three coordinates of:

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

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

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

## Representations

A great rhombicosidodecahedral prism has the following Coxeter diagrams:

• x x5x3x () (full symmetry)
• xx5xx3xx&#x (bases considered separately)