# Great rhombicosidodecahedral prism

The great rhombicosidodecahedral prism or 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).

Great rhombicosidodecahedral prism
Rank4
TypeUniform
SpaceSpherical
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${\displaystyle \sqrt{8+3\sqrt5} ≈ 3.83513}$
Hypervolume${\displaystyle 5(19+10\sqrt5) ≈ 206.80340}$
Dichoral anglesCube–4–hip: ${\displaystyle \arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) ≈ 159.09484°}$
Cube–4–dip: ${\displaystyle \arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253°}$
Hip–4–dip: ${\displaystyle \arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 142.62263°}$
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

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

The great rhombicosidodecahedral pirsm 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:

• ${\displaystyle \left(±\frac12,\,±\frac12,\,±\frac{3+2\sqrt5}{2},\,±\frac12\right),}$

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

• ${\displaystyle \left(±\frac12,\,±\frac{2+\sqrt5}{2},\,±\frac{4+\sqrt5}{2},\,±\frac12\right),}$
• ${\displaystyle \left(±1,\,±\frac{3+\sqrt5}{4},\,±\frac{7+3\sqrt5}{4},\,±\frac12\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,±3\frac{1+\sqrt5}{4},\,±\frac{3+\sqrt5}{2},\,±\frac12\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{2},\,±\frac{5+3\sqrt5}{4},\,±\frac{5+\sqrt5}{4},\,±\frac12\right).}$

## Representations

A great rhombicosidodecahedral prism has the following Coxeter diagrams:

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