# Great rhombicosidodecahedral prism

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 pieces64
Level of complexity24
Related polytopes
ArmyGriddip
RegimentGriddip
DualDisdyakis triacontahedral tegum
ConjugateGreat quasitruncated icosidodecahedral prism
Abstract properties
Flag count5760
Euler characteristic0
Topological properties
OrientableYes
Properties
SymmetryH3×A1, order 240
ConvexYes
NatureTame

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).

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)