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
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Griddip
Coxeter diagram	x x5x3x ()
Elements
Cells	30 cubes, 20 hexagonal prisms, 12 decagonal prisms, 2 great rhombicosidodecahedra
Faces	60+60+60+60 squares, 40 hexagons, 24 decagons
Edges	120+120+120+120
Vertices	240
Vertex figure	Irregular tetrahedron, edge lengths √2, √3, √(5+√5)/2 (base), √2 (legs)
Measures (edge length 1)
Circumradius	$\sqrt{8+3\sqrt5} ≈ 3.83513$
Hypervolume	$5(19+10\sqrt5) ≈ 206.80340$
Dichoral angles	Cube–4–hip: $\arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) ≈ 159.09484°$
	Cube–4–dip: $\arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253°$
	Hip–4–dip: $\arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 142.62263°$
	Grid–10–dip: 90°
	Grid–6–hip: 90°
	Grid–4–cube: 90°
Height	1
Central density	1
Number of external pieces	64
Level of complexity	24
Related polytopes
Army	Griddip
Regiment	Griddip
Dual	Disdyakis triacontahedral tegum
Conjugate	Great quasitruncated icosidodecahedral prism
Abstract & topological properties
Flag count	5760
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H3×A1, order 240
Convex	Yes
Nature	Tame

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.

Gallery

• 

  Card with cell counts, verf, and cross-sections

• 

  Segmentochoron display, grid atop grid

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  Net

Vertex coordinates

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

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

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

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

Representations

A great rhombicosidodecahedral prism has the following Coxeter diagrams:

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

External links

• Bowers, Jonathan. "Category 19: Prisms" (#945).

• Klitzing, Richard. "Griddip".

• Wikipedia Contributors. "Truncated icosidodecahedral prism".