# Octagonal-great rhombicosidodecahedral duoprism

Octagonal-great rhombicosidodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymOgrid
Coxeter diagramx8o x5x3x
Elements
Tera30 square-octagonal duoprisms, 20 hexagonal-octagonal duoprisms, 12 octagonal-decagonal duoprisms, 8 great rhombicosidodecahedral prisms
Cells240 cubes, 160 hexagonal prisms, 60+60+60 octagonal prisms, 96 decagonal prisms, 8 great rhombicosidodecahedra
Faces240+480+480+480 squares, 160 hexagons, 120 octagons, 96 decagons
Edges480+480+480+960
Vertices960
Vertex figureMirror-symmetric pentachoron, edge lengths 2, 3, (5+5)/2 (base triangle), 2+2 (top edge), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {35+2{\sqrt {2}}+12{\sqrt {5}}}}{2}}\approx 4.02061}$
Hypervolume${\displaystyle 10(19+19{\sqrt {2}}+10{\sqrt {5}}+10{\sqrt {10}})\approx 998.53514}$
Diteral anglesSodip–op–hodip: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
Sodip–op–odedip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
Hodip–op–odedip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Griddip–grid–griddip: 135°
Sodip–cube–griddip: 90°
Hodip–hip–griddip: 90°
Odedip–dip–griddip: 90°
Central density1
Number of external pieces70
Level of complexity60
Related polytopes
ArmyOgrid
RegimentOgrid
DualOctagonal-disdyakis triacontahedral duotegum
ConjugatesOctagrammic-great rhombicosidodecahedral duoprism, Octagonal-great quasitruncated icosidodecahedral duoprism, Octagrammic-great quasitruncated icosidodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(8), order 1920
ConvexYes
NatureTame

The octagonal-great rhombicosidodecahedral duoprism or ogrid is a convex uniform duoprism that consists of 8 great rhombicosidodecahedral prisms, 12 octagonal-decagonal duoprisms, 20 hexagonal-octagonal duoprisms, and 30 square-octagonal duoprisms. Each vertex joins 2 great rhombicosidodecahedral prisms, 1 square-octagonal duoprism, 1 hexagonal-octagonal duoprism, and 1 octagonal-decagonal duoprism.

This polyteron can be alternated into a square-snub dodecahedral duoantiprism, although it cannot be made uniform. The octagons can also be alternated into long rectangles to create a snub dodecahedral-square prismantiprismoid, which is also nonuniform.

## Vertex coordinates

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

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

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

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

## Representations

An octagonal-great rhombicosidodecahedral duoprism has the following Coxeter diagrams:

• x8o x5x3x (full symmetry)
• x4x x5x3x (octagons as ditetragons)