Octagonalgreat rhombicosidodecahedral duoprism 


Rank  5 

Type  Uniform 

Notation 

Bowers style acronym  Ogrid 

Coxeter diagram  x8o x5x3x 

Elements 

Tera  30 squareoctagonal duoprisms, 20 hexagonaloctagonal duoprisms, 12 octagonaldecagonal duoprisms, 8 great rhombicosidodecahedral prisms 

Cells  240 cubes, 160 hexagonal prisms, 60+60+60 octagonal prisms, 96 decagonal prisms, 8 great rhombicosidodecahedra 

Faces  240+480+480+480 squares, 160 hexagons, 120 octagons, 96 decagons 

Edges  480+480+480+960 

Vertices  960 

Vertex figure  Mirrorsymmetric pentachoron, edge lengths √2, √3, √(5+√5)/2 (base triangle), √2+√2 (top edge), √2 (side edges) 

Measures (edge length 1) 

Circumradius  ${\frac {\sqrt {35+2{\sqrt {2}}+12{\sqrt {5}}}}{2}}\approx 4.02061$ 

Hypervolume  $10(19+19{\sqrt {2}}+10{\sqrt {5}}+10{\sqrt {10}})\approx 998.53514$ 

Diteral angles  Sodip–op–hodip: $\arccos \left({\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }$ 

 Sodip–op–odedip: $\arccos \left({\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }$ 

 Hodip–op–odedip: $\arccos \left({\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }$ 

 Griddip–grid–griddip: 135° 

 Sodip–cube–griddip: 90° 

 Hodip–hip–griddip: 90° 

 Odedip–dip–griddip: 90° 

Central density  1 

Number of external pieces  70 

Level of complexity  60 

Related polytopes 

Army  Ogrid 

Regiment  Ogrid 

Dual  Octagonaldisdyakis triacontahedral duotegum 

Conjugates  Octagrammicgreat rhombicosidodecahedral duoprism, Octagonalgreat quasitruncated icosidodecahedral duoprism, Octagrammicgreat quasitruncated icosidodecahedral duoprism 

Abstract & topological properties 

Euler characteristic  2 

Orientable  Yes 

Properties 

Symmetry  H_{3}×I2(8), order 1920 

Convex  Yes 

Nature  Tame 

The octagonalgreat rhombicosidodecahedral duoprism or ogrid is a convex uniform duoprism that consists of 8 great rhombicosidodecahedral prisms, 12 octagonaldecagonal duoprisms, 20 hexagonaloctagonal duoprisms, and 30 squareoctagonal duoprisms. Each vertex joins 2 great rhombicosidodecahedral prisms, 1 squareoctagonal duoprism, 1 hexagonaloctagonal duoprism, and 1 octagonaldecagonal duoprism.
This polyteron can be alternated into a squaresnub dodecahedral duoantiprism, although it cannot be made uniform. The octagons can also be alternated into long rectangles to create a snub dodecahedralsquare prismantiprismoid, which is also nonuniform.
The vertices of an octagonalgreat rhombicosidodecahedral duoprism of edge length 1 are given by all permutations of the last three coordinates of:
 $\left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right),$
 $\left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right),$
along with all even permutations of the last three coordinates of:
 $\left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right),$
 $\left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right),$
 $\left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm 1,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right),$
 $\left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm 1,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right),$
 $\left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),$
 $\left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),$
 $\left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right),$
 $\left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right).$
An octagonalgreat rhombicosidodecahedral duoprism has the following Coxeter diagrams:
 x8o x5x3x (full symmetry)
 x4x x5x3x (octagons as ditetragons)