Decagonal-great rhombicosidodecahedral duoprism

Decagonal-great rhombicosidodecahedral duoprism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Dagrid
Coxeter diagram	x10o x5x3x ()
Elements
Tera	30 square-decagonal duoprisms, 20 hexagonal-decagonal duoprisms, 12 decagonal duoprisms, 10 great rhombicosidodecahedral prisms
Cells	300 cubes, 200 hexagonal prisms, 60+60+60+120 decagonal prisms, 200 hexagonal prisms, 10 great rhombicosidodecahedra
Faces	300+600+600+600 squares, 200 hexagons, 120+120 decagons
Edges	600+600+600+1200
Vertices	1200
Vertex figure	Mirror-symmetric pentachoron, edge lengths √2, √3, √(5+√5)/2 (base triangle), √(5+√5)/2 (top edge), √2 (side edges)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {37+14{\sqrt {5}}}}{2}}\approx 4.13234$
Hypervolume	$25{\frac {\sqrt {8125+3622{\sqrt {5}}}}{2}}\approx 1591.18854$
Diteral angles	Squadedip–dip–hadedip: $\arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }$
	Squadedip–dip–dedip: $\arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }$
	Griddip–grid–griddip: 144°
	Hadedip–dip–dedip: $\arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }$
	Squadedip–cube–griddip: 90°
	Hadedip–hip–griddip: 90°
	Dedip–dip–griddip: 90°
Central density	1
Number of external pieces	72
Level of complexity	60
Related polytopes
Army	Dagrid
Regiment	Dagrid
Dual	Decagonal-disdyakis triacontahedral duotegum
Conjugate	Decagrammic-great quasitruncated icosidodecahedral duoprism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	H3×I2(10), order 2400
Convex	Yes
Nature	Tame

The decagonal-great rhombicosidodecahedral duoprism or dagrid is a convex uniform duoprism that consists of 10 great rhombicosidodecahedral prisms, 12 decagonal duoprisms, 20 hexagonal-decagonal duoprisms, and 30 square-decagonal duoprisms. Each vertex joins 2 great rhombicosidodecahedral prisms, 1 square-decagonal duoprism, 1 hexagonal-decagonal duoprism, and 1 decagonal duoprism.

This polyteron can be alternated into a pentagonal-snub dodecahedral duoantiprism, although it cannot be made uniform.

Vertex coordinates

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

• $\left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right),$

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

• $\left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right),$
• $\left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm 1,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm 1,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,\pm 1,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right),$
• $\left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),$
• $\left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right).$

Representations

A decagonal-great rhombicosidodecahedral duoprism has the following Coxeter diagrams:

• x10o x5x3x () (full symmetry)
• x5x x5x3x () (decagons as dipentagons)