Triangular-great rhombicosidodecahedral duoprism

Triangular-great rhombicosidodecahedral duoprism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Tragrid
Coxeter diagram	x3o x5x3x
Elements
Tera	30 triangular-square duoprisms, 20 triangular-hexagonal duoprisms, 12 triangular-decagonal duoprisms
Cells	60+60+60 triangular prisms, 90 cubes, 60 hexagonal prisms, 36 decagonal prisms, 3 great rhombicosidodecahedra
Faces	120 triangles, 90+180+180+180 squares, 60 hexagons, 36 decagons
Edges	180+180+180+360
Vertices	360
Vertex figure	Mirror-symmetric pentachoron, edge lengths √2, √3, √(5+√5)/2 (base triangle), 1 (top edge), √2 (side edges)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {97+36{\sqrt {5}}}{12}}}\approx 3.84598$
Hypervolume	$5{\frac {19{\sqrt {3}}+10{\sqrt {15}}}{4}}\approx 89.54850$
Diteral angles	Tisdip–trip–thiddip: $\arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }$
	Tisdip–trip–tradedip: $\arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }$
	Thiddip–trip–tradedip: $\arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }$
	Tisdip–cube–griddip: 90°
	Thiddip–hip–griddip: 90°
	Tradedip–dip–griddip: 90°
	Griddip–grid–griddip: 60°
Height	${\frac {\sqrt {3}}{2}}\approx 0.86603$
Central density	1
Number of external pieces	65
Level of complexity	60
Related polytopes
Army	Tragrid
Regiment	Tragrid
Dual	Triangular-disdyakis triacontahedral duotegum
Conjugate	Triangular-great quasitruncated icosidodecahedral duoprism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	H3×A2, order 720
Convex	Yes
Nature	Tame

The triangular-great rhombicosidodecahedral duoprism or tragrid is a convex uniform duoprism that consists of 3 great rhombicosidodecahedral prisms, 12 triangular-decagonal duoprisms, 20 triangular-hexagonal duoprisms, and 30 triangular-square duoprisms. Each vertex joins 2 great rhombicosidodecahedral prisms, 1 triangular-square duoprism, 1 triangular-hexagonal duoprism, and 1 triangular-decagonal duoprism. It is a duoprism based on a triangle and a great rhombicosidodecahedron, which makes it a convex segmentoteron.

Vertex coordinates

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

• $\left(0,\,{\frac {\sqrt {3}}{3}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right),$

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

• $\left(0,\,{\frac {\sqrt {3}}{3}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right),$
• $\left(0,\,{\frac {\sqrt {3}}{3}},\,\pm 1,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,\pm 1,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right),$
• $\left(0,\,{\frac {\sqrt {3}}{3}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),$
• $\left(0,\,{\frac {\sqrt {3}}{3}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right).$

External links

• Klitzing, Richard. "tragrid".