# Heptagonal-great rhombicosidodecahedral duoprism

Heptagonal-great rhombicosidodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHegrid
Coxeter diagramx7o x5x3x
Elements
Tera30 square-heptagonal duoprisms, 20 hexagonal-heptagonal duoprisms, 12 heptagonal-decagonal duoprisms, 7 great rhombicosidodecahedral prisms
Cells210 cubes, 140 hexagonal prisms, 60+60+60 heptagonal prisms, 84 cubes, 7 great rhombicosidodecahedra
Faces210+420+420+420 squares, 140 hexagons, 120 heptagons, 84 decagons
Edges420+420+420+840
Vertices840
Vertex figureMirror-symmetric pentachoron, edge lengths 2, 3, (5+5)/2 (base triangle), 2cos(π/7) (top edge), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {31+12{\sqrt {5}}+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 3.97318}$
Hypervolume${\displaystyle 35{\frac {19+10{\sqrt {5}}}{4\tan {\frac {\pi }{7}}}}\approx 751.50544}$
Diteral anglesSquahedip–hep–haheddip: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
Squahedip–hep–hedadip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
Haheddip–hep–hedadip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Griddip–grid–griddip: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Squahedip–cube–griddip: 90°
Haheddip–hip–griddip: 90°
Central density1
Number of external pieces69
Level of complexity60
Related polytopes
ArmyHegrid
RegimentHegrid
DualHeptagonal-disdyakis triacontahedral duotegum
ConjugatesHeptagrammic-great rhombicosidodecahedral duoprism, Great heptagrammic-great rhombicosidodecahedral duoprism, Heptagonal-great quasitruncated icosidodecahedral duoprism, Heptagrammic-great quasitruncated icosidodecahedral duoprism, Great heptagrammic-great quasitruncated icosidodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(7), order 1680
ConvexYes
NatureTame

The heptagonal-great rhombicosidodecahedral duoprism or hegrid is a convex uniform duoprism that consists of 7 great rhombicosidodecahedral prisms, 12 heptagonal-decagonal duoprisms, 20 hexagonal-heptagonal duoprisms, and 30 square-heptagonal duoprisms. Each vertex joins 2 great rhombicosidodecahedral prisms, 1 square-heptagonal duoprism, 1 hexagonal-heptagonal duoprism, and 1 heptagonal-decagonal duoprism.

## Vertex coordinates

The vertices of a heptagonal-great rhombicosidodecahedral duoprism of edge length 2sin(π/7) are given by all permutations of the last three coordinates of:

• ${\displaystyle \left(1,\,0,\,\pm \sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}},\,(3+2{\sqrt {5}})\sin {\frac {\pi }{7}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,\pm \sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}},\,(3+2{\sqrt {5}})\sin {\frac {\pi }{7}}\right),}$

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

• ${\displaystyle \left(1,\,0,\,\pm \sin {\frac {\pi }{7}},\,\pm (2+{\sqrt {5}})\sin {\frac {\pi }{7}},\,\pm (4+{\sqrt {5}})\sin {\frac {\pi }{7}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,\pm \sin {\frac {\pi }{7}},\,\pm (2+{\sqrt {5}})\sin {\frac {\pi }{7}},\,\pm (4+{\sqrt {5}})\sin {\frac {\pi }{7}}\right),}$
• ${\displaystyle \left(1,\,0,\,\pm 2\sin {\frac {\pi }{7}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm {\frac {(7+3{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,\pm 2\sin {\frac {\pi }{7}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm {\frac {(7+3{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),}$
• ${\displaystyle \left(1,\,0,\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm 3{\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm (3+{\sqrt {5}})\sin {\frac {\pi }{7}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm 3{\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm (3+{\sqrt {5}})\sin {\frac {\pi }{7}}\right),}$
• ${\displaystyle \left(1,\,0,\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{7}},\,\pm {\frac {(5+3{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm {\frac {(5+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{7}},\,\pm {\frac {(5+3{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm {\frac {(5+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),}$

where j = 2, 4, 6.