Heptagonal-great rhombicosidodecahedral duoprism Rank 5 Type Uniform Notation Bowers style acronym Hegrid Coxeter diagram x7o x5x3x Elements Tera 30 square-heptagonal duoprisms , 20 hexagonal-heptagonal duoprisms , 12 heptagonal-decagonal duoprisms , 7 great rhombicosidodecahedral prisms Cells 210 cubes , 140 hexagonal prisms , 60+60+60 heptagonal prisms , 84 cubes , 7 great rhombicosidodecahedra Faces 210+420+420+420 squares , 140 hexagons , 120 heptagons , 84 decagons Edges 420+420+420+840 Vertices 840 Vertex figure Mirror-symmetric pentachoron , edge lengths √2 , √3 , √(5+√5 )/2 (base triangle), 2cos(π/7) (top edge), √2 (side edges) Measures (edge length 1) Circumradius
31
+
12
5
+
1
sin
2
π
7
2
≈
3.97318
{\displaystyle {\frac {\sqrt {31+12{\sqrt {5}}+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 3.97318}
Hypervolume
35
19
+
10
5
4
tan
π
7
≈
751.50544
{\displaystyle 35{\frac {19+10{\sqrt {5}}}{4\tan {\frac {\pi }{7}}}}\approx 751.50544}
Diteral angles Squahedip–hep–haheddip:
arccos
(
−
3
+
15
6
)
≈
159.09484
∘
{\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}
Squahedip–hep–hedadip:
arccos
(
−
5
+
5
10
)
≈
148.28253
∘
{\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}
Haheddip–hep–hedadip:
arccos
(
−
5
+
2
5
15
)
≈
142.62263
∘
{\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}
Griddip–grid–griddip:
5
π
7
≈
128.57143
∘
{\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}
Squahedip–cube–griddip: 90° Haheddip–hip–griddip: 90° Hedadip–dip–griddip: 90° Central density 1 Number of external pieces 69 Level of complexity 60 Related polytopes Army Hegrid Regiment Hegrid Dual Heptagonal-disdyakis triacontahedral duotegum Conjugates Heptagrammic-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 characteristic 2 Orientable Yes Properties Symmetry H3 ×I2(7) , order 1680Convex Yes Nature Tame
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.
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:
(
1
,
0
,
±
sin
π
7
,
±
sin
π
7
,
(
3
+
2
5
)
sin
π
7
)
,
{\displaystyle \left(1,\,0,\,\pm \sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}},\,(3+2{\sqrt {5}})\sin {\frac {\pi }{7}}\right),}
(
cos
(
j
π
7
)
,
±
sin
(
j
π
7
)
,
±
sin
π
7
,
±
sin
π
7
,
(
3
+
2
5
)
sin
π
7
)
,
{\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:
(
1
,
0
,
±
sin
π
7
,
±
(
2
+
5
)
sin
π
7
,
±
(
4
+
5
)
sin
π
7
)
,
{\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),}
(
cos
(
j
π
7
)
,
±
sin
(
j
π
7
)
,
±
sin
π
7
,
±
(
2
+
5
)
sin
π
7
,
±
(
4
+
5
)
sin
π
7
)
,
{\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),}
(
1
,
0
,
±
2
sin
π
7
,
±
(
3
+
5
)
sin
π
7
2
,
±
(
7
+
3
5
)
sin
π
7
2
)
,
{\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),}
(
cos
(
j
π
7
)
,
±
sin
(
j
π
7
)
,
±
2
sin
π
7
,
±
(
3
+
5
)
sin
π
7
2
,
±
(
7
+
3
5
)
sin
π
7
2
)
,
{\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),}
(
1
,
0
,
±
(
3
+
5
)
sin
π
7
2
,
±
3
(
1
+
5
)
sin
π
7
2
,
±
(
3
+
5
)
sin
π
7
)
,
{\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),}
(
cos
(
j
π
7
)
,
±
sin
(
j
π
7
)
,
±
(
3
+
5
)
sin
π
7
2
,
±
3
(
1
+
5
)
sin
π
7
2
,
±
(
3
+
5
)
sin
π
7
)
,
{\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),}
(
1
,
0
,
±
(
1
+
5
)
sin
π
7
,
±
(
5
+
3
5
)
sin
π
7
2
,
±
(
5
+
5
)
sin
π
7
2
)
,
{\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),}
(
cos
(
j
π
7
)
,
±
sin
(
j
π
7
)
,
±
(
1
+
5
)
sin
π
7
,
±
(
5
+
3
5
)
sin
π
7
2
,
±
(
5
+
5
)
sin
π
7
2
)
,
{\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.
