Heptagonal-truncated dodecahedral duoprism Rank 5 Type Uniform Notation Bowers style acronym Hetid Coxeter diagram x7o x5x3o Elements Tera 20 triangular-heptagonal duoprisms , 12 heptagonal-decagonal duoprisms , 7 truncated dodecahedral prisms Cells 140 triangular prisms , 30+60 heptagonal prisms , 84 decagonal prisms , 7 truncated dodecahedra Faces 140 triangles , 210+420 squares , 60 heptagons , 84 decagons Edges 210+420+420 Vertices 420 Vertex figure Digonal disphenoidal pyramid , edge lengths 1, √(5+√5 )/2 , √(5+√5 )/2 (base triangle), cos(π/7) (top), √2 (side edges)Measures (edge length 1) Circumradius ${\sqrt {\frac {37+15{\sqrt {5}}+{\frac {2}{\sin ^{2}{\frac {\pi }{7}}}}}{8}}}\approx 3.18522$ Hypervolume $35{\frac {99+47{\sqrt {5}}}{48\tan {\frac {\pi }{7}}}}\approx 309.02670$ Diteral angles Theddip–hep–hedadip: $\arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }$ Tiddip–tid–tiddip: ${\frac {5\pi }{7}}\approx 128.57143^{\circ }$ Hedadip–hep–hedadip: $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$ Theddip–trip–tiddip: 90° Hedadip–dip–tiddip: 90° Central density 1 Number of external pieces 39 Level of complexity 30 Related polytopes Army Hetid Regiment Hetid Dual Heptagonal-triakis icosahedral duotegum Conjugates Heptagrammic-truncated dodecahedral duoprism , Great heptagrammic-truncated dodecahedral duoprism , Heptagonal-quasitruncated great stellated dodecahedral duoprism , Heptagrammic-quasitruncated great stellated dodecahedral duoprism , Great heptagrammic-quasitruncated great stellated dodecahedral duoprism Abstract & topological properties Euler characteristic 2 Orientable Yes Properties Symmetry H_{3} ×I2(7) , order 1680Convex Yes Nature Tame

The heptagonal-truncated dodecahedral duoprism or hetid is a convex uniform duoprism that consists of 7 truncated dodecahedral prisms , 12 heptagonal-decagonal duoprisms , and 20 triangular-heptagonal duoprisms . Each vertex joins 2 truncated dodecahedral prisms, 1 triangular-heptagonal duoprism, and 2 heptagonal-decagonal duoprisms.

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

$\left(1,\,0,\,0,\,\pm \sin {\frac {\pi }{7}},\,\pm {\frac {(5+3{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),$
$\left(1,\,0,\,\pm \sin {\frac {\pi }{7}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm (3+{\sqrt {5}})\sin {\frac {\pi }{7}}\right),$
$\left(1,\,0,\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{7}},\,\pm (2+{\sqrt {5}})\sin {\frac {\pi }{7}}\right),$
$\left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,0,\,\pm \sin {\frac {\pi }{7}},\,\pm {\frac {(5+3{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),$
$\left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,\pm \sin {\frac {\pi }{7}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm (3+{\sqrt {5}})\sin {\frac {\pi }{7}}\right),$
$\left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{7}},\,\pm (2+{\sqrt {5}})\sin {\frac {\pi }{7}}\right),$
where j = 2, 4, 6.