Heptagonal-truncated icosahedral duoprism Rank 5 Type Uniform Notation Bowers style acronym Heti Coxeter diagram x7o o5x3x ( ) Elements Tera 12 pentagonal-heptagonal duoprisms , 20 hexagonal-heptagonal duoprisms , 7 truncated icosahedral prisms Cells 84 pentagons , 140 hexagons , 30+60 heptagonal prisms , 7 truncated icosahedra Faces 210+420 squares , 84 pentagons , 140 hexagons , 60 heptagons Edges 210+420+420 Vertices 420 Vertex figure Digonal disphenoidal pyramid , edge lengths (1+√5 )/2, √3 , √3 (base triangle), 2cos(π/7) (top), √2 (side edges)Measures (edge length 1) Circumradius ${\sqrt {\frac {29+9{\sqrt {5}}+{\frac {2}{\sin ^{2}{\frac {\pi }{7}}}}}{8}}}\approx 2.73287$ Hypervolume $7{\frac {125+43{\sqrt {5}}}{16\tan {\frac {\pi }{7}}}}\approx 200.91077$ Haheddip–hep–haheddip: $\arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18968^{\circ }$ Tipe–ti–tipe: ${\frac {5\pi }{7}}\approx 128.57143^{\circ }$ Pheddip–pip–tipe: 90° Haheddip–hip–tipe: 90° Central density 1 Number of external pieces 39 Level of complexity 30 Related polytopes Army Heti Regiment Heti Dual Heptagonal-pentakis dodecahedral duotegum Conjugates Heptagrammic-truncated icosahedral duoprism , Great heptagrammic-truncated icosahedral duoprism , Heptagonal-truncated great icosahedral duoprism , Heptagrammic-truncated great icosahedral duoprism , Great heptagrammic-truncated great icosahedral duoprism Abstract & topological properties Euler characteristic 2 Orientable Yes Properties Symmetry H_{3} ×I_{2} (7) , order 1680Convex Yes Nature Tame

The heptagonal-truncated icosahedral duoprism or heti is a convex uniform duoprism that consists of 7 truncated icosahedral prisms , 20 hexagonal-heptagonal duoprisms , and 12 pentagonal-heptagonal duoprisms . Each vertex joins 2 truncated icosahedral prisms, 1 pentagonal-heptagonal duoprism, and 2 hexagonal-heptagonal duoprisms.

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

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

Klitzing, Richard. "n-ti-dip" .