# Heptagonal-truncated icosahedral duoprism

Heptagonal-truncated icosahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHeti
Coxeter diagramx7o o5x3x ()
Elements
Tera12 pentagonal-heptagonal duoprisms, 20 hexagonal-heptagonal duoprisms, 7 truncated icosahedral prisms
Cells84 pentagons, 140 hexagons, 30+60 heptagonal prisms, 7 truncated icosahedra
Faces210+420 squares, 84 pentagons, 140 hexagons, 60 heptagons
Edges210+420+420
Vertices420
Vertex figureDigonal disphenoidal pyramid, edge lengths (1+5)/2, 3, 3 (base triangle), 2cos(π/7) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {29+9{\sqrt {5}}+{\frac {2}{\sin ^{2}{\frac {\pi }{7}}}}}{8}}}\approx 2.73287}$
Hypervolume${\displaystyle 7{\frac {125+43{\sqrt {5}}}{16\tan {\frac {\pi }{7}}}}\approx 200.91077}$
Haheddip–hep–haheddip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18968^{\circ }}$
Tipe–ti–tipe: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Pheddip–pip–tipe: 90°
Haheddip–hip–tipe: 90°
Central density1
Number of external pieces39
Level of complexity30
Related polytopes
ArmyHeti
RegimentHeti
DualHeptagonal-pentakis dodecahedral duotegum
ConjugatesHeptagrammic-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 characteristic2
OrientableYes
Properties
SymmetryH3×I2(7), order 1680
ConvexYes
NatureTame

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.

## Vertex coordinates

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:

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

where j = 2, 4, 6.