# Hexagonal-heptagonal duoprism

Hexagonal-heptagonal duoprism
Rank4
TypeUniform
Notation
Bowers style acronymHahedip
Coxeter diagramx6o x7o ()
Elements
Cells7 hexagonal prisms, 6 heptagonal prisms
Faces42 squares, 7 hexagons, 6 heptagons
Edges42+42
Vertices42
Vertex figureDigonal disphenoid, edge lengths 3 (base 1), 2cos(π/7) (base 2), and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {4+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 1.52577}$
Hypervolume${\displaystyle {\frac {21{\sqrt {3}}}{8\tan {\frac {\pi }{7}}}}\approx 9.44118}$
Dichoral anglesHip–6–hip: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Hep–7–hep: 120°
Hip–4–hep: 90°
Central density1
Number of external pieces13
Level of complexity6
Related polytopes
ArmyHahedip
RegimentHahedip
DualHexagonal-heptagonal duotegum
ConjugatesHexagonal-heptagrammic duoprism, Hexagonal-great heptagrammic duoprism
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryG2×I2(7), order 168
ConvexYes
NatureTame

The hexagonal-heptagonal duoprism or hahedip, also known as the 6-7 duoprism, is a uniform duoprism that consists of 6 heptagonal prisms and 7 hexagonal prisms, with two of each joining at each vertex.

## Vertex coordinates

The coordinates of a hexagonal-heptagonal duoprism, centered at the origin and with edge length 2sin(π/7), are given by:

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

where j = 2, 4, 6.

## Representations

A hexagonal-heptagonal duoprism has the following Coxeter diagrams:

• x6o x7o (full symmetry)
• x3x x7o () (hexagons as ditrigons)