# Hexagonal-heptagonal duoprismatic prism

Hexagonal-heptagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymHahep
Coxeter diagramx x6o x7o
Elements
Tera7 square-hexagonal duoprisms, 6 square-heptagonal duoprisms, 2 hexagonal-heptagonal duoprisms
Cells42 cubes, 6+12 heptagonal prisms, 7+14 hexagonal prisms
Faces42+42+84 squares, 14 hexagons, 12 heptagons
Edges42+84+84
Vertices84
Vertex figureDigonal disphenoidal pyramid, edge lengths 3 (disphenoid base 1), 2cos(π/7) (disphenoid base 2), 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {5+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 1.60561}$
Hypervolume${\displaystyle {\frac {21{\sqrt {3}}}{8\tan {\frac {\pi }{7}}}}\approx 9.44118}$
Diteral anglesShiddip–hip–shiddip: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Squahedip–hep–squahedip: 120°
Squahedip–cube–shiddip: 90°
Haheddip–hip–shiddip: 90°
Squahedip–hep–haheddip: 90°
Height1
Central density1
Number of external pieces15
Level of complexity30
Related polytopes
ArmyHahep
RegimentHahep
DualHexagonal-heptagonal duotegmatic tegum
ConjugatesHexagonal-heptagrammic duoprismatic prism, Hexagonal-great heptagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryG2×I2(7)×A1, order 336
ConvexYes
NatureTame

The hexagonal-heptagonal duoprismatic prism or hahep, also known as the hexagonal-heptagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 hexagonal-heptagonal duoprisms, 6 square-heptagonal duoprisms, and 7 square-hexagonal duoprisms. Each vertex joins 2 square-hexagonal duoprisms, 2 square-heptagonal duoprisms, and 1 hexagonal-heptagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

## Vertex coordinates

The vertices of a hexagonal-heptagonal duoprismatic prism of edge length 2sin(π/7) are given by:

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

where j = 2, 4, 6.

## Representations

A hexagonal-heptagonal duoprismatic prism has the following Coxeter diagrams:

• x x6o x7o (full symmetry)
• x x3x x7o (hexagons as ditrigons)
• xx6oo xx7oo&#x (hexagonal-heptagonal duoprism atop hexagonal-heptagonal duoprism)
• xx3xx xx7oo&#x