# Heptagonal-hexagonal antiprismatic duoprism

Heptagonal-hexagonal antiprismatic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHehap
Coxeter diagramx7o s2s12o
Elements
Tera7 hexagonal antiprismatic prisms, 12 triangular-heptagonal duoprisms, 2 hexagonal-heptagonal duoprisms
Cells84 triangular prisms, 14 hexagonal prisms, 7 hexagonal antiprisms, 12+12 heptagonal prisms
Faces84 triangles, 84+84 squares, 14 hexagons, 12 heptagons
Edges84+84+84
Vertices84
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 1, 1, 3 (base trapezoid), 2cos(π/7) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {3+{\sqrt {3}}+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 1.58461}$
Hypervolume${\displaystyle {\frac {7{\sqrt {2+2{\sqrt {3}}}}}{4\tan {\frac {\pi }{7}}}}\approx 8.49442}$
Diteral anglesTheddip–hep–theddip: = ${\displaystyle \arccos \left({\frac {1-2{\sqrt {3}}}{3}}\right)\approx 145.22189^{\circ }}$
Happip–hap–happip: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Theddip–hep–haheddip: = ${\displaystyle \arccos \left({\frac {3-2{\sqrt {3}}}{3}}\right)\approx 98.89943^{\circ }}$
Theddip–trip–happip: 90°
Haheddip–hip–happip: 90°
Height${\displaystyle {\sqrt {{\sqrt {3}}-1}}\approx 0.85560}$
Central density1
Number of external pieces21
Level of complexity40
Related polytopes
ArmyHehap
RegimentHehap
DualHeptagonal-hexagonal antitegmatic duotegum
ConjugatesHeptagrammic-hexagonal antiprismatic duoprism, Great heptagrammic-hexagonal antiprismatic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(7)×I2(12)×A1+, order 336
ConvexYes
NatureTame

The heptagonal-hexagonal antiprismatic duoprism or hehap is a convex uniform duoprism that consists of 7 hexagonal antiprismatic prisms, 2 hexagonal-heptagonal duoprisms, and 12 triangular-heptagonal duoprisms. Each vertex joins 2 hexagonal antiprismatic prisms, 3 triangular-heptagonal duoprisms, and 1 hexagonal-heptagonal duoprism.

## Vertex coordinates

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

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

where j = 2, 4, 6.

## Representations

A heptagonal-hexagonal antiprismatic duoprism has the following Coxeter diagrams:

• x7o s2s12o (full symmetry; hexagonal antiprisms as alternated dodecagonal prisms)
• x7o s2s6s (hexagonal antiprisms as alternated dihexagonal prisms)