# Heptagonal-dodecagonal duoprism

Heptagonal-dodecagonal duoprism
Rank4
TypeUniform
Notation
Coxeter diagramx7o x12o ()
Elements
Cells12 heptagonal prisms, 7 dodecagonal prisms
Faces84 squares, 12 heptagons, 7 dodecagons
Edges84+84
Vertices84
Vertex figureDigonal disphenoid, edge lengths 2cos(π/7) (base 1), (2+6)/2 (base 2), and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {2+{\sqrt {3}}+{\frac {1}{4\sin ^{2}{\frac {\pi }{7}}}}}}\approx 2.24945}$
Hypervolume${\displaystyle {\frac {21(2+{\sqrt {3}})}{4\tan {\frac {\pi }{7}}}}\approx 40.68584}$
Dichoral anglesHep–7–hep: 150°
Twip–12–twip: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Hep–4–twip: 90°
Central density1
Number of external pieces19
Level of complexity6
Related polytopes
DualHeptagonal-dodecagonal duotegum
ConjugatesHeptagonal-dodecagrammic duoprism, Heptagrammic-dodecagonal duoprism, Heptagrammic-dodecagrammic duoprism, Great heptagrammic-dodecagonal duoprism, Great heptagrammic-dodecagrammic duoprism
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryI2(7)×I2(12), order 336
ConvexYes
NatureTame

The heptagonal-dodecagonal duoprism or hetwadip, also known as the 7-12 duoprism, is a uniform duoprism that consists of 7 dodecagonal prisms and 12 heptagonal prisms, with two of each joining at each vertex.

## Vertex coordinates

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

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

where j = 2, 4, 6.

## Representations

A heptagonal-dodecagonal duoprism has the following Coxeter diagrams:

• x7o x12o (full symmetry)
• x6x x7o () (dodecagons as dihexagons)