Heptagonal-dodecagonal duoprism

Heptagonal-dodecagonal duoprism
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Hetwadip
Coxeter diagram	x7o x12o ()
Elements
Cells	12 heptagonal prisms, 7 dodecagonal prisms
Faces	84 squares, 12 heptagons, 7 dodecagons
Edges	84+84
Vertices	84
Vertex figure	Digonal disphenoid, edge lengths 2cos(π/7) (base 1), (√2+√6)/2 (base 2), and √2 (sides)
Measures (edge length 1)
Circumradius	${\sqrt {2+{\sqrt {3}}+{\frac {1}{4\sin ^{2}{\frac {\pi }{7}}}}}}\approx 2.24945$
Hypervolume	${\frac {21(2+{\sqrt {3}})}{4\tan {\frac {\pi }{7}}}}\approx 40.68584$
Dichoral angles	Hep–7–hep: 150°
	Twip–12–twip: ${\frac {5\pi }{7}}\approx 128.57143^{\circ }$
	Hep–4–twip: 90°
Central density	1
Number of external pieces	19
Level of complexity	6
Related polytopes
Army	Hetwadip
Regiment	Hetwadip
Dual	Heptagonal-dodecagonal duotegum
Conjugates	Heptagonal-dodecagrammic duoprism, Heptagrammic-dodecagonal duoprism, Heptagrammic-dodecagrammic duoprism, Great heptagrammic-dodecagonal duoprism, Great heptagrammic-dodecagrammic duoprism
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(7)×I2(12), order 336
Convex	Yes
Nature	Tame

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:

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

where j = 2, 4, 6.

Representations

A heptagonal-dodecagonal duoprism has the following Coxeter diagrams:

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

External links

• Bowers, Jonathan. "Category A: Duoprisms".

• Klitzing, Richard. "n-m-dip".