Heptagonal-dodecagrammic duoprism

Heptagonal-dodecagrammic duoprism
(OFF file)
Rank	4
Type	Uniform
Notation
Coxeter diagram	x7o x12/5o ()
Elements
Cells	12 heptagonal prisms, 7 dodecagrammic prisms
Faces	84 squares, 12 heptagons, 7 dodecagrams
Edges	84+84
Vertices	84
Vertex figure	Digonal disphenoid, edge lengths 2cos(π/7) (base 1), (√6–√2)/2 (base 2), √2 (sides)
Measures (edge length 1)
Circumradius	${\sqrt {2-{\sqrt {3}}+{\frac {1}{4\sin ^{2}{\frac {\pi }{7}}}}}}\approx 1.26330$
Hypervolume	${\frac {42-21{\sqrt {3}}}{4\tan {\frac {\pi }{7}}}}\approx 2.92111$
Dichoral angles	Stwip–12/5–stwip: ${\frac {5\pi }{7}}\approx 128.57143^{\circ }$
	Hep–4–stwip: 90°
	Hep–7–hep: 30°
Central density	5
Number of external pieces	31
Level of complexity	12
Related polytopes
Army	Semi-uniform hetwadip
Dual	Heptagonal-dodecagrammic duotegum
Conjugates	Heptagonal-dodecagonal duoprism, Heptagrammic-dodecagonal duoprism, Heptagrammic-dodecagrammic duoprism
Abstract & topological properties
Flag count	2016
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(7)×I2(12), order 336
Convex	No
Nature	Tame

The heptagonal-dodecagrammic duoprism, also known as the 7-12/5 duoprism, is a uniform duoprism that consists of 12 heptagonal prisms and 7 dodecagrammic prisms, with 2 of each at each vertex.

Vertex coordinates

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

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

where j = 2, 4, 6.

Representations

A heptagonal-dodecagrammic duoprism has the following Coxeter diagrams:

• x7o x12/5o () (full symmetry)
• x6/5x x7o () (G₂×I₂(7) symmetry, dodecagrams as dihexagrams)

External links

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

• Klitzing, Richard. "nd-mb-dip".