Heptagrammic-octagrammic duoprism

Heptagrammic-octagrammic duoprism
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Shestodip
Coxeter diagram	x7/2o x8/3o ()
Elements
Cells	8 heptagrammic prisms, 7 octagrammic prisms
Faces	56 squares, 8 heptagrams, 7 octagrams
Edges	56+56
Vertices	56
Vertex figure	Digonal disphenoid, edge lengths 2cos(2π/7) (base 1), √2–√2 (base 2), √2 (sides)
Measures (edge length 1)
Circumradius	${\sqrt {{\frac {2-{\sqrt {2}}}{2}}+{\frac {1}{4\sin ^{2}{\frac {2\pi }{7}}}}}}\approx 0.83779$
Hypervolume	$7{\frac {{\sqrt {2}}-1}{2\tan {\frac {2\pi }{7}}}}\approx 1.15614$
Dichoral angles	Ship–4–stop: 90°
	Stop–8/3–stop: ${\frac {3\pi }{7}}\approx 77.14286^{\circ }$
	Ship–7/2–ship: 45°
Central density	6
Number of external pieces	30
Level of complexity	24
Related polytopes
Army	Semi-uniform heodip
Regiment	Shestodip
Dual	Heptagrammic-octagrammic duotegum
Conjugates	Heptagonal-octagonal duoprism, Heptagonal-octagrammic duoprism, Heptagrammic-octagonal duoprism, Great heptagrammic-octagonal duoprism, Great heptagrammic-octagrammic duoprism
Abstract & topological properties
Flag count	1344
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(7)×I2(8), order 224
Convex	No
Nature	Tame

The heptagrammic-octagrammic duoprism, also known as shestodip or the 7/2-8/3 duoprism, is a uniform duoprism that consists of 8 heptagrammic prisms and 7 octagrammic prisms, with 2 of each at each vertex.

The name can also refer to the great heptagrammic-octagrammic duoprism.

Vertex coordinates

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

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

where j = 2, 4, 6.

Representations

A heptagrammic-octagrammic duoprism has the following Coxeter diagrams:

• x7/2o x8/3o () (full symmetry)
• x4/3x x7/2o () (B₂×I₂(7) symmetry, octagrams as ditetragrams)

External links

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

• Klitzing, Richard. "nd-mb-dip".