Heptagonal-octagonal duoprism

Heptagonal-octagonal duoprism
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Heodip
Coxeter diagram	x7o x8o ()
Elements
Cells	8 heptagonal prisms, 7 octagonal prisms
Faces	56 squares, 8 heptagons, 7 octagons
Edges	56+56
Vertices	56
Vertex figure	Digonal disphenoid, edge lengths 2cos(π/7) (base 1), √2+√2 (base 2), and √2 (sides)
Measures (edge length 1)
Circumradius	${\sqrt {{\frac {2+{\sqrt {2}}}{2}}+{\frac {1}{4\sin ^{2}{\frac {\pi }{7}}}}}}\approx 1.74215$
Hypervolume	${\frac {7(1+{\sqrt {2}})}{2\tan {\frac {\pi }{7}}}}\approx 17.54608$
Dichoral angles	Hep–7–hep: 135°
	Op–8–op: ${\frac {5\pi }{7}}\approx 128.57143^{\circ }$
	Hep–4–op: 90°
Central density	1
Number of external pieces	15
Level of complexity	6
Related polytopes
Army	Heodip
Regiment	Heodip
Dual	Heptagonal-octagonal duotegum
Conjugates	Heptagonal-octagrammic duoprism, Heptagrammic-octagonal duoprism, Heptagrammic-octagrammic 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
Flag orbits	6
Convex	Yes
Nature	Tame

The heptagonal-octagonal duoprism or heodip, also known as the 7-8 duoprism, is a uniform duoprism that consists of 7 octagonal prisms and 8 heptagonal prisms, with two of each joining at each vertex.

Gallery

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  Net

Vertex coordinates

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

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

where j = 2, 4, 6.

Representations

A heptagonal-octagonal duoprism has the following Coxeter diagrams:

• x7o x8o () (full symmetry)
• x4x x7o () (B₂×I₂(7) symmetry, octagons as ditetragons)

External links

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

• Klitzing, Richard. "n-m-dip".