Heptagonal-decagonal duoprism

Heptagonal-decagonal duoprism
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Hedadip
Coxeter diagram	x7o x10o ()
Elements
Cells	10 heptagonal prisms, 7 decagonal prisms
Faces	70 squares, 10 heptagons, 7 decagons
Edges	70+70
Vertices	70
Vertex figure	Digonal disphenoid, edge lengths 2cos(π/7) (base 1), √(5+√5)/2 (base 2), and √2 (sides)
Measures (edge length 1)
Circumradius	${\sqrt {{\frac {3+{\sqrt {5}}}{2}}+{\frac {1}{4\sin ^{2}{\frac {\pi }{7}}}}}}\approx 1.98646$
Hypervolume	${\frac {35{\sqrt {5+2{\sqrt {5}}}}}{8\tan {\frac {\pi }{7}}}}\approx 27.96008$
Dichoral angles	Hep–7–hep: 144°
	Dip–10–dip: ${\frac {5\pi }{7}}\approx 128.57143^{\circ }$
	Hep–4–dip: 90°
Central density	1
Number of external pieces	17
Level of complexity	6
Related polytopes
Army	Hedadip
Regiment	Hedadip
Dual	Heptagonal-decagonal duotegum
Conjugates	Heptagonal-decagrammic duoprism, Heptagrammic-decagonal duoprism, Heptagrammic-decagrammic duoprism, Great heptagrammic-decagonal duoprism, Great heptagrammic-decagrammic duoprism
Abstract & topological properties
Flag count	1680
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(7)×I2(10), order 280
Convex	Yes
Nature	Tame

The heptagonal-decagonal duoprism or hedadip, also known as the 7-10 duoprism, is a uniform duoprism that consists of 7 decagonal prisms and 10 heptagonal prisms, with two of each joining at each vertex.

Vertex coordinates

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

• $\left(1,0,\pm \sin {\frac {\pi }{7}},\pm {\sqrt {5+2{\sqrt {5}}}}\sin {\frac {\pi }{7}}\right)$,
• $\left(1,0,\pm {\frac {3+{\sqrt {5}}}{2}}\sin {\frac {\pi }{7}},\pm {\sqrt {\frac {5+{\sqrt {5}}}{2}}}\sin {\frac {\pi }{7}}\right)$,
• $\left(1,0,\left(1+{\sqrt {5}}\right)\sin {\frac {\pi }{7}},0\right)$,
• $\left(\cos \left({\frac {j\pi }{7}}\right),\pm \sin \left({\frac {j\pi }{7}}\right),\pm \sin {\frac {\pi }{7}},\pm {\sqrt {5+2{\sqrt {5}}}}\sin {\frac {\pi }{7}}\right)$,
• $\left(\cos \left({\frac {j\pi }{7}}\right),\pm \sin \left({\frac {j\pi }{7}}\right),\pm {\frac {3+{\sqrt {5}}}{2}}\sin {\frac {\pi }{7}},\pm {\sqrt {\frac {5+{\sqrt {5}}}{2}}}\sin {\frac {\pi }{7}}\right)$,
• $\left(\cos \left({\frac {j\pi }{7}}\right),\pm \sin \left({\frac {j\pi }{7}}\right),\left(1+{\sqrt {5}}\right)\sin {\frac {\pi }{7}},0\right)$,

where j = 2, 4, 6.

Representations

A heptagonal-decagonal duoprism has the following Coxeter diagrams:

• x7o x10o () (full symmetry)
• x5x x7o () (decagons as dipentagons)

External links

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

• Klitzing, Richard. "n-m-dip".