Hexagonal-great heptagrammic duoprism

Hexagonal-great heptagrammic duoprism
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Hagishdip
Coxeter diagram	x6o x7/3o ()
Elements
Cells	7 hexagonal prisms, 6 great heptagrammic prisms
Faces	42 squares, 7 hexagons, 6 great heptagrams
Edges	42+42
Vertices	42
Vertex figure	Digonal disphenoid, edge lengths √3 (base 1), 2cos(3π/7) (base 2), √2 (sides)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {4+{\frac {1}{4\sin ^{2}{\frac {3\pi }{7}}}}}}{2}}\approx 1.12384$
Hypervolume	${\frac {21{\sqrt {3}}}{8\tan {\frac {3\pi }{7}}}}\approx 1.03774$
Dichoral angles	Giship–7/3–giship: 120°
	Hip–4–giship: 90°
	Hip–6–hip: ${\frac {\pi }{7}}\approx 25.71429^{\circ }$
Central density	3
Number of external pieces	20
Level of complexity	12
Related polytopes
Army	Semi-uniform haheddip
Regiment	Hagishdip
Dual	Hexagonal-great heptagrammic duotegum
Conjugates	Hexagonal-heptagonal duoprism, Hexagonal-heptagrammic duoprism
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	G2×I2(7), order 168
Convex	No
Nature	Tame

The hexagonal-great heptagrammic duoprism, also known as hagishdip or the 6-7/3 duoprism, is a uniform duoprism that consists of 7 hexagonal prisms and 6 great heptagrammic prisms, with 2 of each at each vertex.

Vertex coordinates

The coordinates of a hexagonal-great heptagrammic duoprism, centered at the origin and with edge length 2sin(3π/7), are given by:

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

where j = 2, 4, 6.

Representations

A hexagonal-great heptagrammic duoprism has the following Coxeter diagrams:

• x6o x7/3o (full symmetry)
• x3x x7/3o () (A₂×I₂(7) symmetry, hexagons as ditrigons)

External links

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

• Klitzing, Richard. "nd-mb-dip".