Heptagonal duoprismatic prism

Heptagonal duoprismatic prism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Hehep
Coxeter diagram	x x7o x7o
Elements
Tera	14 square-heptagonal duoprisms, 2 heptagonal duoprisms
Cells	49 cubes, 14+28 heptagonal prisms,
Faces	98+98 squares, 28 heptagons
Edges	49+196
Vertices	98
Vertex figure	Tetragonal disphenoidal pyramid, edge lengths 2cos(π/7) (disphenoid bases) and √2 (remaining edges)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {1+{\frac {2}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 1.70469$
Hypervolume	${\frac {49}{16\tan ^{2}{\frac {\pi }{7}}}}\approx 13.20532$
Diteral angles	Squahedip–hep–squahedip: ${\frac {5\pi }{7}}\approx 128.57143^{\circ }$
	Squahedip–cube–squahedip: 90°
	Hedip–hep–squahedip: 90°
Height	1
Central density	1
Number of external pieces	16
Level of complexity	15
Related polytopes
Army	Hehep
Regiment	Hehep
Dual	Heptagonal duotegmatic tegum
Conjugates	Heptagrammic duoprismatic prism, Great heptagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	I2(7)≀S2×A1, order 784
Convex	Yes
Nature	Tame

The heptagonal duoprismatic prism or hehep, also known as the heptagonal-heptagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 heptagonal duoprisms and 14 square-heptagonal duoprisms. Each vertex joins 4 square-heptagonal duoprisms and 1 heptagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

Vertex coordinates

The vertices of a heptagonal duoprismatic prism of edge length 2sin(π/7) are given by:

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

where j, k = 2, 4, 6.

Representations

A heptagonal duoprismatic prism has the following Coxeter diagrams:

• x x7o x7o (full symmetry)
• xx7oo xx7oo&#x (heptagonal duoprism atop heptagonal duoprism)