Heptagonalenneagonal duoprismatic prism 


Rank  5 

Type  Uniform 

Notation 

Bowers style acronym  Heep 

Coxeter diagram  x x7o x9o () 

Elements 

Tera  9 squareheptagonal duoprisms, 7 squareenneagonal duoprisms, 2 heptagonalenneagonal duoprisms 

Cells  63 cubes, 7+14 enneagonal prisms, 9+18 heptagonal prisms 

Faces  63+63+126 squares, 18 heptagons, 14 enneagons 

Edges  63+126+126 

Vertices  126 

Vertex figure  Digonal disphenoidal pyramid, edge lengths $2\cos(\pi /7)$ (disphenoid base 1), $2\cos(\pi /9)$ (disphenoid base 2), ${\sqrt {2}}$ (remaining edges) 

Measures (edge length 1) 

Circumradius  ${\frac {\sqrt {1+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}+{\frac {1}{\sin ^{2}{\frac {\pi }{9}}}}}}{2}}\approx 1.92747$ 

Hypervolume  ${\frac {63}{16\tan {\frac {\pi }{7}}\tan {\frac {\pi }{9}}}}\approx 22.46421$ 

Diteral angles  Squahedip–hep–squahedip: 140° 

 Sendip–ep–sendip: ${\frac {5\pi }{7}}\approx 128.57143^{\circ }$ 

 Sendip–cube–squahedip: 90° 

 Heendip–hep–squahedip: 90° 

 Sendip–ep–heendip: 90° 

Height  1 

Central density  1 

Number of external pieces  18 

Level of complexity  30 

Related polytopes 

Army  Heep 

Regiment  Heep 

Dual  Heptagonalenneagonal duotegmatic tegum 

Conjugates  Heptagonalenneagrammic duoprismatic prism, Heptagonalgreat enneagrammic duoprismatic prism, Heptagrammicenneagonal duoprismatic prism, Heptagrammicenneagrammic duoprismatic prism, Heptagrammicgreat enneagrammic duoprismatic prism, Great heptagrammicenneagonal duoprismatic prism, Great heptagrammicenneagrammic duoprismatic prism, Great heptagrammicgreat enneagrammic duoprismatic prism 

Abstract & topological properties 

Euler characteristic  2 

Orientable  Yes 

Properties 

Symmetry  I_{2}(7)×I_{2}(9)×A_{1}, order 504 

Convex  Yes 

Nature  Tame 

The heptagonalenneagonal duoprismatic prism or heep, also known as the heptagonalenneagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 heptagonalenneagonal duoprisms, 7 squareenneagonal duoprisms, and 9 squareheptagonal duoprisms. Each vertex joins 2 squareheptagonal duoprisms, 2 squareenneagonal duoprisms, and 1 heptagonalenneagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.
The vertices of a heptagonalenneagonal duoprismatic prism of edge length $4\sin(\pi /7)\sin(\pi /9)$ are given by:
 $\left(2\sin {\frac {\pi }{9}},\,0,\,2\sin {\frac {\pi }{7}},\,0,\,\pm 2\sin {\frac {\pi }{7}}\sin {\frac {\pi }{9}}\right),$
 $\left(2\sin {\frac {\pi }{9}},\,0,\,2\cos \left({\frac {k\pi }{9}}\right)\sin {\frac {\pi }{7}},\,\pm 2\sin \left({\frac {k\pi }{9}}\right)\sin {\frac {\pi }{7}},\,\pm 2\sin {\frac {\pi }{7}}\sin {\frac {\pi }{9}}\right),$
 $\left(2\sin {\frac {\pi }{9}},\,0,\,\sin {\frac {\pi }{7}},\,\pm {\sqrt {3}}\sin {\frac {\pi }{7}},\,\pm 2\sin {\frac {\pi }{7}}\sin {\frac {\pi }{9}}\right),$
 $\left(2\cos \left({\frac {j\pi }{7}}\right)\sin {\frac {\pi }{9}},\,\pm 2\sin \left({\frac {j\pi }{7}}\right)\sin {\frac {\pi }{9}},\,2\sin {\frac {\pi }{7}},\,0,\,\pm 2\sin {\frac {\pi }{7}}\sin {\frac {\pi }{9}}\right),$
 $\left(2\cos \left({\frac {j\pi }{7}}\right)\sin {\frac {\pi }{9}},\,\pm 2\sin \left({\frac {j\pi }{7}}\right)\sin {\frac {\pi }{9}},\,2\cos \left({\frac {k\pi }{9}}\right)\sin {\frac {\pi }{7}},\,\pm 2\sin \left({\frac {k\pi }{9}}\right)\sin {\frac {\pi }{7}},\,\pm 2\sin {\frac {\pi }{7}}\sin {\frac {\pi }{9}}\right),$
 $\left(2\cos \left({\frac {j\pi }{7}}\right)\sin {\frac {\pi }{9}},\,\pm 2\sin \left({\frac {j\pi }{7}}\right)\sin {\frac {\pi }{9}},\,\sin {\frac {\pi }{7}},\,\pm {\sqrt {3}}\sin {\frac {\pi }{7}},\,\pm 2\sin {\frac {\pi }{7}}\sin {\frac {\pi }{9}}\right),$
where j = 2, 4, 6 and k = 2, 4, 8.
A heptagonalenneagonal duoprismatic prism has the following Coxeter diagrams:
 x x7o x9o (full symmetry)
 xx7oo xx9oo&#x (heptagonalenneagonal duoprism atop heptagonalenneagonal duoprism)