Heptagonaldecagonal duoprismatic prism 


Rank  5 

Type  Uniform 

Notation 

Bowers style acronym  Heddip 

Coxeter diagram  x x7o x10o 

Elements 

Tera  10 squareheptagonal duoprisms, 7 squaredecagonal duoprisms, 2 heptagonaldecagonal duoprisms 

Cells  70 cubes, 7+14 decagonal prisms, 10+20 heptagonal prisms 

Faces  70+70+140 squares, 20 heptagons, 14 decagons 

Edges  70+140+140 

Vertices  140 

Vertex figure  Digonal disphenoidal pyramid, edge lengths 2cos(π/7) (disphenoid base 1), √(5+√5)/2 (disphenoid base 2), √2 (remaining edges) 

Measures (edge length 1) 

Circumradius  ${\frac {\sqrt {7+2{\sqrt {5}}+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 2.04842$ 

Hypervolume  ${\frac {35{\sqrt {5+2{\sqrt {5}}}}}{8\tan {\frac {\pi }{7}}}}\approx 27.96008$ 

Diteral angles  Squahedip–hep–squahedip: 144° 

 Squadedip–dip–squadedip: ${\frac {5\pi }{7}}\approx 128.57143^{\circ }$ 

 Squadedip–cube–squahedip: 90° 

 Hedadip–hep–squahedip: 90° 

 Squadedip–dip–hedadip: 90° 

Height  1 

Central density  1 

Number of external pieces  19 

Level of complexity  30 

Related polytopes 

Army  Heddip 

Regiment  Heddip 

Dual  Heptagonaldecagonal duotegmatic tegum 

Conjugates  Heptagonaldecagrammic duoprismatic prism, Heptagrammicdecagonal duoprismatic prism, Heptagrammicdecagrammic duoprismatic prism, Great heptagrammicdecagonal duoprismatic prism, Great heptagrammicdecagrammic duoprismatic prism 

Abstract & topological properties 

Euler characteristic  2 

Orientable  Yes 

Properties 

Symmetry  I_{2}(7)×I_{2}(10)×A_{1}, order 560 

Convex  Yes 

Nature  Tame 

The heptagonaldecagonal duoprismatic prism or heddip, also known as the heptagonaldecagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 heptagonaldecagonal duoprisms, 7 squaredecagonal duoprisms, and 10 squareheptagonal duoprisms. Each vertex joins 2 squareheptagonal duoprisms, 2 squaredecagonal duoprisms, and 1 heptagonaldecagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.
The vertices of a heptagonaldecagonal duoprismatic prism of edge length 2sin(π/7) are given by:
 $\left(1,\,0,\,0,\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}}\right),$
 $\left(\cos {\frac {j\pi }{7}},\,\pm \sin {\frac {j\pi }{7}},\,0,\,\pm (1+{\sqrt {5}})\sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}}\right),$
 $\left(1,\,0,\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{2}}}\sin {\frac {\pi }{7}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm \sin {\frac {\pi }{7}}\right),$
 $\left(\cos {\frac {j\pi }{7}},\,\pm \sin {\frac {j\pi }{7}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{2}}}\sin {\frac {\pi }{7}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm \sin {\frac {\pi }{7}}\right),$
 $\left(1,\,0,\,\pm ({\sqrt {5+2{\sqrt {5}}}})\sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}}\right),$
 $\left(\cos {\frac {j\pi }{7}},\,\pm \sin {\frac {j\pi }{7}},\,\pm ({\sqrt {5+2{\sqrt {5}}}})\sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}},\,\pm \sin {\frac {\pi }{7}}\right),$
where j = 2, 4, 6.
A heptagonaldecagonal duoprismatic prism has the following Coxeter diagrams:
 x x7o x10o (full symmetry)
 x x7o x5x (decagons as dipentagons)
 xx7oo xx10oo&#x (heptagonaldecagonal duoprism atop heptagonaldecagonal duoprism)
 xx7oo xx5xx&#x