Heptagonaldodecahedral duoprism 


Rank  5 

Type  Uniform 

Notation 

Bowers style acronym  Hedoe 

Coxeter diagram  x7o x5o3o 

Elements 

Tera  12 pentagonalheptagonal duoprisms, 7 dodecahedral prisms 

Cells  84 pentagonal prisms, 30 heptagonal prisms, 7 dodecahedra 

Faces  210 squares, 84 pentagons, 20 heptagons 

Edges  140+210 

Vertices  140 

Vertex figure  Triangular scalene, edge lengths (1+√5)/2 (base triangle), 2cos(π/7) (top), √2 (sides) 

Measures (edge length 1) 

Circumradius  ${\sqrt {\frac {9+3{\sqrt {5}}+{\frac {2}{\sin ^{2}{\frac {\pi }{7}}}}}{8}}}\approx 1.81425$ 

Hypervolume  ${\frac {7(15+7{\sqrt {5}})}{16\tan {\frac {\pi }{7}}}}\approx 27.84710$ 

Diteral angles  Dope–doe–dope: ${\frac {5\pi }{7}}\approx 128.57143^{\circ }$ 

 Pheddip–hep–pheddip: $\arccos \left({\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$ 

 Pheddip–pip–dope: 90° 

Central density  1 

Number of external pieces  19 

Level of complexity  10 

Related polytopes 

Army  Hedoe 

Regiment  Hedoe 

Dual  Heptagonalicosahedral duotegum 

Conjugates  Heptagrammicdodecahedral duoprism, Great heptagrammicdodecahedral duoprism, Heptagonalgreat stellated dodecahedral duoprism, Heptagrammicgreat stellated dodecahedral duoprism, Great heptagrammicgreat stellated dodecahedral duoprism 

Abstract & topological properties 

Euler characteristic  2 

Orientable  Yes 

Properties 

Symmetry  H_{3}×I2(7), order 1680 

Convex  Yes 

Nature  Tame 

The heptagonaldodecahedral duoprism or hedoe is a convex uniform duoprism that consists of 7 dodecahedral prisms and 12 pentagonalheptagonal duoprisms. Each vertex joins 2 dodecahedral prisms and 3 pentagonalheptagonal duoprisms.
The vertices of a heptagonaldodecahedral duoprism of edge length 2sin(π/7) are given by:
 $\left(1,\,0,\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),$
 $\left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),$
as well as all even permutations of the last three coordinates of:
 $\left(1,\,0,\,0,\,\pm \sin {\frac {\pi }{7}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),$
 $\left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,0,\,\pm \sin {\frac {\pi }{7}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),$
where j = 2, 4, 6.