# Heptagonal-dodecahedral duoprism

Heptagonal-dodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHedoe
Coxeter diagramx7o x5o3o
Elements
Tera12 pentagonal-heptagonal duoprisms, 7 dodecahedral prisms
Cells84 pentagonal prisms, 30 heptagonal prisms, 7 dodecahedra
Faces210 squares, 84 pentagons, 20 heptagons
Edges140+210
Vertices140
Vertex figureTriangular scalene, edge lengths (1+5)/2 (base triangle), 2cos(π/7) (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {9+3{\sqrt {5}}+{\frac {2}{\sin ^{2}{\frac {\pi }{7}}}}}{8}}}\approx 1.81425}$
Hypervolume${\displaystyle {\frac {7(15+7{\sqrt {5}})}{16\tan {\frac {\pi }{7}}}}\approx 27.84710}$
Diteral anglesDope–doe–dope: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Pheddip–hep–pheddip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Pheddip–pip–dope: 90°
Central density1
Number of external pieces19
Level of complexity10
Related polytopes
ArmyHedoe
RegimentHedoe
DualHeptagonal-icosahedral duotegum
ConjugatesHeptagrammic-dodecahedral duoprism, Great heptagrammic-dodecahedral duoprism, Heptagonal-great stellated dodecahedral duoprism, Heptagrammic-great stellated dodecahedral duoprism, Great heptagrammic-great stellated dodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(7), order 1680
ConvexYes
NatureTame

The heptagonal-dodecahedral duoprism or hedoe is a convex uniform duoprism that consists of 7 dodecahedral prisms and 12 pentagonal-heptagonal duoprisms. Each vertex joins 2 dodecahedral prisms and 3 pentagonal-heptagonal duoprisms.

## Vertex coordinates

The vertices of a heptagonal-dodecahedral duoprism of edge length 2sin(π/7) are given by:

• ${\displaystyle \left(1,\,0,\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),}$

as well as all even permutations of the last three coordinates of:

• ${\displaystyle \left(1,\,0,\,0,\,\pm \sin {\frac {\pi }{7}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),}$
• ${\displaystyle \left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,0,\,\pm \sin {\frac {\pi }{7}},\,\pm {\frac {(3+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}}\right),}$

where j = 2, 4, 6.