# Heptagonal-decagonal duoprismatic prism

Heptagonal-decagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymHeddip
Coxeter diagramx x7o x10o
Elements
Tera10 square-heptagonal duoprisms, 7 square-decagonal duoprisms, 2 heptagonal-decagonal duoprisms
Cells70 cubes, 7+14 decagonal prisms, 10+20 heptagonal prisms
Faces70+70+140 squares, 20 heptagons, 14 decagons
Edges70+140+140
Vertices140
Vertex figureDigonal disphenoidal pyramid, edge lengths 2cos(π/7) (disphenoid base 1), (5+5)/2 (disphenoid base 2), 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {7+2{\sqrt {5}}+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 2.04842}$
Hypervolume${\displaystyle {\frac {35{\sqrt {5+2{\sqrt {5}}}}}{8\tan {\frac {\pi }{7}}}}\approx 27.96008}$
Diteral anglesSquahedip–hep–squahedip: 144°
Squadedip–dip–squadedip: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Height1
Central density1
Number of external pieces19
Level of complexity30
Related polytopes
ArmyHeddip
RegimentHeddip
DualHeptagonal-decagonal duotegmatic tegum
ConjugatesHeptagonal-decagrammic duoprismatic prism, Heptagrammic-decagonal duoprismatic prism, Heptagrammic-decagrammic duoprismatic prism, Great heptagrammic-decagonal duoprismatic prism, Great heptagrammic-decagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(7)×I2(10)×A1, order 560
ConvexYes
NatureTame

The heptagonal-decagonal duoprismatic prism or heddip, also known as the heptagonal-decagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 heptagonal-decagonal duoprisms, 7 square-decagonal duoprisms, and 10 square-heptagonal duoprisms. Each vertex joins 2 square-heptagonal duoprisms, 2 square-decagonal duoprisms, and 1 heptagonal-decagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

## Vertex coordinates

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

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

where j = 2, 4, 6.

## Representations

A heptagonal-decagonal duoprismatic prism has the following Coxeter diagrams:

• x x7o x10o (full symmetry)
• x x7o x5x (decagons as dipentagons)
• xx7oo xx10oo&#x (heptagonal-decagonal duoprism atop heptagonal-decagonal duoprism)
• xx7oo xx5xx&#x