# Heptagonal-enneagonal duoprismatic prism

Heptagonal-enneagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymHeep
Coxeter diagramx x7o x9o ()
Elements
Tera9 square-heptagonal duoprisms, 7 square-enneagonal duoprisms, 2 heptagonal-enneagonal duoprisms
Cells63 cubes, 7+14 enneagonal prisms, 9+18 heptagonal prisms
Faces63+63+126 squares, 18 heptagons, 14 enneagons
Edges63+126+126
Vertices126
Vertex figureDigonal disphenoidal pyramid, edge lengths ${\displaystyle 2\cos(\pi /7)}$ (disphenoid base 1), ${\displaystyle 2\cos(\pi /9)}$ (disphenoid base 2), ${\displaystyle {\sqrt {2}}}$ (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {1+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}+{\frac {1}{\sin ^{2}{\frac {\pi }{9}}}}}}{2}}\approx 1.92747}$
Hypervolume${\displaystyle {\frac {63}{16\tan {\frac {\pi }{7}}\tan {\frac {\pi }{9}}}}\approx 22.46421}$
Diteral anglesSquahedip–hep–squahedip: 140°
Sendip–ep–sendip: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Sendip–cube–squahedip: 90°
Heendip–hep–squahedip: 90°
Sendip–ep–heendip: 90°
Height1
Central density1
Number of external pieces18
Level of complexity30
Related polytopes
ArmyHeep
RegimentHeep
DualHeptagonal-enneagonal duotegmatic tegum
ConjugatesHeptagonal-enneagrammic duoprismatic prism, Heptagonal-great enneagrammic duoprismatic prism, Heptagrammic-enneagonal duoprismatic prism, Heptagrammic-enneagrammic duoprismatic prism, Heptagrammic-great enneagrammic duoprismatic prism, Great heptagrammic-enneagonal duoprismatic prism, Great heptagrammic-enneagrammic duoprismatic prism, Great heptagrammic-great enneagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(7)×I2(9)×A1, order 504
ConvexYes
NatureTame

The heptagonal-enneagonal duoprismatic prism or heep, also known as the heptagonal-enneagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 heptagonal-enneagonal duoprisms, 7 square-enneagonal duoprisms, and 9 square-heptagonal duoprisms. Each vertex joins 2 square-heptagonal duoprisms, 2 square-enneagonal duoprisms, and 1 heptagonal-enneagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

## Vertex coordinates

The vertices of a heptagonal-enneagonal duoprismatic prism of edge length ${\displaystyle 4\sin(\pi /7)\sin(\pi /9)}$ are given by:

• ${\displaystyle \left(2\sin {\frac {\pi }{9}},\,0,\,2\sin {\frac {\pi }{7}},\,0,\,\pm 2\sin {\frac {\pi }{7}}\sin {\frac {\pi }{9}}\right),}$
• ${\displaystyle \left(2\sin {\frac {\pi }{9}},\,0,\,2\cos \left({\frac {k\pi }{9}}\right)\sin {\frac {\pi }{7}},\,\pm 2\sin \left({\frac {k\pi }{9}}\right)\sin {\frac {\pi }{7}},\,\pm 2\sin {\frac {\pi }{7}}\sin {\frac {\pi }{9}}\right),}$
• ${\displaystyle \left(2\sin {\frac {\pi }{9}},\,0,\,-\sin {\frac {\pi }{7}},\,\pm {\sqrt {3}}\sin {\frac {\pi }{7}},\,\pm 2\sin {\frac {\pi }{7}}\sin {\frac {\pi }{9}}\right),}$
• ${\displaystyle \left(2\cos \left({\frac {j\pi }{7}}\right)\sin {\frac {\pi }{9}},\,\pm 2\sin \left({\frac {j\pi }{7}}\right)\sin {\frac {\pi }{9}},\,2\sin {\frac {\pi }{7}},\,0,\,\pm 2\sin {\frac {\pi }{7}}\sin {\frac {\pi }{9}}\right),}$
• ${\displaystyle \left(2\cos \left({\frac {j\pi }{7}}\right)\sin {\frac {\pi }{9}},\,\pm 2\sin \left({\frac {j\pi }{7}}\right)\sin {\frac {\pi }{9}},\,2\cos \left({\frac {k\pi }{9}}\right)\sin {\frac {\pi }{7}},\,\pm 2\sin \left({\frac {k\pi }{9}}\right)\sin {\frac {\pi }{7}},\,\pm 2\sin {\frac {\pi }{7}}\sin {\frac {\pi }{9}}\right),}$
• ${\displaystyle \left(2\cos \left({\frac {j\pi }{7}}\right)\sin {\frac {\pi }{9}},\,\pm 2\sin \left({\frac {j\pi }{7}}\right)\sin {\frac {\pi }{9}},\,-\sin {\frac {\pi }{7}},\,\pm {\sqrt {3}}\sin {\frac {\pi }{7}},\,\pm 2\sin {\frac {\pi }{7}}\sin {\frac {\pi }{9}}\right),}$

where j = 2, 4, 6 and k = 2, 4, 8.

## Representations

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

• x x7o x9o (full symmetry)
• xx7oo xx9oo&#x (heptagonal-enneagonal duoprism atop heptagonal-enneagonal duoprism)