# Heptagonal duoprismatic prism

Heptagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymHehep
Coxeter diagramx x7o x7o
Elements
Tera14 square-heptagonal duoprisms, 2 heptagonal duoprisms
Cells49 cubes, 14+28 heptagonal prisms,
Faces98+98 squares, 28 heptagons
Edges49+196
Vertices98
Vertex figureTetragonal disphenoidal pyramid, edge lengths 2cos(π/7) (disphenoid bases) and 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {1+{\frac {2}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 1.70469}$
Hypervolume${\displaystyle {\frac {49}{16\tan ^{2}{\frac {\pi }{7}}}}\approx 13.20532}$
Diteral anglesSquahedip–hep–squahedip: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Squahedip–cube–squahedip: 90°
Hedip–hep–squahedip: 90°
Height1
Central density1
Number of external pieces16
Level of complexity15
Related polytopes
ArmyHehep
RegimentHehep
DualHeptagonal duotegmatic tegum
ConjugatesHeptagrammic duoprismatic prism, Great heptagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(7)≀S2×A1, order 784
ConvexYes
NatureTame

The heptagonal duoprismatic prism or hehep, also known as the heptagonal-heptagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 heptagonal duoprisms and 14 square-heptagonal duoprisms. Each vertex joins 4 square-heptagonal duoprisms and 1 heptagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

## Vertex coordinates

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

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

where j, k = 2, 4, 6.

## Representations

A heptagonal duoprismatic prism has the following Coxeter diagrams:

• x x7o x7o (full symmetry)
• xx7oo xx7oo&#x (heptagonal duoprism atop heptagonal duoprism)