# Heptagonal duoprism

Heptagonal duoprism
Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymHedip
Coxeter diagramx7o x7o ()
Elements
Cells14 heptagonal prisms
Faces49 squares, 14 heptagons
Edges98
Vertices49
Vertex figureTetragonal disphenoid, edge lengths 2cos(π/7) (bases) and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt2}{2\sin\frac\pi7} ≈ 1.62971}$
Inradius${\displaystyle \frac{1}{2\tan\frac\pi7} ≈ 1.03826}$
Hypervolume${\displaystyle \frac{49}{16\tan^2\frac\pi7} ≈ 13.20532}$
Dichoral anglesHep–7–hep: ${\displaystyle \frac{5\pi}{7} ≈ 128.57143°}$
Hep–4–hep: 90°
Central density1
Number of external pieces14
Level of complexity3
Related polytopes
ArmyHedip
RegimentHedip
DualHeptagonal duotegum
ConjugatesHeptagrammic duoprism, Great heptagrammic duoprism
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryI2(7)≀S2, order 392
ConvexYes
NatureTame

The heptagonal duoprism or hedip, also known as the heptagonal-heptagonal duoprism, the 7 duoprism or the 7-7 duoprism, is a noble uniform duoprism that consists of 14 heptagonal prisms, with 4 joining at each vertex. It is also the 14-6 gyrochoron. It is the first in an infinite family of isogonal heptagonal dihedral swirlchora and also the first in an infinite family of isochoric heptagonal hosohedral swirlchora.

## Vertex coordinates

The coordinates of a heptagonal duoprism, centered at the origin and with edge length 2sin(π/7), are given by:

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

where j, k = 2, 4, 6.