# Triangular-heptagonal duoprism

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Triangular-heptagonal duoprism
Rank4
TypeUniform
Notation
Bowers style acronymTheddip
Coxeter diagramx3o x7o ()
Elements
Cells7 triangular prisms, 3 heptagonal prisms
Faces7 triangles, 21 squares, 3 heptagons
Edges21+21
Vertices21
Vertex figureDigonal disphenoid, edge lengths 1 (base 1), 2cos(π/7) (base 2), and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {{\frac {1}{3}}+{\frac {1}{4\sin ^{2}{\frac {\pi }{7}}}}}}\approx 1.28892}$
Hypervolume${\displaystyle {\frac {7{\sqrt {3}}}{16\tan {\frac {\pi }{7}}}}\approx 1.57353}$
Dichoral anglesTrip–3–trip: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Hep–4–trip: 90°
Hep–7–hep: 60°
Height${\displaystyle {\frac {\sqrt {3}}{2}}\approx 0.86603}$
Central density1
Number of external pieces10
Level of complexity6
Related polytopes
ArmyTheddip
RegimentTheddip
DualTriangular-heptagonal duotegum
ConjugatesTriangular-heptagrammic duoprism,
Triangular-great heptagrammic duoprism
Abstract & topological properties
Flag count504
Euler characteristic0
OrientableYes
Properties
SymmetryA2×I2(7), order 84
Flag orbits6
ConvexYes
NatureTame

The triangular-heptagonal duoprism or theddip, also known as the 3-7 duoprism, is a uniform duoprism that consists of 3 heptagonal prisms and 7 triangular prisms, with 2 of each meeting at each vertex. It can also be seen as a convex segmentochoron, being a heptagon atop a heptagonal prism.

## Vertex coordinates

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

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

where j = 2, 4, 6.