Heptagonal-octagonal duoprism

Heptagonal-octagonal duoprism
Rank4
TypeUniform
Notation
Bowers style acronymHeodip
Coxeter diagramx7o x8o ()
Elements
Cells8 heptagonal prisms, 7 octagonal prisms
Faces56 squares, 8 heptagons, 7 octagons
Edges56+56
Vertices56
Vertex figureDigonal disphenoid, edge lengths 2cos(π/7) (base 1), 2+2 (base 2), and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {{\frac {2+{\sqrt {2}}}{2}}+{\frac {1}{4\sin ^{2}{\frac {\pi }{7}}}}}}\approx 1.74215}$
Hypervolume${\displaystyle {\frac {7(1+{\sqrt {2}})}{2\tan {\frac {\pi }{7}}}}\approx 17.54608}$
Dichoral anglesHep–7–hep: 135°
Op–8–op: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Hep–4–op: 90°
Central density1
Number of external pieces15
Level of complexity6
Related polytopes
ArmyHeodip
RegimentHeodip
DualHeptagonal-octagonal duotegum
ConjugatesHeptagonal-octagrammic duoprism,
Heptagrammic-octagonal duoprism,
Heptagrammic-octagrammic duoprism,
Great heptagrammic-octagonal duoprism,
Great heptagrammic-octagrammic duoprism
Abstract & topological properties
Flag count1344
Euler characteristic0
OrientableYes
Properties
SymmetryI2(7)×I2(8), order 224
Flag orbits6
ConvexYes
NatureTame

The heptagonal-octagonal duoprism or heodip, also known as the 7-8 duoprism, is a uniform duoprism that consists of 7 octagonal prisms and 8 heptagonal prisms, with two of each joining at each vertex.

Vertex coordinates

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

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

where j = 2, 4, 6.

Representations

A heptagonal-octagonal duoprism has the following Coxeter diagrams:

• x7o x8o () (full symmetry)
• x4x x7o () (B2×I2(7) symmetry, octagons as ditetragons)