# Heptagonal-octagonal duoprismatic prism

Heptagonal-octagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymHeop
Coxeter diagramx x7o x8o
Elements
Tera8 square-heptagonal duoprisms, 7 square-octagonal duoprisms, 2 heptagonal-octagonal duoprisms
Cells56 cubes, 7+14 octagonal prisms, 8+16 heptagonal prisms
Faces56+56+112 squares, 16 heptagons, 14 octagons
Edges56+112+112
Vertices112
Vertex figureDigonal disphenoidal pyramid, edge lengths 2cos(π/7) (disphenoid base 1), 2+2 (disphenoid base 2), 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {5+2{\sqrt {2}}+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 1.81248}$
Hypervolume${\displaystyle 7{\frac {1+{\sqrt {2}}}{2\tan {\frac {\pi }{7}}}}\approx 17.54608}$
Diteral anglesSquahedip–hep–squahedip: 135°
Sodip–op–sodip: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Sodip–cube–squahedip: 90°
Heodip–hep–squahedip: 90°
Sodip–op–heodip: 90°
Height1
Central density1
Number of external pieces17
Level of complexity30
Related polytopes
ArmyHeop
RegimentHeop
DualHeptagonal-octagonal duotegmatic tegum
ConjugatesHeptagonal-octagrammic duoprismatic prism, Heptagrammic-octagonal duoprismatic prism, Heptagrammic-octagrammic duoprismatic prism, Great heptagrammic-octagonal duoprismatic prism, Great heptagrammic-octagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(7)×I2(8)×A1, order 448
ConvexYes
NatureTame

The heptagonal-octagonal duoprismatic prism or heop, also known as the heptagonal-octagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 heptagonal-octagonal duoprisms, 7 square-octagonal duoprisms, and 8 square-heptagonal duoprisms. Each vertex joins 2 square-heptagonal duoprisms, 2 square-octagonal duoprisms, and 1 heptagonal-octagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

## Vertex coordinates

The vertices of a heptagonal-octagonal duoprismatic prism of edge length 2sin(π/7) are given by all permutations of the third and fourth coordinates of:

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

where j = 2, 4, 6.

## Representations

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

• x x7o x8o (full symmetry)
• x x7o x4x (octagons as ditetragons)
• xx7oo xx8oo&#x (heptagonal-octagonal duoprism atop heptagonal-octagonal duoprism)
• xx7oo xx4xx&#x