# Heptagonal-octahedral duoprism

Heptagonal-octahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHeoct
Coxeter diagramx7o o4o3x
Elements
Tera7 octahedral prisms, 8 triangular-heptagonal duoprisms
Cells56 triangular prisms, 7 octahedra, 12 heptagonal prisms
Faces56 triangles, 84 squares, 6 heptagons
Edges42+84
Vertices42
Vertex figureSquare scalene, edge lengths 1 (base square), 2cos(π/7) (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {2+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 1.35203}$
Hypervolume${\displaystyle {\frac {7{\sqrt {2}}}{12\tan {\frac {\pi }{7}}}}\approx 1.71304}$
Diteral anglesOpe–oct–ope: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Theddip–hep–theddip: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Theddip–trip–ope: 90°
${\displaystyle {\frac {\sqrt {6}}{3}}\approx 0.81650}$
Central density1
Number of external pieces15
Level of complexity10
Related polytopes
ArmyHeoct
RegimentHeoct
DualHeptagonal-cubic duotegum
ConjugatesHeptagrammic-octahedral duoprism, Great heptagrammic-octahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(7), order 672
ConvexYes
NatureTame

The heptagonal-octahedral duoprism or heoct is a convex uniform duoprism that consists of 7 octahedral prisms and 8 triangular-heptagonal duoprisms. Each vertex joins 2 octahedral prisms and 4 triangular-heptagonal duoprisms.

## Vertex coordinates

The vertices of a heptagonal-octahedral duoprism of edge length 2sin(π/7) are given by all permutations and sign changes of the last three coordinates of:

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

where j = 2, 4, 6.