# Heptagonal-cuboctahedral duoprism

Heptagonal-cuboctahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHeco
Coxeter diagramx7o o4x3o ()
Elements
Tera7 cuboctahedral prisms, 8 triangular-heptagonal duoprisms, 6 square-heptagonal duoprisms
Cells56 triangular prisms, 42 cubes, 7 cuboctahedra, 24 heptagonal prisms
Faces56 triangles, 42+168 squares, 12 heptagons
Edges84+168
Vertices84
Vertex figureRectangular scalene, edge lengths 1, 2, 1, 2 (base rectangle), 2cos(π/7) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {4+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 1.52577}$
Hypervolume${\displaystyle {\frac {35{\sqrt {2}}}{12\tan {\frac {\pi }{7}}}}\approx 8.56521}$
Diteral anglesCope–co–cope: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Theddip–hep–squahedip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Theddip–trip–cope: 90°
Squahedip–cube–cope: 90°
Central density1
Number of external pieces21
Level of complexity20
Related polytopes
ArmyHeco
RegimentHeco
DualHeptagonal-rhombic dodecahedral duotegum
ConjugatesHeptagrammic-cuboctahedral duoprism, Great heptagrammic-cuboctahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(7), order 672
ConvexYes
NatureTame

The heptagonal-cuboctahedral duoprism or heco is a convex uniform duoprism that consists of 7 cuboctahedral prisms, 6 square-heptagonal duoprisms, and 8 triangular-heptagonal duoprisms. Each vertex joins 2 cuboctahedral prisms, 2 triangular-heptagonal duoprisms, and 2 square-heptagonal duoprisms.

## Vertex coordinates

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

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

where j = 2, 4, 6.

## Representations

A heptagonal-cuboctahedral duoprism has the following Coxeter diagrams:

• x7o o4x3o (full symmetry)
• x7o x3o3x ()