# Heptagonal-cubic duoprism

Heptagonal-cubic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHecube
Coxeter diagramx7o x4o3o
Elements
Tera7 tesseracts, 6 square-heptagonal duoprisms
Cells7+42 cubes, 12 heptagonal prisms
Faces42+84 squares, 8 heptagons
Edges56+84
Vertices56
Vertex figureTriangular scalene, edge lengths 2cos(π/7) (top), 2 (base triangle and sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {3+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 1.44152}$
Hypervolume${\displaystyle {\frac {7}{4\tan {\frac {\pi }{7}}}}\approx 3.66391}$
Diteral anglesTes–cube–tes: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Tes–cube–squahedip: 90°
Squahedip–hep–squahedip: 90°
Height1
Central density1
Number of external pieces13
Level of complexity10
Related polytopes
ArmyHecube
RegimentHecube
DualHeptagonal-octahedral duotegum
ConjugatesHeptagrammic-cubic duoprism, Great heptagrammic-cubic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(7), order 672
ConvexYes
NatureTame

The heptagonal-cubic duoprism or hecube, also known as a square-heptagonal duoprismatic prism, is a convex uniform duoprism that consists of 7 tesseracts and 6 square-heptagonal duoprisms. Each vertex joins 2 tesseracts and 3 square-heptagonal duoprisms. It is a duoprism based on a square and a heptagonal prism, which makes it a convex segmentoteron.

## Vertex coordinates

The vertices of a heptagonal-cubic duoprism of edge length 2sin(π/7) are given by:

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

where j = 2, 4, 6.

## Representations

A heptagonal-cubic duoprism has the following Coxeter diagrams:

• x7o x4o3o (full symmetry)
• x x4o x7o (square-heptagonal duoprismatic prism)
• x x x x7o (heptagonal prismatic prismatic prism)