# Heptagonal-truncated octahedral duoprism

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Heptagonal-truncated octahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHetoe
Coxeter diagramx7o o4x3x ()
Elements
Tera6 square-heptagonal duoprisms, 8 hexagonal-heptagonal duoprisms, 7 truncated octahedral prisms
Cells42 cubes, 56 hexagonal prisms, 12+24 heptagonal prisms, 7 truncated octahedra
Faces42+84+168 squares, 56 hexagons, 42 heptagons
Edges84+168+168
Vertices168
Vertex figureDigonal disphenoidal pyramid, edge lengths 2, 3, 3 (base triangle), 2cos(π/7) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {10+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 1.95652}$
Hypervolume${\displaystyle {\frac {14{\sqrt {2}}}{\tan {\frac {\pi }{7}}}}\approx 41.11303}$
Diteral anglesTope–toe–tope: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Squahedip–hep–haheddip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Haheddip–hep–haheddip: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Squahedip–cube–tope: 90°
Haheddip–hip–tope: 90°
Central density1
Number of external pieces21
Level of complexity30
Related polytopes
ArmyHetoe
RegimentHetoe
DualHeptagonal-tetrakis hexahedral duotegum
ConjugatesHeptagrammic-truncated octahedral duoprism, Great heptagrammic-truncated octahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(7), order 672
ConvexYes
NatureTame

The heptagonal-truncated octahedral duoprism or hetoe is a convex uniform duoprism that consists of 7 truncated octahedral prisms, 8 hexagonal-heptagonal duoprisms, and 6 square-heptagonal duoprisms. Each vertex joins 2 truncated octahedral prisms, 1 square-heptagonal duoprism, and 2 hexagonal-heptagonal duoprisms.

## Vertex coordinates

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

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

where j = 2, 4, 6.

## Representations

A heptagonal-truncated octahedral duoprism has the following Coxeter diagrams:

• x7o o4x3x (full symmetry)
• x7o x3x3x ()