Heptagonal-truncated octahedral duoprism

Heptagonal-truncated octahedral duoprism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Hetoe
Coxeter diagram	x7o o4x3x ()
Elements
Tera	6 square-heptagonal duoprisms, 8 hexagonal-heptagonal duoprisms, 7 truncated octahedral prisms
Cells	42 cubes, 56 hexagonal prisms, 12+24 heptagonal prisms, 7 truncated octahedra
Faces	42+84+168 squares, 56 hexagons, 42 heptagons
Edges	84+168+168
Vertices	168
Vertex figure	Digonal disphenoidal pyramid, edge lengths √2, √3, √3 (base triangle), 2cos(π/7) (top), √2 (side edges)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {10+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 1.95652$
Hypervolume	${\frac {14{\sqrt {2}}}{\tan {\frac {\pi }{7}}}}\approx 41.11303$
Diteral angles	Tope–toe–tope: ${\frac {5\pi }{7}}\approx 128.57143^{\circ }$
	Squahedip–hep–haheddip: $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$
	Haheddip–hep–haheddip: $\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$
	Squahedip–cube–tope: 90°
	Haheddip–hip–tope: 90°
Central density	1
Number of external pieces	21
Level of complexity	30
Related polytopes
Army	Hetoe
Regiment	Hetoe
Dual	Heptagonal-tetrakis hexahedral duotegum
Conjugates	Heptagrammic-truncated octahedral duoprism, Great heptagrammic-truncated octahedral duoprism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	B3×I2(7), order 672
Convex	Yes
Nature	Tame

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:

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

where j = 2, 4, 6.

Representations

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

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

External links

• Klitzing, Richard. "n-toe-dip".