Heptagonal-cuboctahedral duoprism

Heptagonal-cuboctahedral duoprism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Heco
Coxeter diagram	x7o o4x3o ()
Elements
Tera	7 cuboctahedral prisms, 8 triangular-heptagonal duoprisms, 6 square-heptagonal duoprisms
Cells	56 triangular prisms, 42 cubes, 7 cuboctahedra, 24 heptagonal prisms
Faces	56 triangles, 42+168 squares, 12 heptagons
Edges	84+168
Vertices	84
Vertex figure	Rectangular scalene, edge lengths 1, √2, 1, √2 (base rectangle), 2cos(π/7) (top), √2 (side edges)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {4+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 1.52577$
Hypervolume	${\frac {35{\sqrt {2}}}{12\tan {\frac {\pi }{7}}}}\approx 8.56521$
Diteral angles	Cope–co–cope: ${\frac {5\pi }{7}}\approx 128.57143^{\circ }$
	Theddip–hep–squahedip: $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$
	Theddip–trip–cope: 90°
	Squahedip–cube–cope: 90°
Central density	1
Number of external pieces	21
Level of complexity	20
Related polytopes
Army	Heco
Regiment	Heco
Dual	Heptagonal-rhombic dodecahedral duotegum
Conjugates	Heptagrammic-cuboctahedral duoprism, Great heptagrammic-cuboctahedral duoprism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	B3×I2(7), order 672
Convex	Yes
Nature	Tame

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:

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

where j = 2, 4, 6.

Representations

A heptagonal-cuboctahedral duoprism has the following Coxeter diagrams:

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

External links

Klitzing, Richard. "n-co-dip".