Heptagonal-snub cubic duoprism

Heptagonal-snub cubic duoprism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Hesnic
Coxeter diagram	x7o s4s3s
Elements
Tera	8+24 triangular-heptagonal duoprisms, 6 square-heptagonal duoprisms, 6 snub cubic prisms
Cells	56+168 triangular prisms, 42 cubes, 12+24+24 heptagonal prisms, 7 snub cubes
Faces	56+168 triangles, 42+84+168+168 squares, 24 heptagons
Edges	84+168+168+168
Vertices	168
Vertex figure	Mirror-symmetric pentagonal scalene, edge lengths 1, 1, 1, 1, √2 (base pentagon), 2cos(π/7) (top edge), √2 (side edges)
Measures (edge length 1)
Circumradius	≈ 1.77018
Hypervolume	≈ 28.66968
Diteral angles	Theddip–hep–theddip: ≈ 153.23459°
	Theddip–hep–squahedip: ≈ 142.98343°
	Sniccup–snic–sniccup: ${\frac {5\pi }{7}}\approx 128.57143^{\circ }$
	Theddip–trip–sniccup: 90°
	Squahedip–cube–sniccup: 90°
Central density	1
Number of external pieces	45
Level of complexity	50
Related polytopes
Army	Hesnic
Regiment	Hesnic
Dual	Heptagonal-pentagonal icositetrahedral duotegum
Conjugates	Heptagrammic-snub cubic duoprism, Great heptagrammic-snub cubic duoprism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	B3+×I2(7), order 336
Convex	Yes
Nature	Tame

The heptagonal-snub cubic duoprism or hesnic is a convex uniform duoprism that consists of 7 snub cubic prisms, 6 square-heptagonal duoprisms, and 32 triangular-heptagonal duoprisms of two kinds. Each vertex joins 2 snub cubic prisms, 4 triangular-heptagonal duoprisms, and 1 square-heptagonal duoprism.

Vertex coordinates

The vertices of a heptagonal-snub cubic duoprism of edge length 2sin(π/7) are given by by all even permutations with an even number of sign changes, plus all odd permutations with an odd amount of sign changes, of the last three coordinates of:

• $\left(1,\,0,\,2c_{1}\sin {\frac {\pi }{7}},\,2c_{2}\sin {\frac {\pi }{7}},\,2c_{3}\sin {\frac {\pi }{7}}\right),$
• $\left(\cos \left({\frac {j\pi }{7}}\right),\,\pm \sin \left({\frac {j\pi }{7}}\right),\,2c_{1}\sin {\frac {\pi }{7}},\,2c_{2}\sin {\frac {\pi }{7}},\,2c_{3}\sin {\frac {\pi }{7}}\right),$

where

• j = 2, 4, 6,
• $c_{1}={\sqrt {{\frac {1}{12}}\left(4-{\sqrt[{3}]{17+3{\sqrt {33}}}}-{\sqrt[{3}]{17-3{\sqrt {33}}}}\right)}},$
• $c_{2}={\sqrt {{\frac {1}{12}}\left(2+{\sqrt[{3}]{17+3{\sqrt {33}}}}+{\sqrt[{3}]{17-3{\sqrt {33}}}}\right)}},$
• $c_{3}={\sqrt {{\frac {1}{12}}\left(4+{\sqrt[{3}]{199+3{\sqrt {33}}}}+{\sqrt[{3}]{199-3{\sqrt {33}}}}\right)}}.$