Heptagonal-snub dodecahedral duoprism

Heptagonal-snub dodecahedral duoprism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Hesnid
Coxeter diagram	x7o s5s3s
Elements
Tera	20+60 triangular-heptagonal duoprisms, 12 pentagonal-heptagonal duoprisms, 7 snub dodecahedral prisms
Cells	140+420 triangular prisms, 84 pentagonal prisms, 30+60+60 heptagonal prisms, 7 snub dodecahedra
Faces	140+420 triangles, 210+420+420 squares, 84 pentagons, 60 heptagons
Edges	210+420+420+420
Vertices	420
Vertex figure	Mirror-symmetric pentagonal scalene, edge lengths 1, 1, 1, 1, (1+√5)/2 (base pentagon), 2cos(π/7) (top edge), √2 (side edges)
Measures (edge length 1)
Circumradius	≈ 2.44451
Hypervolume	≈ 136.69561
Diteral angles	Theddip–hep–theddip: ≈ 164.17537°
	Theddip–hep–pheddip: ≈ 152.92992°
	Sniddip–snid–sniddip: ${\frac {5\pi }{7}}\approx 128.57143^{\circ }$
	Theddip–trip–sniddip: 90°
	Pheddip–pip–sniddip: 90°
Central density	1
Number of external pieces	99
Level of complexity	50
Related polytopes
Army	Hesnid
Regiment	Hesnid
Dual	Heptagonal-pentagonal hexecontahedral duotegum
Conjugates	Heptagonal-great snub icosidodecahedral duoprism, Heptagonal-great inverted snub icosidodecahedral duoprism, Heptagonal-great inverted retrosnub icosidodecahedral duoprism, Heptagrammic-snub dodecahedral duoprism, Heptagrammic-great snub icosidodecahedral duoprism, Heptagrammic-great inverted snub icosidodecahedral duoprism, Heptagrammic-great inverted retrosnub icosidodecahedral duoprism, Great heptagrammic-snub dodecahedral duoprism, Great heptagrammic-great snub icosidodecahedral duoprism, Great heptagrammic-great inverted snub icosidodecahedral duoprism, Great heptagrammic-great inverted retrosnub icosidodecahedral duoprism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	H3+×I2(7), order 840
Convex	Yes
Nature	Tame

The heptagonal-snub dodecahedral duoprism or hesnid is a convex uniform duoprism that consists of 7 snub dodecahedral prisms, 12 pentagonal-heptagonal duoprisms, and 80 triangular-heptagonal duoprisms of two kinds. Each vertex joins 2 snub dodecahedral prisms, 4 triangular-heptagonal duoprisms, and 1 pentagonal-heptagonal duoprism.

Vertex coordinates

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

• $\left(1,\,0,\,\phi {\sqrt {\phi \left(\xi -1-{\frac {1}{\xi }}\right)}}\sin {\frac {\pi }{7}},\,\xi \phi {\sqrt {3-\xi ^{2}}}\sin {\frac {\pi }{7}},\,\phi {\sqrt {\xi (\xi +\phi )+1}}\sin {\frac {\pi }{7}}\right),$
• $\left(\cos {\frac {j\pi }{7}},\,\pm \sin {\frac {j\pi }{7}},\,\phi {\sqrt {\phi \left(\xi -1-{\frac {1}{\xi }}\right)}}\sin {\frac {\pi }{7}},\,\xi \phi {\sqrt {3-\xi ^{2}}}\sin {\frac {\pi }{7}},\,\phi {\sqrt {\xi (\xi +\phi )+1}}\sin {\frac {\pi }{7}}\right),$
• $\left(1,\,0,\,\phi {\sqrt {3-\xi ^{2}}}\sin {\frac {\pi }{7}},\,\xi \phi {\sqrt {1-\xi +{\frac {1+\phi }{\xi }}}}\sin {\frac {\pi }{7}},\,\phi {\sqrt {\xi (\xi +1)}}\sin {\frac {\pi }{7}}\right),$
• $\left(\cos {\frac {j\pi }{7}},\,\pm \sin {\frac {j\pi }{7}},\,\phi {\sqrt {3-\xi ^{2}}}\sin {\frac {\pi }{7}},\,\xi \phi {\sqrt {1-\xi +{\frac {1+\phi }{\xi }}}}\sin {\frac {\pi }{7}},\,\phi {\sqrt {\xi (\xi +1)}}\sin {\frac {\pi }{7}}\right),$
• $\left(1,\,0,\,\xi ^{2}\phi {\sqrt {\phi \left(\xi -1-{\frac {1}{\xi }}\right)}}\sin {\frac {\pi }{7}},\,\phi {\sqrt {\xi +1-\phi }}\sin {\frac {\pi }{7}},\,{\sqrt {\xi ^{2}(1+2\phi )-\phi }}\sin {\frac {\pi }{7}}\right),$
• $\left(\cos {\frac {j\pi }{7}},\,\pm \sin {\frac {j\pi }{7}},\,\xi ^{2}\phi {\sqrt {\phi \left(\xi -1-{\frac {1}{\xi }}\right)}}\sin {\frac {\pi }{7}},\,\phi {\sqrt {\xi +1-\phi }}\sin {\frac {\pi }{7}},\,{\sqrt {\xi ^{2}(1+2\phi )-\phi }}\sin {\frac {\pi }{7}}\right),$

as well as all even permutations with an even number of sign changes of the last three coordinates of:

• $\left(1,\,0,\,\xi ^{2}\phi {\sqrt {3-\xi ^{2}}}\sin {\frac {\pi }{7}},\,\xi \phi {\sqrt {\phi (\xi -1-{\frac {1}{\xi }})}}\sin {\frac {\pi }{7}},\,{\frac {\phi ^{2}{\sqrt {\xi (\xi +\phi )+1}}\sin {\frac {\pi }{7}}}{\xi }}\right),$
• $\left(\cos {\frac {j\pi }{7}},\,\pm \sin {\frac {j\pi }{7}},\,\xi ^{2}\phi {\sqrt {3-\xi ^{2}}}\sin {\frac {\pi }{7}},\,\xi \phi {\sqrt {\phi (\xi -1-{\frac {1}{\xi }})}}\sin {\frac {\pi }{7}},\,{\frac {\phi ^{2}{\sqrt {\xi (\xi +\phi )+1}}\sin {\frac {\pi }{7}}}{\xi }}\right),$
• $\left(1,\,0,\,{\sqrt {\phi (\xi +2)+2}}\sin {\frac {\pi }{7}},\,\phi {\sqrt {1-\xi +{\frac {1+\phi }{\xi }}}}\sin {\frac {\pi }{7}},\,\xi {\sqrt {\xi (1+\phi )-\phi }}\sin {\frac {\pi }{7}}\right),$
• $\left(\cos {\frac {j\pi }{7}},\,\pm \sin {\frac {j\pi }{7}},\,{\sqrt {\phi (\xi +2)+2}}\sin {\frac {\pi }{7}},\,\phi {\sqrt {1-\xi +{\frac {1+\phi }{\xi }}}}\sin {\frac {\pi }{7}},\,\xi {\sqrt {\xi (1+\phi )-\phi }}\sin {\frac {\pi }{7}}\right),$

where

• j = 2, 4, 6,
• $\phi ={\frac {1+{\sqrt {5}}}{2}},$
• $\xi ={\sqrt[{3}]{\frac {\phi +{\sqrt {\phi -{\frac {5}{27}}}}}{2}}}+{\sqrt[{3}]{\frac {\phi -{\sqrt {\phi -{\frac {5}{27}}}}}{2}}}.$