Snub_dodecadodecahedron Knowpia

Snub dodecadodecahedron

Type	Uniform star polyhedron
Elements	F = 84, E = 150 V = 60 (χ = −6)
Faces by sides	60{3}+12{5}+12{5/2}
Coxeter diagram
Wythoff symbol	\| 2 5/2 5
Symmetry group	I, [5,3]⁺, 532
Index references	U₄₀, C₄₉, W₁₁₁
Dual polyhedron	Medial pentagonal hexecontahedron
Vertex figure	3.3.5/2.3.5
Bowers acronym	Siddid

In geometry, the snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as $U 40$ . It has 84 faces (60 triangles, 12 pentagons, and 12 pentagrams), 150 edges, and 60 vertices.^[1] It is given a Schläfli symbol $sr{.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}5⁄2,5},$ as a snub great dodecahedron.

Cartesian coordinates edit

Cartesian coordinates for the vertices of an inverted snub dodecadodecahedron are all the even permutations of

{\begin{array}{crrrc}{\Bigl (}&\pm \,2\alpha \ ,&\pm \,2\ ,&\pm \,2\beta \ &{\Bigr )},\\[2pt]{\Bigl (}&\pm {\bigl [}\alpha +{\frac {\beta }{\varphi }}+\varphi {\bigr ]},&\pm {\bigl [}-\alpha \varphi +\beta +{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}{\frac {\alpha }{\varphi }}+\beta \varphi -1{\bigr ]}&{\Bigr )},\\[2pt]{\Bigl (}&\pm {\bigl [}-{\frac {\alpha }{\varphi }}+\beta \varphi +1{\bigr ]},&\pm {\bigl [}-\alpha +{\frac {\beta }{\varphi }}-\varphi {\bigr ]},&\pm {\bigl [}\alpha \varphi +\beta -{\frac {1}{\varphi }}{\bigr ]}&{\Bigr )},\\[2pt]{\Bigl (}&\pm {\bigl [}-{\frac {\alpha }{\varphi }}+\beta \varphi -1{\bigr ]},&\pm {\bigl [}\alpha -{\frac {\beta }{\varphi }}-\varphi {\bigr ]},&\pm {\bigl [}\alpha \varphi +\beta +{\frac {1}{\varphi }}{\bigr ]}&{\Bigr )},\\[2pt]{\Bigl (}&\pm {\bigl [}\alpha +{\frac {\beta }{\varphi }}-\varphi {\bigr ]},&\pm {\bigl [}\alpha \varphi -\beta +{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}{\frac {\alpha }{\varphi }}+\beta \varphi +1{\bigr ]}&{\Bigr )},\end{array}}

with an even number of plus signs, where

\beta ={\frac {\ \ {\frac {\alpha ^{2}}{\varphi }}+\varphi \ \ }{\ \alpha \varphi -{\frac {1}{\varphi }}}}\ ,

\varphi ={\tfrac {1+{\sqrt {5}}}{2}}

is the golden ratio, and

α

is the positive real root of

\varphi \alpha ^{4}-\alpha ^{3}+2\alpha ^{2}-\alpha -{\frac {1}{\varphi }}\quad \implies \quad \alpha \approx 0.7964421.

Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one. Taking

α

to be the negative root gives the inverted snub dodecadodecahedron.

Related polyhedra edit

Medial pentagonal hexecontahedron edit

Medial pentagonal hexecontahedron

Type	Star polyhedron
Face
Elements	F = 60, E = 150 V = 84 (χ = −6)
Symmetry group	I, [5,3]⁺, 532
Index references	DU₄₀
dual polyhedron	Snub dodecadodecahedron 3D model of a medial pentagonal hexecontahedron The medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces. See also edit List of uniform polyhedra Inverted snub dodecadodecahedron References edit ^ Maeder, Roman. "40: snub dodecadodecahedron". MathConsult. Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208 External links edit Weisstein, Eric W. "Medial pentagonal hexecontahedron". MathWorld. Weisstein, Eric W. "Snub dodecadodecahedron". MathWorld. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip exeateure conquat. Duis aute irure dolor in esse. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip exeateure conquat. Duis aute EDITOR’S PICK TripAdvisor Users Reviews Apr 23 2020 By Super User Carbonated Ice Cream Rise Apr 23 2020 By Super User Costco Is The Least Busy Apr 23 2020 By Super User Volkswagen Cars Products Apr 23 2020 By Super User RANDOM NEWS TripAdvisor Users Reviews Apr 23 2020 By Super User Homeless Shelters For Cold Apr 23 2020 By Super User Lawsuit says Sherwood Apr 23 2020 By Super User Jian Ghomeshi Speaks Apr 23 2020 By Super User POST GALLERY By using this platform, you agree to the Terms of Use ©2020 Knowpia About How it Works Privacy Disclaimer Contact FAQ

[1] Maeder, Roman. "40: snub dodecadodecahedron". MathConsult.

[1]

Summary

Cartesian coordinates edit

Related polyhedra edit

Medial pentagonal hexecontahedron edit

See also edit

References edit

External links edit