Great_inverted_snub_icosidodecahedron Knowpia

Great inverted snub icosidodecahedron

Type	Uniform star polyhedron
Elements	F = 92, E = 150 V = 60 (χ = 2)
Faces by sides	(20+60){3}+12{5/2}
Coxeter diagram
Wythoff symbol	\| 5/3 2 3
Symmetry group	I, [5,3]⁺, 532
Index references	U₆₉, C₇₃, W₁₁₆
Dual polyhedron	Great inverted pentagonal hexecontahedron
Vertex figure	3⁴.5/3
Bowers acronym	Gisid

In geometry, the great inverted snub icosidodecahedron (or great vertisnub icosidodecahedron) is a uniform star polyhedron, indexed as U₆₉. It is given a Schläfli symbol $sr{.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}5⁄3,3},$ and Coxeter-Dynkin diagram . In the book Polyhedron Models by Magnus Wenninger, the polyhedron is misnamed great snub icosidodecahedron, and vice versa.

Cartesian coordinates edit

Cartesian coordinates for the vertices of a great inverted snub icosidodecahedron are all the even permutations of

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

with an even number of plus signs, where

{\begin{aligned}\alpha &=\xi -{\frac {1}{\xi }}\\[4pt]\beta &=-{\frac {\xi }{\varphi }}+{\frac {1}{\varphi ^{2}}}-{\frac {1}{\xi \varphi }},\end{aligned}}

where $\varphi ={\tfrac {1+{\sqrt {5}}}{2}}$ is the golden ratio and $ξ$ is the greater positive real solution to:

\xi ^{3}-2\xi =-{\tfrac {1}{\varphi }}\quad \implies \quad \xi \approx 1.2224727.

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

The circumradius for unit edge length is

R={\frac {1}{2}}{\sqrt {\frac {2-x}{1-x}}}=0.816081\dots

where

x

is the appropriate root of

x^{3}+2x^{2}=\left({\frac {1\pm {\sqrt {5}}}{2}}\right)^{2}.

The four positive real roots of the sextic in

R 2

,

4096R^{12}-27648R^{10}+47104R^{8}-35776R^{6}+13872R^{4}-2696R^{2}+209=0

are the circumradii of the snub dodecahedron (U₂₉), great snub icosidodecahedron (U₅₇), great inverted snub icosidodecahedron (U₆₉), and great retrosnub icosidodecahedron (U₇₄).

Related polyhedra edit

Great inverted pentagonal hexecontahedron edit

Great inverted pentagonal hexecontahedron

Type	Star polyhedron
Face
Elements	F = 60, E = 150 V = 92 (χ = 2)
Symmetry group	I, [5,3]⁺, 532
Index references	DU₆₉
dual polyhedron	Great inverted snub icosidodecahedron 3D model of a great inverted pentagonal hexecontahedron The great inverted pentagonal hexecontahedron (or petaloidal trisicosahedron) is a nonconvex isohedral polyhedron. It is composed of 60 concave pentagonal faces, 150 edges and 92 vertices. It is the dual of the uniform great inverted snub icosidodecahedron. Proportions edit Denote the golden ratio by $\phi$ . Let $\xi \approx 0.252\,780\,289\,27$ be the smallest positive zero of the polynomial $P=8x^{3}-8x^{2}+\phi ^{-2}$ . Then each pentagonal face has four equal angles of $\arccos(\xi )\approx 75.357\,903\,417\,42^{\circ }$ and one angle of $360^{\circ }-\arccos(-\phi ^{-1}+\phi ^{-2}\xi )\approx 238.568\,386\,330\,33^{\circ }$ . Each face has three long and two short edges. The ratio $l$ between the lengths of the long and the short edges is given by $l={\frac {2-4\xi ^{2}}{1-2\xi }}\approx 3.528\,053\,034\,81$ . The dihedral angle equals $\arccos(\xi /(\xi +1))\approx 78.359\,199\,060\,62^{\circ }$ . Part of each face lies inside the solid, hence is invisible in solid models. The other two zeroes of the polynomial $P$ play a similar role in the description of the great pentagonal hexecontahedron and the great pentagrammic hexecontahedron. See also edit List of uniform polyhedra Great snub icosidodecahedron Great retrosnub icosidodecahedron References edit Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208 p. 126 External links edit Weisstein, Eric W. "Great inverted pentagonal hexecontahedron". MathWorld. Weisstein, Eric W. "Great inverted snub icosidodecahedron". 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

Summary

Cartesian coordinates edit

Related polyhedra edit

Great inverted pentagonal hexecontahedron edit

Proportions edit

See also edit

References edit

External links edit