Great inverted snub icosidodecahedron
| 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} |
| 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 |
File:Great inverted snub icosidodecahedron.stl
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{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
Cartesian coordinates for the vertices of a great inverted snub icosidodecahedron are all the even permutations of [math]\displaystyle{ \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} }[/math] with an even number of plus signs, where [math]\displaystyle{ \begin{align} \alpha &= \xi - \frac{1}{\xi} \\[4pt] \beta &= -\frac{\xi}{\varphi} + \frac{1}{\varphi^2} - \frac{1}{\xi\varphi}, \end{align} }[/math]
where [math]\displaystyle{ \varphi = \tfrac{1+ \sqrt 5}{2} }[/math] is the golden ratio and ξ is the greater positive real solution to: [math]\displaystyle{ \xi^3 - 2\xi = -\tfrac{1}{\varphi} \quad \implies \quad \xi \approx 1.2224727. }[/math] 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 [math]\displaystyle{ R = \frac12\sqrt{\frac{2-x}{1-x}} = 0.816081\dots }[/math] where x is the appropriate root of [math]\displaystyle{ x^3 + 2x^2 = \left( \frac{1\pm\sqrt5}2 \right)^2. }[/math] The four positive real roots of the sextic in R2, [math]\displaystyle{ 4096R^{12} - 27648R^{10} + 47104R^8 - 35776R^6 + 13872R^4 - 2696R^2 + 209 = 0 }[/math] are the circumradii of the snub dodecahedron (U29), great snub icosidodecahedron (U57), great inverted snub icosidodecahedron (U69), and great retrosnub icosidodecahedron (U74).
Related polyhedra
Great inverted pentagonal hexecontahedron
| 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 |
File:Great inverted pentagonal hexecontahedron.stl 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
Denote the golden ratio by [math]\displaystyle{ \phi }[/math]. Let [math]\displaystyle{ \xi\approx 0.252\,780\,289\,27 }[/math] be the smallest positive zero of the polynomial [math]\displaystyle{ P = 8x^3-8x^2+\phi^{-2} }[/math]. Then each pentagonal face has four equal angles of [math]\displaystyle{ \arccos(\xi)\approx 75.357\,903\,417\,42^{\circ} }[/math] and one angle of [math]\displaystyle{ 360^{\circ}-\arccos(-\phi^{-1}+\phi^{-2}\xi)\approx 238.568\,386\,330\,33^{\circ} }[/math]. Each face has three long and two short edges. The ratio [math]\displaystyle{ l }[/math] between the lengths of the long and the short edges is given by
- [math]\displaystyle{ l = \frac{2-4\xi^2}{1-2\xi}\approx 3.528\,053\,034\,81 }[/math].
The dihedral angle equals [math]\displaystyle{ \arccos(\xi/(\xi+1))\approx 78.359\,199\,060\,62^{\circ} }[/math]. Part of each face lies inside the solid, hence is invisible in solid models. The other two zeroes of the polynomial [math]\displaystyle{ P }[/math] play a similar role in the description of the great pentagonal hexecontahedron and the great pentagrammic hexecontahedron.
See also
References
- Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5 p. 126
External links
- Weisstein, Eric W.. "Great inverted pentagonal hexecontahedron". http://mathworld.wolfram.com/GreatInvertedPentagonalHexecontahedron.html.
- Weisstein, Eric W.. "Great inverted snub icosidodecahedron". http://mathworld.wolfram.com/GreatInvertedSnubIcosidodecahedron.html.
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