View source for Great inverted snub icosidodecahedron

{{Short description|Polyhedron with 92 faces}}
{{Uniform polyhedra db|Uniform polyhedron stat table|Gisid}}
[[File:Great inverted snub icosidodecahedron.stl|thumb|3D model of a great inverted snub icosidodecahedron]]
In [[Geometry|geometry]], the '''great inverted snub icosidodecahedron''' (or '''great vertisnub icosidodecahedron''') is a [[Uniform star polyhedron|uniform star polyhedron]], indexed as U<sub>69</sub>. It is given a [[Schläfli symbol]] {{math|sr{{mset|{{Frac|5|3}},3}},}} and Coxeter-Dynkin diagram {{CDD|node_h|5|rat|d3|node_h|3|node_h}}. In the book ''Polyhedron Models'' by Magnus Wenninger, the polyhedron is misnamed ''[[Great snub icosidodecahedron|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 display=block>\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 display=block>\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>\varphi = \tfrac{1+ \sqrt 5}{2}</math> is the [[Golden ratio|golden ratio]] and
{{mvar|ξ}} is the greater positive real solution to:
<math display=block>\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 [[Chirality (mathematics)|enantiomorph]] of the other one.

The circumradius for unit edge length is
<math display=block>R = \frac12\sqrt{\frac{2-x}{1-x}} = 0.816081\dots</math>
where {{mvar|x}} is the appropriate root of 
<math display=block>x^3 + 2x^2 = \left( \frac{1\pm\sqrt5}2 \right)^2.</math> 
The four positive real roots of the sextic in {{math|''R''{{sup|2}}}},
<math display=block>4096R^{12} - 27648R^{10} + 47104R^8 - 35776R^6 + 13872R^4 - 2696R^2 + 209 = 0</math>
are the circumradii of the [[Snub dodecahedron|snub dodecahedron]] (U<sub>29</sub>), [[Great snub icosidodecahedron|great snub icosidodecahedron]] (U<sub>57</sub>), great inverted snub icosidodecahedron (U<sub>69</sub>), and [[Great retrosnub icosidodecahedron|great retrosnub icosidodecahedron]] (U<sub>74</sub>).
{{-}}

== Related polyhedra ==

=== Great inverted pentagonal hexecontahedron ===
{{Uniform polyhedra db|Uniform dual polyhedron stat table|Gisid}}
[[File:Great inverted pentagonal hexecontahedron.stl|thumb|3D model of a great inverted pentagonal hexecontahedron]]
The '''great inverted pentagonal hexecontahedron''' (or '''petaloidal trisicosahedron''') is a nonconvex [[Isohedral figure|isohedral]] [[Polyhedron|polyhedron]]. It is composed of 60 [[Concave polygon|concave]] pentagonal faces, 150 edges and 92 vertices.

It is the [[Dual polyhedron|dual]] of the [[Uniform star polyhedron|uniform]] great inverted snub icosidodecahedron.
{{-}}

==Proportions==

Denote the [[Golden ratio|golden ratio]] by <math>\phi</math>. Let <math>\xi\approx 0.252\,780\,289\,27</math> be the smallest positive  zero of the polynomial <math>P = 8x^3-8x^2+\phi^{-2}</math>. Then each pentagonal face has four equal angles of <math>\arccos(\xi)\approx 75.357\,903\,417\,42^{\circ}</math> and one angle of <math>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>l</math> between the lengths of the long and the short edges is given by
:<math>l = \frac{2-4\xi^2}{1-2\xi}\approx 3.528\,053\,034\,81</math>. 
The [[Dihedral angle|dihedral angle]] equals <math>\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>P</math> play a similar role in the description of the great pentagonal hexecontahedron and the great pentagrammic hexecontahedron.

== See also ==
* [[List of uniform polyhedra]]
* [[Great snub icosidodecahedron]]
* [[Great retrosnub icosidodecahedron]]

==References==
*{{Citation | last1=Wenninger | first1=Magnus |  title=Dual Models | publisher={{wipe|Cambridge University Press}} | isbn=978-0-521-54325-5 |mr=730208 | year=1983}} p.&nbsp;126

== External links ==
* {{mathworld | urlname = GreatInvertedPentagonalHexecontahedron| title =Great inverted pentagonal hexecontahedron}}
* {{mathworld | urlname = GreatInvertedSnubIcosidodecahedron  | title = Great inverted snub icosidodecahedron}}

[[Category:Uniform polyhedra]]



{{Sourceattribution|Great inverted snub icosidodecahedron}}