View source for Great snub icosidodecahedron

{{short description|Polyhedron with 92 faces}}
{{Uniform polyhedra db|Uniform polyhedron stat table|Gosid}}
[[File:Great snub icosidodecahedron.stl|thumb|3D model of a great snub icosidodecahedron]]
In [[geometry]], the '''great snub icosidodecahedron''' is a [[nonconvex uniform polyhedron]], indexed as U<sub>57</sub>. It has 92 faces (80 [[Triangle|triangles]] and 12 [[Pentagram|pentagrams]]), 150 edges, and 60 vertices.<ref>{{Cite web|url=https://www.mathconsult.ch/static/unipoly/57.html|title=57: great snub icosidodecahedron|last=Maeder|first=Roman|website=MathConsult}}</ref> It can be represented by a [[Schläfli symbol]] sr{{{Frac|5|2}},3}, and [[Coxeter-Dynkin diagram]] {{CDD|node_h|5|rat|d2|node_h|3|node_h}}.

This polyhedron is the [[snub (geometry)|snub]] member of a family that includes the [[great icosahedron]], the [[great stellated dodecahedron]] and the [[great icosidodecahedron]].

In the book ''[[Polyhedron Models]]'' by [[Magnus Wenninger]], the polyhedron is misnamed ''[[great inverted snub icosidodecahedron]]'', and vice versa.

==Cartesian coordinates==

Let <math>\xi\approx 0.3990206456527105</math> be the positive zero of the polynomial <math>x^3+2x^2-\phi^{-2}</math>, where <math>\phi</math> is the [[golden ratio]]. Let the point <math>p</math> be given by
:<math>p=
\begin{pmatrix}
        \xi \\
        \phi^{-2}-\phi^{-2}\xi \\
        -\phi^{-3}+\phi^{-1}\xi+2\phi^{-1}\xi^2
\end{pmatrix}
</math>.
Let the matrix <math>M</math> be given by
:<math>M=
\begin{pmatrix}
         1/2  & -\phi/2 & 1/(2\phi) \\
         \phi/2   & 1/(2\phi)     & -1/2 \\
         1/(2\phi)     & 1/2  & \phi/2
\end{pmatrix}
</math>.
<math>M</math> is the rotation around the axis <math>(1, 0, \phi)</math>  by an angle of <math>2\pi/5</math>, counterclockwise. Let the linear transformations <math>T_0, \ldots, T_{11}</math> 
be the transformations which send a point <math>(x, y, z)</math> to the [[parity of a permutation|even permutations]] of <math>(\pm x, \pm y, \pm z)</math>  with an even number of minus signs. 
The transformations <math>T_i</math> constitute the group of rotational symmetries of a [[regular tetrahedron]].
The transformations <math>T_i M^j</math> <math>(i = 0,\ldots, 11</math>, <math>j = 0,\ldots, 4)</math> constitute the group of rotational symmetries of a [[regular icosahedron]].
Then the 60 points <math>T_i M^j p</math>  are the vertices of a great snub icosahedron. The edge length equals <math>2\xi\sqrt{1-\xi}</math>, the circumradius equals <math>\xi\sqrt{2-\xi}</math>, and the midradius equals <math>\xi</math>.

For a great snub icosidodecahedron whose edge length is 1,
the circumradius is
:<math>R = \frac12\sqrt{\frac{2-\xi}{1-\xi}} \approx 0.8160806747999234</math>
Its midradius is
:<math>r=\frac{1}{2}\sqrt{\frac{1}{1-\xi}} \approx 0.6449710596467862</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, in order, the circumradii of the [[great retrosnub icosidodecahedron]] (U<sub>74</sub>), great snub icosidodecahedron (U<sub>57</sub>), [[great inverted snub icosidodecahedron]] (U<sub>69</sub>) and [[snub dodecahedron]] (U<sub>29</sub>).
{{-}}

== Related polyhedra==

=== Great pentagonal hexecontahedron ===
{{Uniform polyhedra db|Uniform dual polyhedron stat table|Gosid}}
[[File:Great pentagonal hexecontahedron.stl|thumb|3D model of a great pentagonal hexecontahedron]]
The '''great pentagonal hexecontahedron''' (or '''great petaloid ditriacontahedron''') is a nonconvex [[Isohedral figure|isohedral]] [[polyhedron]] and [[Dual polyhedron|dual]] to the uniform ''great snub icosidodecahedron''. It has 60 intersecting irregular pentagonal faces, 120 edges, and 92 vertices.

==Proportions==

Denote the [[golden ratio]] by <math>\phi</math>. Let <math>\xi\approx -0.199\,510\,322\,83</math> be the negative 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 101.508\,325\,512\,64^{\circ}</math> and one angle of <math>\arccos(-\phi^{-1}+\phi^{-2}\xi)\approx 133.966\,697\,949\,42^{\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 1.315\,765\,089\,00</math>. 
The [[dihedral angle]] equals <math>\arccos(\xi/(\xi+1))\approx 104.432\,268\,611\,86^{\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 inverted pentagonal hexecontahedron]] and the [[great pentagrammic hexecontahedron]].

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

==References==
*{{Citation | last1=Wenninger | first1=Magnus | author1-link=Magnus Wenninger | title=Dual Models | publisher=[[Cambridge University Press]] | isbn=978-0-521-54325-5 |mr=730208 | year=1983}}
{{Reflist}}

== External links ==
* {{mathworld | urlname = GreatPentagonalHexecontahedron| title =Great pentagonal hexecontahedron}}
* {{mathworld | urlname = GreatSnubIcosidodecahedron| title = Great snub icosidodecahedron}}

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[[Category:Uniform polyhedra]]