In his remarkable book _The Theory of Singularities and its Applications_, Vladimir Arnol'd discussed a conjecture of A. B. Givental, which asserts that the symmetry group of the icosahedron is secretly lurking in the problem of finding the shortest path from one point to another in a region of the plane. Via [Huyghen's principle][1], this problem is connected to the motion of waves in such a region.
Arnol'd nicely expresses the awe mathematicians feel when they discover connections like this:
> Thus the propagation of waves, on a 2-manifold with boundary, is controlled by an icosahedron hidden at an inflection point at the boundary. This icosahedron is hidden, and it is difficult to find it even if its existence is known.
Unfortunately, even now a full proof of Givental's conjecture seems hard to find! **Can you find one, or give one?**
Let me sketch the idea here. For more details, see:
* [Involutes of a cubical parabola](http://blogs.ams.org/visualinsight/2016/05/01/involutes-of-a-cubical-parabola/), _Visual Insight_, 1 May 2016.
* [Discriminant of the icosahedral group](http://blogs.ams.org/visualinsight/2016/05/15/discriminant-of-the-icosahedral-group/), 15 May 2016.
[![Discriminant of the icosahedral group][2]][2]
This image, created by [Greg Egan](http://www.gregegan.net/), shows the discriminant of the symmetry group of the icosahedron, a 120-element group known as $\mathrm{H}_3$.
This group acts as linear transformations of $\mathbb{R}^3$, and thus also $\mathbb{C}^3$. By a theorem of Chevalley, the space of orbits of this group action is again isomorphic to $\mathbb{C}^3$. Each point in the surface shown here corresponds to a **nongeneric** orbit: an orbit with fewer than the maximal number of points. More precisely, the space of nongeneric orbits forms a complex surface in $\mathbb{C}^3$, called the **discriminant** of $\mathrm{H}_3$, whose intersection with $\mathbb{R}^3$ is shown above.
The following image, created by [Marshall Hampton](http://blogs.ams.org/visualinsight/2016/05/01/involutes-of-a-cubical-parabola/), shows the [involutes](https://en.wikipedia.org/wiki/Involute) of the curve $y = x^3$:
[![enter image description here][3]][3]
Loosely speaking, an involute of a plane curve $C$ is a new plane curve $D$ obtained by attaching one end of a taut string to a point $p$ on $C$ and tracing the path of the string's free end as you wind the string onto $C$ There are different involutes for different choices of $p$ and different lengths of string. I am ignoring some important nuances here, some of which are discussed on my [_Visual Insight_](http://blogs.ams.org/visualinsight/2016/05/01/involutes-of-a-cubical-parabola/) post.
But here's the point: the involutes, shown in blue, look like _slices_ of the discriminant of the icosahedral group.
Indeed, Arnol'd claimed this is true! In _The Theory of Singularities and its Applications_, he wrote:
> The discriminant of the group $\mathrm{H}_3$ is shown in Fig. 18. Its singularities were studied by O. V. Lyashko (1982) with the help of a computer. This surface has two smooth cusped edges, one of order 3/2 and the other of order 5/2. Both are cubically tangent at the origin. Lyashko has also proved that this surface is diffeomorphic to the set of polynomials $x^5 + ax^4 + bx^2 + c$ having a multiple root.
> The comparison of this discriminant with the patterns of the propagation of the perturbations on a manifold with boundary (studied as early as in the textbook of L'Hopital in the form of the theory of evolutes of plane curves), has led A. B. Givental to the conjecture (later proven by O. P. Shcherbak) that this discriminant it locally diffeomorphic to the graph of the multivalued time function in the plane problem on the shortest path, on a manifold with boundary, which is a generic plane curve.
(Figure 18 is a hand-drawn version of the picture at the top of this post.)
This seems like an exciting claim worthy of a nice conceptual proof. Unfortunately, I haven't been able to find a complete proof the literature, not even in these promising-looking papers:
* O. P. Shcherbak, Singularities of a family of evolvents in the neighbourhood of a point of inflection of a curve, and the group $\mathrm{H}_3$ generated by reflections, _[Funktsional. Anal. i Prilozhen.][2]_ **17:4** (1983), 70–72. English translation in _[Functional Analysis and its Applications](http://link.springer.com/article/10.1007%2FBF01076721)_ **17:4** (1983), 301–303.
* O. P. Shcherbak, Wavefronts and reflection groups, _[Uspekhi Mat. Nauk][4]_ <b>43:3</b> (1988), 125–160. English translation in _[Russian Mathematical Surveys][5]_ <b>43:3</b> (1988), 1497–194.
(The Russian versions are open-access; the English versions are not.)
Perhaps an expert could construct a full proof based on the ideas in these papers! For more clues and references, see my _Visual Insight_ posts.
[1]: https://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_principle
[2]: https://i.sstatic.net/mPkey.png
[3]: https://i.sstatic.net/6MN7y.gif
[4]: http://blogs.ams.org/visualinsight/files/2016/03/involutes_of_cubical_parabola.gif
[5]: http://iopscience.iop.org/article/10.1070/RM1988v043n03ABEH001741