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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.

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.

• 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. 17:4 (1983), 70–72. English translation in Functional Analysis and its Applications 17:4 (1983), 301–303.

• O. P. Shcherbak, Wavefronts and reflection groups, Uspekhi Mat. Nauk 43:3 (1988), 125–160. English translation in Russian Mathematical Surveys 43:3 (1988), 1497–194.

(The Russian versions are open-access; the English versions are not.)

• 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. 17:4 (1983), 70–72. English translation in Functional Analysis and its Applications 17:4 (1983), 301–303; free version available here.

• O. P. Shcherbak, Wavefronts and reflection groups, Uspekhi Mat. Nauk 43:3 (1988), 125–160. English translation in Russian Mathematical Surveys 43:3 (1988), 1497–194.

The Russian versions are open-access; the English versions are not, but the first paper is currently available for free online.

• Involutes of a cubical parabola, Visual Insight, 1 May 2016.

• Discriminant of the icosahedral group, 15 May 2016.

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 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:

• Involutes of a cubical parabola, Visual Insight, 1 May 2016.

• Discriminant of the icosahedral group, Visual Insight, 15 May 2016.

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 post.

But here's the point: the involutes, shown in blue, look like slices of the discriminant of the icosahedral group!

Indeed, Givental conjectured this is true, and Arnol'd says it's been proved. In The Theory of Singularities and its Applications, he wrote: