This Demonstration shows one of the most frequently occurring objects in singularity theory: the swallowtail singularity. It is realized here as the subspace of consisting of all points such that has multiple real roots. In addition, a section of the singularity surface by a plane is shown as a blue line. Such a section is a plane curve. As the level varies, the curve undergoes a metamorphosis (or a perestroika), which is exactly the same as that of a wave front on a plane.

The swallowtail surface has many equivalent definitions. Among them: the swallowtail is the codimension-three stratum of the caustic (and the Maxwell set) of the space of all smooth mappings from the real line to itself. It is also a singularity that appears during the propagation of a generic smooth wave front in three-dimensional space. The swallowtail singularity remains stable under small perturbations. Its sections by generic horizontal planes are plane curves describing wave fronts on a plane, and so on.

The swallowtail surface was the subject of the last painting of Salvadore Dali, entitled The Swallow's Tail, inspired by René Thom's lectures on catastrophe theory.

Reference

[1] V. I. Arnold, The Theory of Singularities and Its Applications, Cambridge, UK: Cambridge University Press, 1993.