Details

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.