The Swallowtail is one of the very few surfaces of which there are several quite different historical models, and at the same time, it is one of the surfaces which appear in many different areas of mathematics.
The shape of the one we show looks close to models around 1890 in Germany and the Netherlands. Most of the classical models, however, consisted of a metal structure with strings.
The Swallowtail is known as the so-called discriminant of polynomials of degree four. More recently, the swallowtail became important in very different areas of mathematics, such as singularity theory and catastrophy theory.