1. Introduction

When writing about ovals the first thing to do is to make sure that the reader knows what you are talking about. The word oval has, both in common and in technical language, an ambiguous meaning. It may be any shape resembling a circle stretched from two opposite sides, sometimes even more to one side than to the other. When it comes to mathematics you have to be precise if you don’t want to talk about ellipses, or about non-convex shapes, or about forms with a single symmetry axis. Polycentric ovals are convex, with two symmetry axes, and are made of arcs of circle connected in a way that allows for a common tangent at every connection point. This form doesn’t have an elegant equation as do the ellipse, Cassini’s Oval, or Cartesian Ovals. But it has been used probably more than any other similar shape to build arches, bridges, amphitheatres, churches and windows whenever the circle was considered not convenient or simply uninteresting. The ellipse is nature, it is how the planets move, while the oval is human, it is imperfect. It has often been an artist’s attempt to approximate the ellipse, to come close to perfection. But the oval allows for freedom, because choices of properties and shapes to inscribe or circumscribe can be made by the creator. The fusion between the predictiveness of the circle and the arbitrariness of how and when this changes into another circle is described in the biography of the violin-maker Martin Schleske: “Ovals describe neither a mathematical function (as the ellipse does) nor an arbitrary shape. [...] Two elements mesh here in a fantastic dialectic: familiarity and surprise. They form a harmonic contrast. […] In this shape the one cannot exist without the other.” (our translation from the German, [15], pp. 47–48).