## Abstract

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, keels of boats 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, [17], pp. 47–48).

## References

- 1.AA.VV: Il Colosseo. Studi e ricerche (Disegnare idee immagini X(18-19)). Gangemi, Roma (1999)Google Scholar
- 2.Dotto, E.: Note sulle costruzioni degli ovali a quattro centri. Vecchie e nuove costruzioni dell’ovale. Disegnare Idee Immagini.
**XII**(23), 7–14 (2001)Google Scholar - 3.Dotto, E.: Il Disegno Degli Ovali Armonici. Le Nove Muse, Catania (2002)Google Scholar
- 4.Duvernoy, S., Rosin, P.L.: The compass, the ruler and the computer. In: Duvernoy, S., Pedemonte, O. (eds.) Nexus VI—Architecture and Mathematics, pp. 21–34. Kim Williams, Torino (2006)Google Scholar
- 5.Gómez-Collado, M.d.C., Calvo Roselló, V., Capilla Tamborero, E.: Mathematical modeling of oval arches. A study of the George V and Neuilly Bridges. J. Cult. Herit.
**32**, 144–155 (2018)CrossRefGoogle Scholar - 6.Huerta, S.: Oval domes, geometry and mechanics. Nexus Netw. J.
**9**(2), 211–248 (2007)CrossRefGoogle Scholar - 7.Lluis i Ginovart, J., Toldrà Domingo, J.M., Fortuny Anguera, G., Costa Jover, A., de Sola Morales Serra, P.: The ellipse and the oval in the design of Spanish military defense in the eighteenth century. Nexus Netw. J.
**16**(3), 587–612 (2014)CrossRefGoogle Scholar - 8.López Mozo, A.: Oval for any given proportion in architecture: a layout possibly known in the sixteenth century. Nexus Netw. J.
**13**(3), 569–597 (2011)CrossRefGoogle Scholar - 9.Mazzotti, A.A.: What Borromini might have known about ovals. Ruler and compass constructions. Nexus Netw. J.
**16**(2), 389–415 (2014)CrossRefGoogle Scholar - 10.Paris, L., Ricci M., Roca De Amicis A.: Con più difficoltà. La scala ovale di Ottaviano Mascarino nel Palazzo del Quirinale. Campisano Editore, Roma (2016)Google Scholar
- 11.Ragazzo, F.: Geometria delle figure ovoidali. Disegnare idee immagini.
**VI**(11), 17–24 (1995)Google Scholar - 12.Ragazzo, F.: Curve Policentriche. Sistemi di raccordo tra archi e rette. Prospettive, Roma (2011)Google Scholar
- 13.Rosin, P.L.: A survey and comparison of traditional piecewise circular approximations to the ellipse. Comput. Aided Geomet. Des.
**16**(4), 269–286 (1999)MathSciNetCrossRefGoogle Scholar - 14.Rosin, P.L.: A family of constructions of approximate ellipses. Int. J. Shape Model.
**8**(2), 193–199 (1999)CrossRefGoogle Scholar - 15.Rosin, P.L.: On Serlio’s constructions of ovals. Math. Intel.
**23**(1), 58–69 (2001)MathSciNetCrossRefGoogle Scholar - 16.Rosin, P.L., Pitteway, M.L.V.: The ellipse and the five-centred arch. Math. Gaz.
**85**(502), 13–24 (2001)CrossRefGoogle Scholar - 17.Schleske, M.: Der Klang: Vom unerhörten Sinn des Lebens. Kösel, München (2010)Google Scholar
- 18.Trevisan, C.: Sullo schema geometrico costruttivo degli anfiteatri romani: gli esempi del Colosseo e dell’arena di Verona. Disegnare Idee Immagini.
**X**(18–19), 117–132 (2000)Google Scholar - 19.Zerlenga, O.: La “forma ovata” in architettura. Rappresentazione geometrica. Cuen, Napoli (1997)Google Scholar