All Sides to an Oval

Before picking up this book, I can’t say that I had given much thought to ovals beyond thinking of ellipses.  Mazzotti certainly has.  

 

The book begins with a careful definition of exactly which objects are under scrutiny here. These are polycentric ovals: arcs of circles that are joined together smoothly (that is, with a common tangent line at the intersection point) to form a closed, convex curve, and which have two orthogonal axes of symmetry. And under scrutiny they are.  Five chapters detail all sorts of constructions and the properties of these geometric figures. We learn how to construct ovals with a ruler and compass with desired parameters. We learn how to inscribe and circumscribe ovals with rectangles.  We learn how to deduce equations for some of the necessary parameters in terms of other specified parameters.

 

The last three chapters explore ovals for design.  Chapter 6 introduces some “remarkable ovals”, remarkable for their architectural use and/or beauty, and chapters 7 and 8 consider specific architecture. This is where Mazzotti has been heading, as his subtitle “Properties, Parameters and Borromini’s Mysterious Construction” indicated.  Chapter 7 is devoted to the ovals in the dome of the San Carlo alle Quattro Fontane church in Rome, designed by the architect Borromini.  In fact, this is the second edition of the 2017 book precisely because new information about this dome came to light after the original publication.  Some of the questions Mazzotti and Margherita Caputo (who co-wrote this chapter) aim to address in this chapter are: “How did the project of the dome surmounting the church develop? Which parameters are involved in its shape and in the shape of the coffers? How did he work on the building site?” The final chapter concerns ovals in the design of the Colosseum, and the arches of the Neuilly-sur-Seine bridge. The author attempts to answer how the architects of the Colosseum determined which four-center oval and eight-center ovals to use in the construction. Likewise, Mazzotti addresses how the arches of the bridge (with 11 centers) were designed in the 18th century.

 

The book is well documented.  Each chapter contains numerous references and citations. There are numerous pictures throughout the text, as is to be expected, and the formulae need only high school algebra and trigonometry. This book is perhaps of primary interest to architects and others in design fields.  

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Michele Intermont is an Associate Professor of Mathematics at Kalamazoo College.  Her main area of interest is in algebraic topology.