It is not difficult to define mathematically what an ellipse is. Its Cartesian equation is well known. It is however less clear what an oval is. Most people will come up with the condition that it looks like an ellipse. It is a smooth convex closed curve of the plane with two orthogonal symmetry axes. But how to be more precise? Since antiquity, ovals have been used in architecture. So what was the construction used by the architects?

There are the Cartesian and the Cassini ovals, that have a simple Cartesian equation, but they do not always have the symmetry (in th first case) or are not convex (second case). Historically however the ones that have been used most in arts, especially in architecture, are the polycentric ovals that consist of circular arcs that are stitched together in a smooth way. This is the only kind of ovals that is considered in this book.

The easiest and most popular one that has been studied thoroughly is the so-called four center oval. It consists of four circular arcs that fit together smoothly. Two arcs with smaller radius are at the tops of the long symmetry axis and the ones at the end points of the shorter symmetry axis have lesser curvature because they belong to a circle with a longer radius. The crux is of course to choose these four centers of the circles in such a way that the arcs fit smoothly together at the four connection points. How does one have to select these centers and how large should the sector angle be that supports the arcs so that one does indeed get this smooth transition? Because of the symmetry, two centers are located symmetrically on the long axis and two on the shorter one. So it suffices to consider only a quarter of the oval and find two of the centers to define the arcs and the connection point where the arcs meet in that quadrant. Once the length of the axes are given, an ellipse is completely defined. For an oval, one needs at least one more parameter, like the distance from one of the centers of the circles to the center of the oval or the distance of the connection point to one of the symmetry axes.

Once these arguments have been formulated, it needs some analysis of the geometry of the problem. And that is where this book gets started. The author has, besides other interests, a knack for polycentric curves like eggs or ovals. This book is restricted to ovals, and the first chapter analyses the properties that will enable us to relate the different parameters. Once this is cleared out, the construction with ruler and compass of an oval (actually a quarter of an oval, because the rest follows by symmetry) is given step by step depending on which parameters are prescribed. So one might choose three of the six possible parameters in many different combinations and that gives rise to twenty different ways to define and construct an oval satisfying the data. Some are more complex and some have more restrictive conditions than others. The solution may not always be unique. Everything is clearly explained and the many illustrations produced with geogebra are crystal clear. It might however be interesting to have a look at the associated website www.mazzottiangelo.eu/en/pcc.asp where you find links to YouTube videos showing animated geogebra constructions. The link goes both ways: you may consider this book as a manual for the online site, or the online site as an illustration for the book.

Besides the parameters described above, one might also choose for one of the radii of the arcs or the ratio of the axes or the angle formed by a symmetry axis and the line joining the circular centers of the arcs. With all ten parameters, there are a total of 116 possibilities to construct the ovals, many of which, but not all, reduce to the twenty constructions mentioned before. Some of the constructions are historical and often pretty old, but others are surprisingly recent. For particular choices of the parameters, the construction may simplify considerably or the oval may have especially pleasing esthetic properties, which are discussed in a separate chapter.

Towards more practical applications of stadium design, one may consider ovals circumscribing or inscribed in a rectangle. If the symmetry axes are the middle-lines of a rectangle and the diagonals of a rhombus, then all previous constructions circumscribe the rhombus and are inscribed in the rectangle. For a stadium one should find an oval circumscribing the inner rectangular field (for example a soccer field) and surround it by ovals like running tracks, all inside an outer rectangle defining the limitations of the stadium. Modern constructs however have straight parts for the running tracks along the long sides.

While the constructions are mostly obvious, it takes more algebra and more formulas to express some parameters as a function of others. This is a short chapter, but essential to find ovals that are optimal in some sense. For example finding the "roundest" oval with given axes. They are also needed in geogebra animations when slider rules are provided allowing to see the effect of changing a parameter.

The last two chapters discuss ovals in two famous architectures in Rome: the dome of the church San Carlo alle Quattro Fontane by the architect Borromini and the ground plan of the Colosseum. A careful study is made of the ovals of the base of the dome in the church, the rings of coffers, and of the lantern. It turns out that there are small defects making them deviate from perfect mathematical ovals. This has long been a mystery. It is suggested that the starting point was a mathematically perfect oval, but that practical restrictions entailed heuristic corrections. The solution that Mazzotti proposes here corresponds remarkably well with Borromini's original drawings.
For the Colosseum, we have to leave the simple ovals with four centers and go to quarter ovals consisting of more than two arcs. Because of symmetry there have to be always $4n$ centers. Again constructions of such ovals are considered. In the case of the Colosseum, $n=2$, i.e., ovals consisting of eight circular arcs seem to match the ground plan perfectly well.

This is a very nice geometric application that requires only simple algebra and that can be easily experimented with. You do not need to be a mathematician to enjoy it. It that sense, it might be interesting to have the geogebra source available somewhere, which is unfortunately not the case. Also historians might be interested in the last two chapters about historical buildings. For the mathematician, it is invaluable because it brings together so much information that was either not known or never writen down or if it was, then at least it was scattered in diverse publications. The graphics are very readable since they use colors (except for the pictures in the last two chapters, only red, green, and blue suffice for the mathematical constructions). As a LaTeX purist, I cannot resist mentioning my irritatin when seeing variables mentioned in roman font when in a sentence, while they are in a different font when used in a formula. Also, I do not understand why the ratio of the half symmetry axes is denoted at least twice as $\frac{p=\overline{OB}}{\overline{OA}}$ (p.20 and 148) and when at the end of a line $p=\frac{\sqrt{2}}{2}$ is split into $p$, which is left dangling at the end, and $=\frac{\sqrt{2}}{2}$ at the beginning of the next line (p.102). These are however minor flaws in an otherwise nice text, and as I am sure, these will disappear in a next edition. Do not let this prevent you from reading this most enjoyable book and you should certainly try out some of the constructions for yourself, either with ruler and compass or with geogebra.