This book provides a thematic introduction to a variety of mathematical topics from the perspective of a single problem. It centres specifically upon Poncelet’s theorem and invokes the use of an exciting range of ideas that are central to algebraic geometry.
Physically, the book is compact and beautifully produced, and its 235 pages and 15 chapters reveal its enormous mathematical scope. In fact, it may be said that Leopold Flatto has created a mathematical gem which (despite my use of mixed of metaphors) brings to mind King Lear’s dream of being bounded in a nutshell and yet finding himself king of infinite space.
Together with appropriate historical observations, the introductory chapter begins with the most concrete version of Poncelet’s theorem concerning closed polygons inscribed in one conic and circumscribed about another. Subsequent chapters lay the foundations for more general statements of the theorem, so that in the complex plane the reference is to smooth conics in general position.
A proof of a more general version of the real theorem is based upon notions from dynamical systems that are introduced in chapter 12. This proof is ascribed to Bertrand and is stated as follows in terms of T-orbits on the outer ellipse C.
If the T-orbit of a point x of C closes after n steps, that is, Tn(x) = x, then the T-orbits of all points of C close after n steps.
However, the central ideas in this book are those of Poncelet correspondence, M, and the Poncelet map, η, whereby the theorem is succinctly stated as:
If ηn has a fixed point, then η is the identity map on M
This concept of Poncelet correspondence is developed from Cayley’s version of the Poncelet theorem, where η is the composition of two involutions of M. In chapter 9, M is identified with a smooth algebraic curve in CP2, thereby classifying it as a Riemann surface. It is then shown to be an elliptic curve endowed with a group structure and, when η is shown to be a translation of this structure, a proof of the above version of Poncelet’s follows readily.
Apart from this geometric application, the key ideas of Poncelet map and Poncelet correspondence are used to unite ideas from aspects of queuing theory and billiards in an ellipse, thereby achieving the book’s overall aim.
As for the intended readership, it is suggested that the material should be accessible to anyone who has taken the standard courses in undergraduate mathematics. I feel this claim is rather optimistic and needs to some qualification. For a start, the introductions to projective geometry, Riemann surfaces, elliptic functions and elliptic curves are covered in the space of 120 pages. Consequently, the treatment, although very interesting and mathematically cogent, may be too condensed for students with little or no prior knowledge of such topics.
Taking account of this, it could serve as an excellent basis for an introductory postgraduate course in algebraic geometry. But this unique and highly interesting book will undoubtedly be of interest to a much wider readership than that.
Prior to reading this book, Peter Ruane had unwittingly regarded Poncelet’s theorem and Steiner’s porism as being mathematically analogous. Leopold Flatto has corrected that misapprehension.