Conics

I loved this book! It transported me back to my undergraduate and graduate student days and reminded me of why I became a mathematician. Using the ubiquitous topic of the conic sections as its unifying theme, it’s a breathtaking virtual tour through a variety of topics from projective geometry, complex variables, plane geometry, polynomial equations, differential equations, celestial mechanics and electromagnetic theory. It’s an astonishing synthesis of an enormous body of mathematics, most of it at the upper-division undergraduate or early graduate school level.

The topic of conic sections has a distinguished pedigree. Apollonius and other ancient Greeks defined them by cutting a double-napped cone with a plane. Depending on the angle of the cut, one gets an ellipse, a parabola or a hyperbola. From its inception through the advent of analytic geometry, the sections were studied synthetically. The utility of coordinate geometry was demonstrated in part by the ease with which the tricky geometric propositions of Apollonius could be derived by manipulating second-order equations. However, whether you’re treating the conics geometrically or algebraically, it seems like the three types of conics are essentially different from one another, and must be studied separately. There are even attributes, such as the eccentricity of an ellipse, which are classically presented as pertaining only to some conics and not to others. This is a troubling state of affairs for a family of curves with such a simple, uniform geometric definition. Keith Kendig’s goal in writing Conics is to provide a consistent, unified and beautiful framework for studying the conic sections.

The text is written in the form of dialogs among three characters: Philosopher is a mathematically sophisticated outsider who is driven by a desire to understand the conic sections in a unified way, Teacher is a mathematician who is well-versed in all branches of mathematics and who may have been quite happy with the traditional view of conic sections, but is also sympathetic to the Philosopher’s quest, and Student, who is the enthusiastic minor character that asks good questions, occasionally provides nice explanations, and with whom student readers are probably meant to identify.

The use of dialog form begs a comparison to Hofstadter’s Gödel, Escher, Bach . Hofstadter’s book has much broader appeal, of course, using art and music to court a wide audience of non-mathematicians, many of whom are baffled by the mathematics in the chapters between the dialogs. By comparison, this book is only about mathematics and will never reach a general audience. Indeed, it is more technically demanding than most popular mathematics books on the market today, but also much more rewarding to the reader who wants both to learn new material and to discover fresh perspective on material that s/he already knows. One of the stylistic differences between Kendig’s book and Hofstadter’s is that Kendig never leaves the dialog mode. Many people have told me that in Gödel, Escher, Bach, they only read the dialogs, skipping the technical discussion in between. In Kendig’s book, the technical discussion is front and center in the dialogs, coming mostly out of Teacher’s mouth. Furthermore, if you skipped the dialogs in Conics there would be nothing left except for the boxes, frequently containing historical background, and the extensive captions on many of the illustrations.

Illustrations are important in a subject as visual as conic sections. But traditional illustrations are static, while a big part of what leads to a unified view of the conic sections is their behavior at infinity in the projective plane. To aid the imagination, Kendig includes a CD-ROM with 36 applets that provide colorful and interactive illumination of the some important examples and illustrations. The applets run in any Java-enabled browser, so no special software is needed, and you can load the CD and run the applets on almost any computer. A number of the applets illustrate transformations of the projective plane using a rotating sphere, whose motion can be manipulated using the computer’s mouse. Other applets assist the reader in visualizing subspaces of two-dimensional complex space by means of various representations of R³. And for those who are even more technically savvy (or demanding), Maple code is supplied in some places for performing some of the trickier calculations.

Who is this book for? Certainly, mathematics professors and other professionals in the mathematical sciences will have the necessary background and will quite possibly have as much fun with it as I did. There is much here for physicists as well. Most graduate students of mathematics would likely enjoy it too. However, since it’s an expository book, the main strength of which is in providing new insight into familiar material, it’s best thought of as recreational reading and not something to further their thesis research. The promotional material says that many undergraduates can self-study the book, as long as they have some exposure to complex variables and linear algebra. I thought it would be valuable to test that claim as part of my reviewing process, so I supervised an independent study of this book by a student at my university.

My experience was mixed, but I’m optimistic that some students will succeed in learning this material largely on their own, with perhaps the sort of weekly meeting that I had with my student. It will work for the sort of student who would be drawn in by the friendly, conversational style of the book and encouraged by a series of little mathematical epiphanies. To succeed, though, the student would have to have a strong background in the standard mathematics curriculum, some exposure to non-Euclidean geometry and/or topology, superior algebraic skills, good study habits and genuine intellectual curiosity. Talented majors at good liberal arts schools, even those who are not bound for graduate school in mathematics, will probably have both the background and the disposition to succeed in a self-study. I feel that the average math major at a comprehensive university would be much more likely to struggle with it.

My feeling is that the best way to use this book in an undergraduate setting is in the sort of small-scale, seminar-style capstone course that more and more colleges and universities seem to be instituting. This setting would give the students a real opportunity to engage in self-study, but with the support of a professor and of other students. With 400 pages and 19 chapters and appendices, there’s plenty of material for both a common core of material and a selection of special topics which students could present in the seminar.

Rob Bradley is professor of mathematics at Adelphi University. His main research interest is the history of mathematics. He is currently the president of the Euler Society and past president of the Canadian Society for the History and Philosophy of Mathematics.