The Universe of Conics

Conics may well be on the verge of becoming an endangered species in a variety of mathematical curricula around the world. At a time firmly implanted in the heartland of high-school and undergraduate mathematics (and physics), conics could easily find themselves relegated to the fringes of courses nowadays. We may be more certain regarding the decline and near-obliteration of projective geometry within undergraduate curricula.

If you are an undergraduate in the U.S. today, you may have seen conics either in high school, or possibly in a (remedial) entry-level college course in your freshman year. Some readers may remember them from multivariable calculus if their instructor had time to demonstrate that the orbits of planets are ellipses. And perhaps a handful of linear algebra instructors return their students to (either dreams or nightmares of) conics via quadratic forms, while struggling to fit whatever else makes the last week or two of a first, and sadly often only, course. A fortunate few may study an upper-level course or an independent study in undergraduate classical algebraic geometry and may have there (re)discovered conics as examples of affine varieties. That may well be the extent of it.

The text being reviewed — The Universe of Conics: From the ancient Greeks to 21st century developments — was written and beautifully illustrated by conical connoisseurs Georg Glaeser, Hellmuth Stachel and Boris Odehnal. The authors are Viennese geometers who have published extensively in computational geometry, computer graphics and allied areas. Their book is a visual delight. The hundreds of meticulously crafted and beautifully rendered figures (that adorn almost three out of every four pages) form the most attractive component of the book. In the authors’ words: “The book has been written for people who love geometry and it is mainly based on figures and synthetic conclusions rather than on pure analytic calculations. In many proofs, illustrations help to explain ideas and to support the argumentation, and in a few cases, the picture can display a theorem at a glance together with its proof.”

After a brief historical introduction, Chapter 2 begins with the classical definitions of conics and describes a variety of mechanical linkages that generate conics. Chapter 3 discusses differential geometric properties, and Chapter 4 studies conics as generated by the planar intersection of a surface in Euclidean 3-space. There is a shift in sophistication in Chapters 5 through 7 that study conics in the framework of projective geometry, followed by an affine perspective in Chapter 8. There appears to be an implicit demand for “more mathematical maturity” for this set of chapters. The last two chapters are a compendium of more esoteric results. Chapter 9 presents a potpourri of special problems such as pedal points and Poncelet porisms. The book ends with a final Chapter 10 on conics in non-Euclidean geometries.

The writing is uneven with regard to detail, and there could have been some more hand-holding through the more sophisticated portions of the text. Though my preferences are biased in favor of the strongly geometric and visual approach taken by the authors in explaining the bulk of the material, plenty of (our?) students who lack basic visualization skills and an aptitude for geometric reasoning will struggle with the book.

Sadly, the reader is left completely in the dark regarding the production of the beautiful and instructive figures that adorn almost every page. A separate resource on how to create such figures may be extremely well-received, and poised to revitalize classes on geometry, especially those aimed at training teachers at the elementary, middle and high school levels. Such a text/course may radically enhance the creation and use of manipulative materials and technology in the setting of geometry education. We hope that the authors will provide some documentation and further aids (perhaps on their websites) to empower readers to create and manipulate such figures on their own.

Higher eduction aside, I hope that the text will be taken up by the community of K–12 researchers and educators here in the US and abroad, if only to aid the return of younger minds to this lush area of classical mathematics that plays “a fundamental role in numerous fields of mathematics and physics, with applications to mechanical engineering, architecture, astronomy, design and computer graphics”.

Tushar Das is an Assistant Professor of Mathematics at the University of Wisconsin–La Crosse.