Did you know you could turn a hyperbola into a loop?

Porter Square Books’ penchant for choosing surprisingly excellent books for it’s 2-shelf math collection has remained undimmed over the five years that I’ve known the place. A few months ago, I discovered a book called Elliptic Tales, written by a couple of professors at Boston College. Avner Ash and Robert Gross have taken upon themselves the difficult task of bringing the elegant theory of elliptic curves to the genre of popular mathematical literature.

The writers ask of their readers nothing more than a high school mathematics background, curiosity and the patience to process the beautiful but not immediately obvious results in number theory. It is remarkable to me that one can, in the authors’ opinion, start the book not knowing anything about complex numbers and finish it with a decent grasp of the Birch and Swinnerton-Dyer conjecture, an unsolved problem at the frontier of number theory. Obviously, one of the reasons for the authors’ success at communicating such a difficult topic is their choice of explanatory material. Often, there is more than one approach to a given result, and the writers have judiciously chosen the approach that allows them to reach readers with limited mathematical training, even though that may not necessarily be the most popular approach in the field. While explaining the projective plane, for instance, they do mention the geometric approach and provide a reference to Courant and Robbins’ What is Mathematics? *. However, they treat the geometric approach no further, instead choosing to explain the concept of the projective plane algebraically, using the first principles of rings, fields and abelian groups.

Equally important, I believe, is the fact that they are not shy about using mathematics “as is” when it is warranted. As was the case with most of Sarah and David Flannery’s engaging story In Code, the writers have enough faith that the interested reader will persevere and eventually understand the concepts that are best communicated concisely, systematically and quite precisely using mathematical notation. They are showing us the process by which real mathematicians approach difficult problems. There is another lesson in this, I think. Too often, books with the honorable intention of bringing math to popular audiences take the path of replacing the language of math by everyday metaphors, thinking that this will make for an easier learning curve. The result is often immediate, but deceptively so. Most readers can, in fact, grasp the metaphors readily and understand the patterns at play in the mathematical problem. However this victory is shortlived. The metaphor, by definition, turns out to be an inept analogy, and it can only get a reader so far, before it fails to provide any new insight. Here lies the math writer’s conundrum: if a reader is interested enough at this impasse she will take the trouble to consult a proper text and run with the subject for as long as she has the will and the means. On the other hand, if the reader has grasped the analogy but is still not intrigued enough, the book has failed its didactic mission. Worse still, the reader is now left with a paper-thin understanding of the subject, shored up by pretty English analogies that don’t apply beyond their specific use cases.

The difficult alternative is to instruct a reader, from the very beginning, with a judicious mixture of analogy and mathematical language. The learning curve is now steeper, but progress – at least in this blogger’s case – is surer and longer lasting. This alternative is harder to execute, it demands patience and application from the reader and therefore, it may result in fewer book sales. That is why I find myself full of admiration for Ash and Gross. This isn’t a mathematics textbook by any means, but they make no excuses for their math and that is the way it should be. Furthermore, when they choose to make analogies, they compare the difficult topic at hand to a simpler, more accessible topic that an average high-school student is already familiar with. Therefore, the analogies tend to share a common language, are more precise, and go a longer way in aiding and extending understanding than a random analogy from human emotions**.

I am a little less than half way through the book now, and much of it has been devoted to the following abstract conceit: Let us try to develop new mathematics that allows us to say that a line intersects a curve of degree k in k points. Among the concepts developed to enable us to think the above thought, are complex numbers and intersection multiplicities, which are not that hard to grasp. However, the development of the projective plane was tougher, and though I can safely say that I got the main ideas, I am still not comfortable in my head about the mysterious points at infinity. I’ve worked with coordinate transformations before, yet it completely blew my mind that a simple coordinate transformation like x -> x/z as z tends to zero causes such a profound change in the geometry of curves that I thought I was quite familiar with. It starts off being merely funny: a line and a parabola loop over at infinity under the transformation. But then, things become weird: The two disjoint elements of a hyperbola connect and loop over at infinity too! I’ve had many happy moments traversing this loopy hyperbola in my mind’s eye, and returning to the starting point. At the moment, I am coasting along more or less comfortably, through the material on abelian groups. Elliptic curves have made their brief introductory appearance, but they are still mysterious and I am waiting impatiently for the book to tell me how elliptic curves are related to abelian groups and the whole curves-and-intersections story in the first third of the book.

I recommend the book highly to anyone who likes mathematics (for its own sake).


 What is Mathematics?  is a superb book. I would classify it as a proper math “overview” textbook rather than a representative of the popular science genre.

** The use of stupid analogies is the biggest problem with the books of David Berlinski. If there was ever a clear example of vacuous  mathematical writing, his is it.


1 thought on “Did you know you could turn a hyperbola into a loop?”

  1. I tried in vain to finish Berlinski’s _A Tour of the Calculus_. I find him tediously rapt by himself. I loved _Unknown Quantity_ even if I heard rumor’s of Derbyshire’s maniacal political views.

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