By DAVID IZRAELEVITZ
Los Alamos
I inherited my interest in mathematics from my Dad. He had studied engineering for a year in Uruguay before having to drop out and get a job, but his love for the subject remained. Even as a factory worker, he would work through geometry puzzles during his work breaks.
Uruguay’s technical university system was modeled on the French system, heavy on abstract mathematics and cruelly reliant on culling students through difficult written and subsequent oral exams. One of the last subjects he struggled through was synthetic projective geometry, a subject last relevant when engineers used drafting tables and T-squares, but pretty much obsolete these days of computer-aided drafting programs. I suspect many LANL mathematicians reading this column have never heard of this obscure corner of geometry.
Thirty years after that challenging projective geometry class, I was searching for a science project topic that didn’t involve being neat or patient, neither of which were my strong suits. The math category fit the bill, and Dad suggested I learn something from his old nemesis. I was intrigued by the name and his explanation of mysterious “points at infinity,” so I visited our local library and, due to some very weird public library purchasing policy, I found a college textbook on projective geometry.
Chapter One discussed the so-called axioms of incidence upon which this type of geometry is constructed, such as the postulates that two points determine a single line and that any two lines intersect at a single point (even parallel lines, see reference above to “points at infinity”). That these relationships are preserved under shadow-casting and perspective drawing took me some time staring into space, but I eventually got there.
The second chapter, meaty and challenging, derived a foundational result upon which much of the rest of the book rested: Desargues’ Theorem. Getting through the derivation took me quite a while, and, figuring that a ninth-grader getting through the first two chapters of a college-level textbook was effort enough for me, I decided to make Monsieur Desargues’ life’s work my culminating topic. After putting together a poster featuring color-coded lines and triangles, I was ready for the first round of the science fair competition at our local community college.
Once I started my presentation, it became apparent that the judges were not that interested in Monsieur Desargues’ legacy. My droning on about an obscure theorem in an esoteric subject lulled them into a peaceful slumber, and I easily advanced to the regional competition at Stony Brook University and my first visit to a university campus.
I confidently started my well-rehearsed presentation, but I grew increasingly nervous as I realized the math professors in front of me were not dozing, but strangely familiar and interested in the topic. Rather than drooping as I neared the climactic end of my proof, they alternated between nodding and squinting in concentration, as if recalling a long-forgotten class.
As I culminated my salute to Mr. Desargues’ crowning achievement, those judges were still awake, still nodding, and still squinting. After an ominous pause, one of the professors finally asked me a question. “David, I want to ask you a fundamental question about projective geometry. Do you know what the word incidence means?”
Where did this vocabulary quiz come from? I had no idea what the word ‘incidence’ meant. What kind of stupid math question is that? I thought in indignation, followed by embarrassment. Co-incidence is when two things overlap, like a point on a line. Projective geometry is fundamentally the analysis of when such co-incidences occur, and Desargues’ theorem was fundamental to developing a whole mathematical structure that applied not only to lines and points but also to much more complicated relationships. I had no idea of the broader mathematical scheme. The word Incidence was one of the most common ones in the textbook, and even part of the fundamental axioms and definitions. I could recite theorems, but I had no idea why they were essential and where they were headed. I meekly answered “no” and rolled up my color-coded poster.
It might surprise some readers that mathematics can provide life lessons deeper than the imperative that you should spend less than what you earn. My embarrassing experience at Stony Brook taught me that rote learning is a trap one can fall into. We can learn the “how” but not understand the “what”, and more importantly, the “why” of what we are doing.
My three sons hated it when I helped them with their math homework, because I wanted them to understand the underlying concepts, not just follow the steps to get to the answer. Don’t just memorize sine and cosine formulas… Why are you learning trigonometry? They, in turn, wanted to finish the math problem set so they could get on with the English reading assignment.
Well, dear children, maybe the word “incidence” showed up on your SATs. I could have helped you with that one.