Richard Trudeau begins The Non-Euclidean Revolution with an admission: that he first studied non-Euclidean geometry only when he was asked to develop a course on the subject. He then went on to write a fascinating look at this area of mathematics.
I came to non-Euclidean geometry the same way — as did, I suspect, many of us who teach geometry courses for future secondary teachers. I didn’t, however, think to roll that experience — which has become an unexpected area of expertise, as I’ve taught that course fourteen times at two different colleges — into writing a book.
Which is no loss, because I wouldn’t have done any better than Trudeau has. In The Non-Euclidean Revolution, we have a mathematically rigorous explanation of this sea change in mathematics which is at the same time suitable for any educated reader. Yes, the mathematics is present and at the center of the exposition, but great mathematical proficiency is no prerequisite for benefitting from this book.
I teach an honors class called “Seven Big Ideas That Shaped Science” that examines, in as much detail as is possible in one semester, a major idea from each of seven area of science. In and among such ideas as evolution, DNA structure, and plate tectonics, I include a three-week unit on non-Euclidean geometry that uses Trudeau’s book. It’s a little more suited for a general audience that the textbook (Greenberg's ) I use for my upper-division geometry course. My honors students come from throughout my college; few if any are mathematics majors or even all that interested in mathematics. Nonetheless, I have found that this book achieves my goal of bringing some serious mathematics into our honors program and Trudeau’s goal of bringing his readers through the 19th century revolution brought about by an alternative to Euclid’s geometry.
Indeed, one major advantage of representing mathematics with non-Euclidean geometry is that the students have that opportunity to live through a revolution in their own understanding of mathematics. While they are all on the “modern” side of such scientific advances as the Big Bang theory, the periodic Law, or the structure of the atom, none of them has seen non-Euclidean geometry before — in Trudeau’s words, they are “committed Euclideans”. As a result, they get a chance to experience a change in mathematics of a magnitude comparable to how chemistry has been affected by Mendeleev’s periodic table. Trudeau’s book shines as a guide to that revolution.
Mark Bollman (firstname.lastname@example.org) is an associate professor of mathematics at Albion College in Michigan. His mathematical interests include number theory, probability, and geometry. His claim to be the only Project NExT fellow (Forest dot, 2002) who has taught both English composition and organic chemistry to college students has not, to his knowledge, been successfully contradicted. If it ever is, he is sure that his experience teaching introductory geology will break the deadlock.