Tuesday, July 31, 2007

Philosophers love lines

This is the last of my posts on math books for a while (following this one and this one). The last one was on John Stillwell's number theory book. This one is on his Four Pillars of Geometry. This is, as the title suggests, an introduction to geometry, from Euclid through some non-Euclidean geometries. There is, in my opinion, a lot to recommend this book. It begins with Euclid, moving to coordinate planes and vectors, then to perspective, projective geometry, and transformations, and closes with hyperbolic geometry (the thing most commonly associated with the phrase "non-Euclidean geometry" if I'm not mistaken.). The sections on Euclid are quite nice, with lots of exercises and constructions to get you used to doing things his way. There is also a decent explanation about how to do arithmetic with constructions of line segments instead of numbers, which is generalized to work in projective geometry. Unfortunately, there is no proof of the infinitude of primes done entirely in line segments, which, I am told, was how the original was done. There is a lot of discussion of Pappas's theorem and Desargues's theorem in both old Euclidean geometry (plane geometry, I think it is called) and projective geometry and how the two theorems relate. The chapter on transformations explains some of Klein's Erlanger Program (geometry is the study of invariants of groups of transformations) and gives a good sense of what transformations are for various geometries. There are lots of (glorious) diagrams (yay for diagrams) in addition to there being a good selection of exercises. Also, as David Corfield of n-category Cafe likes, there is a decently cohesive narrative element connecting each chapter, giving you the "why"s as well as the "what"s of the proofs and chapters, including a couple of excursions into art to demonstrate the evolution of perspectival ideas. One of the narrative sections prompted an earlier post on the history of the parallel postulate.

Possibly the lowest point of the book (for me) is the discussion of quaternions. They weren't that motivated compared to vectors, which they followed. I don't have any other big complaints though.

In summary, I'd recommend this as an introduction to geometry. I'm not sure what else one would want from an introductory book apart from more stuff on hyperbolic geometry. The book does a very good job of going through conceptual relations in geometry.

Geometry is a bit of a hot topic around these parts. Both Pitt and CMU had dissertations recently on geometry and there was a class offered at Pitt last fall on philosophical issues in geometry. I don't think I'll be joining the bandwagon, but there is certainly philosophical mileage to be found in geometry.

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