Review of Hidden Harmonies

I’ve started perusing the New Items shelves at the public library when we make our weekly visits, and a few weeks ago this nerdy book caught my eye – Hidden Harmonies: The Lives and Times of the Pythagorean Theorem by Robert & Ellen Kaplan. I remembered watching a silly video in high school geometry class about the Pythagorean theorem (you know, the old a2 + b2 = c2), and the epiphany of learning that this equation meant you could draw squares from the sides of a right triangle and that their respective areas were related. I also remembered being astonished at both the number of vastly different applications of the theorem and the number of vastly different methods that existed for proving the thing. This book looked possibly interesting along those lines, so I checked it out and read it.

The book is loosely divided into three parts: speculation about the discovery of the Pythagorean theorem, a discussion of many of the proofs of the theorem, and further mathematical applications and implications of the theorem. I did not find the first couple chapters to be all that interesting, as they had a lot to do with hypothetical meanderings about various historical characters and cultures and what they may or may not have figured out about right triangles.

It started to get more interesting around page 54 with stuff about ratios and the search for integers within triangles. In college I had seen the proof that the square root of two is irrational, but I found it quite pleasant to step through again. Then the book began diving through the rich waters of Pythagorean proofs. I generally skimmed through the heavily computational parts whenever I didn’t feel like thinking too hard, but the combinations of pictures, steps, and commentary made most of the proofs quite enjoyable. I skipped through Euclid’s proportional posturings and Loomis’s trigonomical troves, but the simple elegance of Thabit’s transpositions brought a smile to my face; his method seemed much more intuitive to me than Guido’s apparently similar method. And the tiling riff was even more fun!

As the authors explored the proofs across the centuries and continents, they asked good questions about whether or not the subtle differences between some of these proofs really counted as distinct proofs or not, with no attempt to provide an answer. Is it all tautology? The delight I experienced in discovering the different ones was good enough for me. Da Vinci’s quadrilaterals were tantalizing, though I had to accept one step on faith that I couldn’t prove for myself just staring at it. The trapezoid of Garfield – yes, the former U.S. President – was clever. The blind Coolidge’s shape comparison and transfer to algebra was interesting as well. I was always fascinated by the way the theorem holds true for shapes besides squares, although this was only briefly touched on in the book (but really, how much can you say about it?).

After racing through these various halls of Pythagorus, the rest of the book explores other wonders that have to do with the famous equation or that emerge from it, from multi-dimensional play to parity rules to the fascinating integer triple generator! The book’s style is much more casual than one might expect coming from two Harvard professors, and there is a copious amount of metaphors and humorous analogies that risk misunderstanding but for the most part work very well and (to use a metaphor myself) help the meaty content go down a little smoother. Consider this sentence from page 156: “So precariously is mathematics balanced on the edge of tautology that the gentlest push can tumble it to the depths.” Or this gem from the discussion about rational numbers on page 179: “There are incomparably many more irrationals than rationals. But this makes those whole number solutions blaze even more brightly among the dark numbers, as stars do in a universe made mostly of dark matter.”

Hidden Harmonies: The Lives And Times of the Pythagorean Theorem is not a book for the non-nerdy, but it’s also not too esoterically academic for the college graduate who wants to keep learning yet avoid trudging through more horrid proofs! If you’ve been fortunate enough to catch a glimpse of the sacred delight that lies hidden in the planes of mathematics, this work by the Kaplans unearths a little more…