Beauty to one can terrify another

kw: book reviews, nonfiction, mathematics, history

This image is a Calabi-Yao Manifold, one of many. The image itself is pretty, and to some, the math behind it is beautiful. To most of us, the math is incomprehensible, and perhaps most of us would not recognize it as mathematics were it presented in any form other than such an image.

Ian Stewart, a professor of mathematics, has written Why Beauty is Truth: A History of Symmetry, and a monochrome version of this image graces page 253. It seems this is a projection onto the page of some aspect of a 6-dimensional analogue of a torus; I suppose 6-D creatures could get these at Dunkin' or Krispy-Kreme.

The book is a historical survey of the continual extension of the concept of symmetry to higher and higher abstractions. However, symmetry as most of us understand it is left behind about a quarter of the way through the book. Because of a background in crystallography, via mineralogy, I suppose I have a bit more of a concept of symmetry than most blokes, but it is a stretch for me simply to understand most of the book in a mostly foggy way.

This is not to say that Dr. Stewart does not explain himself well. His writing is excellent, and a great value of the book is that it tells the stories, not just of some math concepts, but of the people who developed them.

The chapters have charming titles such as "The Luckless Revolutionary" or "The Drunken Vandal" (the first is Galois, who introduced the concept of a Group to mathematics, forever changing the discipline—or set of deeply nested disciplines—; the second is Hamilton, who had a flash of insight while walking, and promptly carved it into the nearest bridge. I didn't get exposed to Galois groups while in school, but 4-dimensional "Hamiltonian Tensors" had a lot to do with me flunking out of Physics—and taking up Geology—as a Senior).

Though I call myself a working mathematician, I must confess that I am really an algorithmicist. If I can figure out the steps behind an operation, I can either perform it analytically or program it digitally. I tend toward the latter...

The author, beginning with the Babylonian solution of quadratic equations (!- this bane of Algebra 101 students is truly 3,000 years old !!), goes through step after step in the development of mathematical concepts, showing how each either solves a problem of symmetry posed at a "lower" level, or introduces a new kind of symmetry needed for progressing to the next step.

It is really a tour de force, and I really wish I had a better background for understanding it all. Though I think the title is a bit hyperbolic, I know why it is there, and Stewart's explanations helped me gain a little more understanding, not necessarily of the beauty of all these incomprehensible beasts, but of the soul of a mathematician who finds them beautiful.