A Mathematical Gallery

by Lisl Gaal

I often remember how years ago I spontaneously bought a book after just glancing at the first page. It was Yoshida Kenko's Essays in Idleness. I was struck by the first two sentences. The first one came as a validation of my mood; the other - in contrast - was a mood setter. I still have and treasure the book, though never before or afterwards I was so light-headed when buying a book.

In 45 years since, I bought hundreds of books but always after taking time to read, browse through, trying to get a sense whether I was going to enjoy the book or not.

The book that I am going to say a few words about, would have broken the decades old habit. It could not because I received it for review. But it would have ... I tested this on my wife who chuckled with raised eyebrows on reading Author's Introduction.

The greatest part of Introduction is an assay by author's granddaughter, written when she was eleven years old. It is precious. Not to belittle my impression of the book proper, I would buy it just to have a chance to revisit that assay at will. What if math were a color? What if math were a sound? What if math were a taste, an emotion, a texture? The girl asked and gave her answers to those questions; imaginative, insightful answers. Math teachers should internalize the spirit of the assay and aspire to enrich their students with that sensation of familiarity and appreciation of what mathematics is.

This is what the book does gently, with beautiful lithographs, colorful diagrams, simple explanations, and always from unexpected angles.

The small book consists of 13 chapters, charmingly illustrated. Like Kinko's book, comprising small pieces that lack a story line, and are only connected by the personality of the author and his historical background, so the 13 chapters of the present book are unrelated to each other but linked with the refined taste, imagination and artistic mastery of the author.

The story line actually emerges as if set up in the introduction. Mathematics is conveyed via colors, the graphical images, the quiet narrative, the unexpected turns, the implied mystery, even dynamics. Can you imagine that shifting triangles in a proof by dissection of the Pythagorean theorem is motivated by the comfort of a cute dog squeezed in a couple of triangles? Can you imagine that the absence of commutativity of the ordinal numbers multiplication is illustrated by the emergence of rivulets in a magically looking waterfall?

Other topics introduced in the book include counting, Desargues' theorem, permutations, Pascal's triangle, volume of a pyramid and that of a sphere, area of a circle, Gaussian integers, Markov processes, calculus, seven-circles theorem. However, true to the spirit of Introduction, the book is less about specific topics and more about seeing mathematics as you may not have seen it before. The book is small but Lisl Gaal's insights and artistry make it great. The book made me think of other topics, close my eyes and ask, What do I see?

P.S. The book comes with three maps of equal right pyramids that together comprise a cube. This illsutrates the formula for the volume of the pyramid.

A Mathematical Gallery, by Lisl Gaal. AMS, 2017. Softcover, 64 pp, $25. ISBN 978-1470441593.

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