Beautiful Geometry, by Eli Maor and Eugen Jost
Tony Mann admires a book where art and mathematics are in perfect harmony
Mathematicians and artists have much in common. Both work with beauty and truth. While the popular impression may be that art is more concerned with the former and mathematics with the latter, many examples suggest the contrary. The hostile reception by many mathematicians of the ugly, un-mathematical 1976 computer-generated proof of the then 124-year-old four colour theorem shows that, for many practitioners, beauty is fundamental to good mathematics.
Here we have a collaboration between Eli Maor, an academic and a prolific popular writer on mathematics, and the Swiss artist Eugen Jost. Fifty-one sections present attractive colour artworks by Jost, accompanied by Maor’s account of the mathematics underlying each. The explanations are clear, and cover the background to the paintings in a manner that will be appreciated by readers whatever their level of mathematical knowledge.
But it is Jost’s images that are the main attraction. Some were created digitally, while others are acrylics on canvas. They are generally sensitive and interesting responses to the mathematics. I particularly admire the artist’s use of colour: even at first glance, before one is drawn into the mathematical content, these are beautiful paintings.
The subjects are diverse. We have familiar ancient geometry – Thales, Pythagoras, the doubling of the cube and the squaring of the circle. As one might expect, Archimedes’ spiral, the golden ratio and fractals all feature. There is number theory – the digits of π and Euler’s number e, and nice knot-like designs relating to the numbers 11 and 50.
But the artist is also inspired by some less well-known mathematics. I particularly liked the images relating to Chladni and Lissajous figures, which are generated by the stationary points of vibrating solids and interacting compound vibrations, while Gothic Rose is a stunning representation of different symmetries.
While much of the mathematics featured is ancient, it’s good to be reminded that not all of geometry was known to Euclid and Apollonius. “Steiner’s porism” illustrates the remarkable result of the 19th-century Swiss mathematician Jakob Steiner. If one has two circles, with one contained in the other, then if the construction of a series of circles, each of which touches the original two and its predecessor, yields a closed chain for one choice of starting circle, then necessarily the chain is closed for any choice. Maor tells us how this theorem had previously been discovered in 1784 in Japan by Ajima Chokuyen. From the 20th century, Morley’s theorem (that the angle trisectors of any triangle form an equilateral triangle), and Pick’s theorem, about integer lattice points lying inside a polygon, provoke useful descriptions and attractive pictures. While this is not really, as Maor suggests, a survey of the history of geometry, it does give a feel for how mathematicians’ interests have changed over the centuries.
The art generally engages fully with the mathematics, providing insights, and sometimes taking surprising directions. There is wit, as in “25 + 25 = 49”, which displays a right-angled triangle with sides 7, 7 and 10; this may not work mathematically, but the artist quotes the painter Josef Albers: “In science, one plus one is always two: in art it can also be three or more.” Occasionally, I feel that the artist is less inspired: a couple of images are for me little more than tastefully coloured diagrams; but the vast majority of the 60 colour plates are definite hits.
In short, even though some will object to the title as a tautology, anyone with any interest in visual mathematics will love this book, which, given the quality of the reproductions, is very attractively priced. It will inspire interest in a wide variety of mathematics, and should be a compulsory purchase for sixth-form libraries.
By Eli Maor and Eugen Jost
Princeton University Press, 208pp, £19.95
ISBN 9780691150994 and 9781400848331 (e-book)
Published 29 January 2014
Tony Mann is director of the maths centre, University of Greenwich.