Mathematics and Art: A Cultural History by Lynn Gamwell (review)

This is a monumental work of considerable merit. So this review is also more substantial than normal. The book itself is 576 pages in length but is large format, like many arts books, and weighs 3 kilograms! As the title implies, it is a cultural history and, starting from prehistorical time, traces the influence of art on mathematics and mathematics on art through most of history. It is very well organized, but not always chronologically.

The first chapter, art and geometry, examines the very beginnings of shape and counting. Unlike standard history of mathematics texts, the author starts in prehistory and examines all kinds of ancient symmetry in stone flints, designs in pots and cave paintings, as well as clothing remnants found in ancient graves. It is well known that the Greeks pioneered the abstraction of mathematical ideas, and this idea is well covered. However, the coverage is larger than mathematics, broadening to Democritus’s mechanical universe, astronomy and philosophy. The wide-ranging nature of the book means that the relation to religion is not neglected; neither are contributions from other cultures such as Arabic, Hindu and Chinese. Ideas are beautifully illustrated throughout. The first chapter concludes in the 17th century with the contribution of Galileo, Kepler’s (heliocentric) laws and Newtonian mechanics, but there is more than a nod at art, architecture and theism. Of course the dispute between Galileo and the church is well known, but the often-difficult relationships among religion, government and the new renaissance of science is more deeply explored here. The English Civil War is contemporary with Newton’s birth, but the U.S. Declaration of Independence and the French Revolution happened a little later; however, the author wraps things together nicely to launch into the next chapter on proportion. As the author says, mathematics (she means pure mathematics) born at the time of the Ancient Greeks remains wonderfully immune to all the changes in fashion, opinion and religious beliefs.

There follows an account of linear perspective, possibly the best account this reviewer has seen, all beautifully illustrated and with considerable technical detail. The Golden Section gets a good airing, but it is also well researched: The pitfall all too easy to fall into is finding the Golden Section everywhere, but the author is quick to state correctly that not until the 19th century is it featured deliberately in art; before this time, it is just a proportion that seems to occur because ratios close to φ:1 are pleasurable to the eye [φ = 125 + 1]. The link between φ and the Fibonacci sequence (1,1,2,3,5,8,13,21,35 . . .), although known for hundreds of years, was not again explored by artists until after Darwin and his scientific breakthroughs in the late 19th century. The author makes this point clear with many beautiful examples. She ends the chapter with the geometric link between φ with spirals and kinetic art.

Chapter 3 is about infinity. The Greeks barely acknowledged it, calling it “not finite,” and the author soon skips to the 17th century, calculus and tangents. Then the chapter moves straight to a description of probability that this reviewer thinks slightly odd. Far better is the account of Cantor and the development of transfinite numbers. The relation of this to fractals is quite well handled, although at this stage there is no mention of the Sierpinski or Koch curves, which are met later in Chapter 12. This chapter leads naturally on to the next on formalism, in which we find an account of some of the axiomatic arguments in mathematics—for example, the work of Euclid — followed by the rejection of the parallel postulate and birth of non-Euclidean geometries in the early 19th century. There is more on Cantor and the birth of the attempts to put the whole of mathematics on an axiomatic footing. The later parts of the chapter move away from mathematics and explore Russian formalism in art as well as linguistics...