**Mathematics and Art: A Cultural History**

by Lynn Gamwell

Princeton University Press, Princeton, NJ, 2015

576 pp., Illus., 444 col. Trade, $49.50

ISBN: 978-0691165288.

Reviewed by Phil Dyke

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 pre-historical time traces the influence of art on mathematics and mathematics on art through most of history. It is very well organised 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 as found in ancient graves. It is well known that the Greeks started 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 finishes in the 17^{th} 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 between religion, government, and the new renaissance of science is more deeply explored here. The English civil war is contemporary with Newton, but the USA’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 is now 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 all too easy pitfall to fall into is finding the golden section everywhere, but the author is quick to state correctly that not until the 19^{th} century does it feature 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[].The link between *ϕ *and the Fibonacci sequence (1,1,2,3,5,8,13,21,35…) though known for hundreds of years was again not explored by artists until after Darwin and his scientific breakthroughs in the late 19^{th} century.The author makes this point clear with lots of 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 done though at this stage there is no mention of the Sierpinski or Koch curves that 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 Euclid followed by the rejection of the parallel postulate and birth of non-Euclidean geometries in the early 19^{th} 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.

Chapter 5 on logic takes us through the work of Frege to Russell and Whitehead’s *Principia Mathematica *of 1910. There is a good account of how mathematics coped with the contradictions of self-referencing (Russell’s paradox) and the influence of this on art, sculpture, and literature in the early 20^{th} century. Chapter 6 on intuitionism gives an account of Brouwer’s development of topology and relates this to parallel developments in art. The links here are interesting without being wholly convincing. There follows a chapter on symmetry. The scientific applications here are clear, but the artistic ones more tenuous. Complete symmetry is usually avoided in the visual arts, but they may play a big part in trying to explain the forces of nature. The use of group theory in Swiss concrete art is intriguing.

The next two chapters are more or less chronological and concern post World War I developments. There is quantum mechanics and its bizarre nature, metaphysics following Wittgenstein, and Bauhaus art and architecture.Maybe all are a reflection of a change in culture, but the connections are loose. This is followed by an account of the rapid and substantial developments that took place in the 1930s where we find the Gödel incompleteness theorem, Magritte’s “This is not a pipe” picture of a pipe, the early Escher sketches of the metamorphoses of shapes and tessellations and finally Turing’s 1936 seminal paper on the halting problem in computing and the Turing machine. There is a very powerful sense of the overthrowing of fundamentals that permeates both mathematics and art that reflects the pending onset of World War II. This is gripping stuff indeed. The final few chapters are on the modern era where abstraction thrives, from Rothko and others in art with parallels in literature and poetry.There are some links to mathematics where the development of computers has led to amazing images and CGI in films, but once more these links are not wholly convincing. It is here that there are further examples of computer-generated fractals. Exactly what aspects will be important in the long term remains unknown; time has to pass for a consensus to form so the latter parts of the book are much more the personal views of the author.

To summarise, this is a very deep, well researched, ambitious and by and large successful attempt to capture the major connections between mathematics and art.It deserves a place on your bookshelf or coffee table, but make sure where it goes is both substantial and load bearing.