The first seven chapters are named after topics —"Arithmetic and Geometry", "Proportion", "Infinity", "Formalism", "Logic", "Intuitionism" and "Symmetry" — but are roughly chronological, taking the story down to the early 20th century. The last six chapters are also roughly chronological: "Utopian Visions after World War I", "The Incompleteness of Mathematics", "Computation", "Geometric Abstraction after World War II", "Computers in Mathematics and Art", and "Platonism in the Postmodern Era". The illustrations are mostly from the period being covered, but are sometimes used to illustrate it. So Boticelli's 1480 Saint Augustine in his Study appears in chapter one, alongside a marginal quote from Wittgenstein, to accompany a discussion of Augustine's attitude to mathematics. And a diagram from Kepler's 1611 On the six-cornered snowflake appears in the penultimate chapter, to illustrate a discussion of the proof of Kepler's sphere-packing conjecture.
I describe two chapters in more detail to illustrate the approach. "Formalism" begins with Euclidean and non-Euclidean geometries, Helmholtz's "learned geometry" (looking at the psychological foundations of space perception), Hilbert's formalisation of geometry, and the modern mathematical notion of "formalism" (the term was coined in 1912 by Brouwer as an insult directed at Hilbert). A brief look at formalism in Russian linguistics and literature (Khlebnikov) leads into the extensively illustrated second half of the chapter, which is devoted to Russian constructivist art (Tatlin and Rodchenko) and Unism in Poland (Strzeminski and Kobro).
"Of the three approaches to the foundations of mathematics — formalism, logicism, and intuitionism — formalism has had by far the most profound and lasting impact on the visual arts, first on Russian Constructivism in the 1910s, then throughout the West in the 1920s, then globally after 1945."
"But people did not forget that for a brief moment after October 1917 art had become an instrument of social change; Rodchenko and Tatlin had applied abstract color and form to practical tasks to make the world a better place."
"Far from being only a formalist, Hilbert used formalism as a tool to provide a foundation for the modern version of Plato's view of mathematics as the cognition of perfect, timeless abstract objects."
"Computers in Mathematics and Art" touches on computer proofs "by exhaustion" (the four-colour theorem and sphere packing), visualization, knots, networks (graphs), origami, recursive algorithms (including cellular automata), and fractal geometry and its applications. Gamwell manages decent introductions to these topics, not attempting too much but trying to give non-mathematicians some feel for them. Some of the photographs directly illustrate these topics, showing for example Julia and Mandelbrot sets or mountains and crystals (exemplifying the fractal geometry of nature), but more are of works of art inspired (more or less directly) by these ideas, from the Book of Kells through to recent work.
"Robert Bosch (American, b. 1963), Knot? 2006. Digital print, 34 x 34 in.
"The American mathematician Robert Bosch drew this continuous line based on the solution to a 5000-city instance of the travelling salesman problem. From a distance, the print appears to depict a black cord against a grey background in the form of a Celtic knot. But on close inspection the apparent 'grey' is actually a continuous white line moving on top of a black background. The white line never crosses itself — it is a network rather than a knot — and so the answer to the title is 'Not.'"
I skipped over almost all the explanations of mathematics, computer science, and physics, and only a little of the history of science was new to me, but it's always nice to get new angles on familiar topics, for example the correspondence between the geometer Coxeter and the artist Escher. The potted histories of different artistic movements were what I enjoyed most: 6 pages (half illustrations) on "Early-Twentieth Century Meta-Art" (covering de Chirico and Magritte), for example, and 26 pages (two-thirds illustrations) spread over two chapters on Swiss Concrete Art (Max Bill, Andreas Speiser, Verena Loewensburg, Karl Gerstner and others).
In a few places Gamwell goes off on tangents. There are seventeen pages on quantum mechanics, arguing for the De Broglie-Bohm interpretation and against the Copenhagen interpretation, which she explains as a strand in German idealism. There are only two small illustrations in this, of two of Antony Gormley's Quantum Cloud sculptures, and the connection to art seems a little tenuous. There's also a bit on music, with seven pages on Schoenberg and serialism and computer music.
Gamwell sensibly refrains from attempting to fit any kind of meta-narrative to the book or even the individual chapters. The links between art and mathematics she describes are mostly local, without larger-scale structure, and it would be hard to make a case for mathematical ideas driving long-term changes in art or aesthetics.
Given how much Gamwell covers and how multi-disciplinary Mathematics and Art is, there were bound to be some errors in it, and I found a good number, albeit mostly minor, just in the material I was familiar with. There's a confusion between the number of possible sequences for a protein and the number of possible structures into which it can fold; there's a reference to "Chinese, the oldest living language" that will make linguists blanch; Charles Darwin may have declared in the Origin of Species that "animals have innate drives for survival and reproduction", but that's hardly the notable idea in the work; Hermann Weyl's 1952 book Symmetry was based on lectures delivered in 1951 and is not the same as a much shorter 1938 article with the same title; and so forth.
Mathematics and Art is not designed for learning mathematics or science from. It works better for learning about artists and schools of art, but its selection and coverage are probably too idiosyncratic for use as a reference. Mathematics and Art is, above all, fun to browse in, probably for people from quite a wide range of backgrounds, and it makes a lovely coffee table book, which provides some substance for those drawn in by the illustrations and which may bring people into contact with new ideas and inspire novel connections.
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