*Mathematics and Art*is distinctly asymmetrical. The larger part of the text describes people, schools and trends in the history of art which were influenced by or connected in some way to mathematics, with a smaller part devoted to the history of mathematics and related sciences. (There is also a little exposition of the latter, much of it in separate boxes, on for example Kepler's laws and the double-slit experiment.) And more than half the space is devoted to glorious full colour reproductions, almost entirely of art works, with only a little space used for diagrams illustrating aspects of mathematics or physics. The captions to these photographs are often quite substantial, sometimes offering mini-biographies of artists as well as discussions of specific works. There are also some fun, if slightly random, quotes in the wide margins.

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.

June 2016

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