Various artists, such as Piet Mondrian and Theo van Doesburg, produced many paintings based upon plane geometric shapes. Naum Gabo, who trained as an engineer, is known for his 3-d constructions of ruled surfaces. Another sculptor, Man Ray, constructed a visual equivalent of Enneper’s surface of constant negative curvature (derived from the pseudo-sphere). But Man Ray said of these objects that their formulae meant nothing to him, which raises the question of how deeply mathematics should be understood by artists for their works to be classified as ‘mathematical’ or ‘geometric’?

This book doesn’t specifically address this question, but it does outline a wide range of mathematical ideas that have been taken up by many 20^{th} century writers and artists — no matter how superficially. For the benefit of non-specialists, Robert Tubbs provides insight into the wide range of associated mathematical ideas. There are introductions to transfinite numbers, Klein forms, axiomatic systems and graph theory and many other topics. The mathematics is accessible to good high school students and mathematically competent artists and writers (other readers will need to catch up).

The literary use of mathematical imagery is illustrated by its appearance in poetry and experimental writing. But sometimes the literary relevance of mathematical ideas seems highly tenuous, as in this poem (Möbius Strip), by Robert Desnos.

The track I'm running on

Won't be the same when I turn back

It's useless to follow it straight

I'll return to another place

I circle around but the sky changes

Yesterday I was a child

I'm a man now

The world's a strange thing

And the rose among the roses

Doesn't resemble another rose.

Chapter 7 (Poetry, Permutations and Zeckendorf’s tTheorem) is more convincing in its use of Fibonacci numbers to analyse Paul Braffort’s poem

This is my work, this is my study.

like Jarry, Cyrano puffy,

to split hairs on Rimbaud

and on willies find booboos.

If it was fair or if it snowed

in Lhassa Emma Sophie Bo-

vary widow of slow carnac

gave herself to the god of wack

Leibniz, saying “Vers…” What an ac-

tor for this superb “Vers…” Oh “nach”!

He aims, Emma, the apoplexy

of those drunk on galaxy.

At the club of “spinach” kings (nay,

Bach never went there, Banach yea!)

Leibnitz – his graph ibo: not six

mus, three nus, one phi, bona xi-

haunts without profit Bonn: “Ach! Gee

if I were great Fibonacci!!!

Much of the commentary in this book concerns art theory and literary criticism; with the aim of attaining mathematical rigor in this field, axioms were introduced to provide rigorous foundations. Raymond Queneau (1903–76) wrote *Foundations of Literature (after David Hilbert)*, where his axioms include ‘A sentence exists containing two given words’, and ‘Any two words are contained in one and only one sentence’ — but no theorems seem to have emerged (what could they possibly be?).

For the visual arts, Richard Hertz (*Philosophical Foundations of Modern Art*, 1978) began with an ‘undecidability axiom’, saying that ‘The distinction between art and non-art is undecidable’. Another axiom declares that ‘Art theory is as important as, or more important than, art works’. This is followed by Theorem 1, which says that ‘Ideas are as important as, or more important than, the fulfilment of those ideas in practice’ (Wow!).

But there is also mathematical analysis of novels, such as Bruce Morisette’s attempt to use the Klein bottle to represent the narrative structure of Robbe-Grillet’s book *In the Labyrinth*. We then have Albert Wachtel’s story ‘Ham’, in which two characters take contrary points of view on the respective cardinalities of \(\mathbb{Q}\) and \(\mathbb{R}/\mathbb{Q}\). This seems to be a metaphor for some philosophical abstraction regarding the law of the excluded middle; but it’s all good fun anyway.

Robert Tubbs immerses the reader in the (intriguing) interdisciplinary world of art, mathematics and literature. Many of the artists and writers were new to me, and I’ve enjoyed reading about their quirky attempts to seek inspiration from mathematics. To my mind, a major omission is discussion of that most mathematical of artists: M. C. Escher. And mention of Edwin Abbott’s *Flatland* might also have been included, had it not been a late 19^{th} century publication.

Final practical point: given the highly visual subject matter, there is a relative dearth of illustrations; and paintings that were originally done in colour are shown in black and white. So I often had to go online to gain proper insight of the works under discussion.

Peter Ruane has taught mathematics across the age-range (5 year olds to 55 year olds). That is, from school arithmetic to transfinite arithmetic. His postgraduate study concerned the application of differential geometry to superconductivity.