In Mathematics and Art Lynn Gamwell continues to explore the relationship between science, art, philosophy and spirituality of Exploring the Invisible (2002). There, Gamwell argued that science served to ground nineteenth- and twentieth-century art's expression of the invisible. Mathematics and Art largely stays within the same time period while arguing that the vector of influence proceeds in a similar direction: from mathematics to art. The foreword by Neil deGrasse Tyson reinforces this perspective by presenting artists and philosophers as ‘interpreters’ of the ‘muse’ of mathematics, not as groups who understand nature directly as scientists do (p. 7). While this appears as Gamwell's main thesis regarding her grand narrative, which stretches back to prehistory, it is not always backed up by the engrossing chronicle she has written.

Gamwell fills her narrative with numerous examples of artists interacting with mathematicians in person or through their texts and, as a result, going on to incorporate mathematical ideas into their work. For instance, in early twentieth-century France, Henri Poincaré’s intuitionist view of geometric axioms as products of convention influenced Marcel Duchamp's equal evaluation of impressionism and Dadaism in art. Later, the development of computers and recursive algorithms on geometrical figures in the mid-twentieth century led to the Koch snowflake and Mandelbrot's fractals, which were eventually used in computer animation in films like Jurassic Park (1993). Brute computation of population and death statistics inspired Tatsuo Miyajima's Megadeath (1999). While evidence such as this in support of her main thesis is strong, it loses strength in the light of several counterexamples which are never fully integrated into her argument.

Gamwell acknowledges directly some counterexamples where art influences mathematics. That Filippo Brunelleschi's development of linear perspective ultimately led to Jean-Victor Poncelet and L.E.J. Brouwer's projective geometry and topology is a rarity and so does not, for Gamwell, undermine her thesis. Yet other examples appear without this explicit acknowledgement. These include Gamwell's accounts of early hominid tools as vehicles for the development of symmetry, of the Pythagoreans’ working out of numerical ratios from musical harmonies, and of Humiaki Huzita's six axioms of origami becoming integrated into topology and eventually applied to biology and the design of interplanetary telescopes. Also suggestive are cases where art developed in parallel with mathematics, as in the use of computation and algorithms. Arnold Schoenberg's twelve-tone music was inspired not by studying mathematics, but by Johann Sebastian Bach's methods in composing his fugues. As with the other examples above, Gamwell leaves Bach's inspiration unexplored and provides no comment connecting it to her stated thesis.

The role of philosophy, religion and mysticism in Gamwell's narrative further complicates her main thesis as they often appear more influential than mathematics. Philosophical debates grounded in Enlightenment and Romantic sensibilities, more than mathematics, in her presentation, appeared to drive the Bauhaus movement. Furthermore, Gamwell cites ancient religion up to Nicholas of Cusa and modern theosophical movements as mystical sources present in both mathematics and art. Gamwell illustrates all this quite thoroughly with her narrative, but the closest she gets to connecting it with her central thesis is by pointing out the underlying Platonism in modern mathematics. Favouring mathematics leads to missed opportunities and leaves space for further explorations that could supplement Gamwell's text by also focusing more on the influence of art on mathematics.

As for the structure and presentation of Mathematics and Art, Gamwell's broad overview in the first three chapters stays largely within European traditions, while giving some attention to Chinese events before focusing on the particular concepts of proportion and infinity. As an added perspective on her material, Gamwell introduces the theoretical distinction between the philosophy (e.g. Plato) and practice (e.g. Euclid) of mathematics and where they intersect (e.g. Leibniz). Gamwell's discussion of proportion is a negative argument focusing on dispelling the notion that the conscious use of the golden ratio began with ancient artists by arguing that it did not come into explicit use until it was popularized by the German psychologist Adolf Zeising in 1854 and was subsequently picked up by Bauhaus artists and others. Gamwell's discussion of proportion holds potential for further historical investigation more fully exploring positive arguments of how mathematics and art expressed notions of higher consciousness and spiritual subjects prior to the nineteenth century.

To frame the remaining ten chapters, Gamwell presents three philosophical understandings applicable strictly to modern mathematics: formalism, logicism and intuitionism. The former two Gamwell identifies as modern versions of Platonism as they assert the reality of mind-independent mathematical objects, while intuitionism asserts that such objects are mind-dependent. Gamwell traces the impact of formalism in the arts on, for example, the Russian poet Velimir Khlebnikov seeing the internal consistency of non-Euclidean geometry in poems. British writers T.S. Eliot's and Virginia Woolf's integration of subjective experiences into settings out of which they would naturally arise reflected the logicism of Bertrand Russell and Alfred North Whitehead's Principia Mathematica (1910). Brouwer's numerically inspired artificial language, Intuitive Significs, came out of his working at Frederik van Eeden's early twentieth-century Dutch commune, which joined those with Calvinist backgrounds with Romantic, theosophical, Buddhist, Hindu, Taoist and Transcendentalist perspectives. Gamwell follows numerous other developments within mathematics and art that she is able to deftly connect with these three traditions.

Gamwell's intention for her text was for ‘non-specialist’ audiences (p. 8), yet she does not avoid the complicatedly rich subjects of her topic, offering skilful explanations all along the way. Each chapter is heavily populated with full-colour images representing both works of art and mathematical/geometrical concepts to illustrate particular points. Numerous sidebars and descriptions of the images expand upon the main text without interrupting the central narrative of each chapter. Despite its theoretical shortcomings, Mathematics and Art is an enjoyable read accessible to anyone interested in the history of mathematics and art.