American Scientist

BLUEPRINTS: How Mathematics Shapes Creativity. Marcus du Sautoy. 384 pp. Basic Books, 2025. $32.

In Blueprints: How Mathematics Shapes Creativity, Marcus du Sautoy explores the mathematics underlying structures in music, art, literature, architecture, and other artistic endeavors. These mathematical topics—what he calls “blueprints”—he defines as “the hidden plans for the art, architecture, music, and stories that humanity has created.” Each chapter focuses on one such blueprint, ranging from prime numbers, Fibonacci numbers, and fractals, to symmetry, hyperbolic geometry, and randomness.

Each chapter begins with a story related to the “blueprint,” often an engaging anecdote about an artist, a moment in time of their work, and their relation to mathematics. I found it especially enjoyable to guess at the connection to the blueprint and then read on to see how it unfolded. The breadth of artistic pieces discussed is wide-ranging, including poetry, painting, crochet, novels, choreography, and musical compositions. For example, a discussion of circles in Bach’s The Musical Offering is particularly compelling, with du Sautoy describing how the piece was written: “Bach is instructing you to play the piano piece from the beginning with one hand and play it in reverse from the end with your other hand.” There is also a fascinating discussion about physicist Richard Taylor, who computed the fractal dimension of Jackson Pollock’s paintings. Comparing these calculations to try to authenticate several paintings found in storage, it was found that “none of them had the characteristic fractal quality that makes Pollock’s paintings so distinctive," and so were deemed to be fakes. 

du Sautoy’s writing conveys both his expertise and his enthusiasm. Proofs that mathematicians and mathematics educators may find familiar, such as the infinitude of primes, are presented in intuitive and conversational ways and paired with artistic works, making them feel fresh. Sometimes the mathematics explains the art, as in the allure of the compositions of 20th century French composer Olivier Messiaen, and at other times the art illuminates the mathematics, such as using the infinite Library of Babel, as written by Jorge Luis Borges, to model a four-dimensional torus. du Sautoy's definition of mathematics as “the study of structure” naturally connects to the artistic world, in which creators constantly respond to (and sometimes resist) structure. As du Sautoy writes, “if mathematics acts as a powerful set of blueprints for human creativity, I also believe that an artistic mindset is an important blueprint for discovering new mathematics.”

Occasionally, connections between the art and the mathematics seem tenuous or wander into tangential topics, such as when the chapter on Fibonacci Numbers detours into bird songs or the irrationality of √2. At times, concepts are mentioned that are difficult to comprehend without a figure to refer to, such as the discussion of the Fibonacci numbers in Modulor Man, the stylized human figure created by the French architect known as Le Corbusier. On the other hand, some linkages are truly spot on, such as the anecdote about measuring the separate compartments in a nautilus shell and uncovering a number close to the golden ratio.

Reading the book while following along with the art and music greatly enhanced my experience. I often listened to the musical pieces described in the text, and du Sautoy’s vivid descriptions allowed me to visualize the symmetry found in Bach’s The Art of Fugue and imagine the rotation of a shape in time with the composition. Doing so allowed me to appreciate the composition on a deeper level.

Du Sautoy also reflects on his own creative projects, such as animations visualizing the blueprints found in pieces by Messiaen and Bach, choreographing an interpretive dance related to a mathematical proof, and finding a tessellation inspired by patterns found in The Alhambra of Grenada, the 9th-century fortress in Spain. He gives a taste of his own mathematical research on creating a shape to encode rational solutions of the elliptic curve y²=x³-x. These additions reinforce his thesis that mathematics and art are intertwined acts of creation.

Much of the mathematics may be unfamiliar to people who aren't mathematicians. For example, I rarely see hyperbolic geometry discussed in recreational mathematics books. With a background in mathematics, I was able to follow the mathematical references, but struggled with some of the more technical discussions in music theory. A musician might well feel the opposite, and a reader without training in either field may find parts of the book challenging. Nonetheless, the writing is accessible, and the explanations are clearly written for a general audience. Equations and proofs are few and far between and are adequately explained when they do arise.

Ultimately, Blueprints succeeds in challenging the stereotype that art and mathematics are worlds apart. By revealing how artists draw inspiration from mathematics and how mathematics can illuminate artistic works, du Sautoy underscores the profound connections between the two. His wealth of knowledge about both the artistic and the mathematical make it hard to deny these connections and allow the reader to delve deeper into whichever field they are most interested. I strongly recommend that like I did, readers view the artwork and architectural pieces and listen to the music that du Sautoy writes about as you read. That way, his vivid writing and the art itself combine into the rich multisensory experience the book points toward, but does not fully provide on its own.