Professor of Computing William Latham recently reviewed the book Mathematics and Art for New Scientist magazine. We reprint his review here, with added hyperlinks and images.

From Renaissance painters’ first use of perspective to artistic algorithms shaping 21st-century works, mathematics and art have a long, rich history. “Cells and tissues, shell and bone, leaf and flower, are so many portions of matter, and it is their obedience to the laws of physics that their particles have been moved, molded and conformed. Their problems of form are in the first instance mathematical problems,” wrote the Scottish polymath D’Arcy Wentworth Thompson in his influential 1917 book, On Growth and Form.

This is a text that the author of the excellent new book, Mathematics and Art, has taken to heart and built on. In 500-plus, sumptuously illustrated pages, Lynn Gamwell has interleaved mathematics and culture (art, in particular) from 3000 BC to the present day, as she works to show how artists have harnessed maths for their own creative goals and how the arts, albeit to a lesser extent, have influenced maths.

There are many telling examples. Take Piero della Francesca’s 1455 painting The Flagellation of Christ, in which he positioned Jesus in a three-dimensional, naturalistic scene rather than an out-of-scale figure on a flat, 2D plane as his early Renaissance predecessors such as Giotto had done. This was a radical and daring innovation. What made it possible was the painter’s use of a set of new mathematical rules, which we now call linear perspective, that had been invented by mathematician and architect Filippo Brunelleschi.

Brunelleschi had himself been influenced by an 11th-century Islamic treatise on optics and visual distortion that had helped shape his ideas on perspective. This single mathematical step was to influence the whole of Western art, as exemplified in works by Leonardo da Vinci, Hans Holbein, Albrecht Dürer, Salvador Dali and, of course, M. C. Escher.

“Early Renaissance artists no longer painted saints floating in a golden mist in a faraway place; linear perspective gave them the tool to depict Jesus and the apostles existing right here, right now before their eyes in the natural world,” writes Gamwell.

There have been many examples of these mathematical cross-overs: think of Mandelbrot’s fractal maths translated into psychedelic-style computer art in the 1980s, or the influence of quantum mechanics on post-modernist painting and sculpture. They may not all be of the same magnitude as Francesca’s use of perspective but they are significant, and it’s illuminating to discover the background to these innovations.

It’s also important to recognise how many mathematical fields inform art. Crystallography, celestial geometry, phyllotaxis, differential calculus – all helped to shape Renaissance art and movements such as surrealism, constructivism, pop art and minimalism.

Mathematics and Art is split in two, with the first section bringing us up to about 1900, and serving as a handbook for readers who want to choose specific topics. Among the mathematical gems and anecdotes, Gamwell cites conversations between da Vinci and Franciscan friar and mathematician Luca Pacioli discussing what would become Pacioli’s book, On The Divine Proportion. There are also reproductions of John Dalton’s rough but extraordinary diagrams of atomic elements from 1806.

The second half, post-1900, has fewer diagrams and works less well as a mathematical handbook. Instead, its strong suit is the presentation of the philosophical relationship between the arts and maths – as when Gamwell discusses the detail of quantum mechanics, taking Antony Gormley’s Quantum Cloud V sculpture as her hook.

Gamwell also dives into the compelling area of how we measure aesthetic value, citing George D. Birkhoff’s attempts in the 1930s to reduce aesthetics to a mathematical formula, M=O:C, or the amount of aesthetic pleasure produced by an object (M) equals the ratio of the object’s order (O) to its complexity (C).

“George D. Birkhoff attempted to reduce aesthetics to a single mathematical formula”
This is particularly relevant to the emerging field of creative robotics, where the goal is, apparently, to create a robot that will create art for its own aesthetic enjoyment, emulating the human creative process.

Gamwell must have had her work cut out deciding what to include and exclude in what aims to be a comprehensive tome. There are casualties. In the computation section, for example, it was right to make much of fractal mathematics, Alan Turing, John Conway’s Game of Life and computer artworks by Roman Verostko, Manfred Mohr and Yoichiro Kawaguchi. But some classic computer graphic algorithms are missing, such as Ken Perlin’s noise texture algorithm or the Blinn-Phong reflection model, which have had a major impact across the arts and in film.

And we really do need more than a brief reference to artist Robert Rauschenberg, composer John Cage and the Experiments in Art and Technology group’s show in 1966 at The Armory in New York. The group was set up to foster collaborations between artists and engineers through direct personal contact rather than through any kind of formal process. The creative talents that came together then helped define the work of a generation – and generations to come.

Overall this is a comprehensive, valuable and detailed book. It is written in an accessible style, with enough mathematics to interest the technical reader without overwhelming one with an arts background. It doesn’t quite rival Douglas Hofstadter’s hugely influential Gödel, Escher, Bach from 1979, but its rich anthology is particularly relevant today, given the explosion of interest in the digital arts and the need for digital artists to use maths creatively. I will definitely be keeping it close at hand.

Postgraduate degrees at Goldsmiths Computing include:

• MFA Computational Art
• MA Computer Games Art & Design