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Mathematics + Art: A Cultural History

Lynn Gamwell
Princeton University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Frank Swetz
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A large, yellow, mailing envelope was delivered to my doorstep. When I picked it up, I realized that it was a book, one of the heaviest books I have recently handled. Upon opening the parcel, I found a beautiful tome, a book of “coffee-table” appearance and quality: Mathematics + Art: A Cultural History, by Lynn Gamwell, a lecturer in the history of art, science, and mathematics at New York’s School of Visual Arts.

The book abounds with colorful illustrations, line drawings and numerous sidebars. The contents of the sidebars clarify and amplify the mathematical concepts referenced in the text. They are very good. The chapter headings point the direction in which the discussion will proceed: 1. “Arithmetic and Geometry”; 2. “Proportion”; 3. “Infinity”; 4. “Formalism”; … 13. “Platonism in the Postmodern Era”. While the title of the book designates an ambitious undertaking, Gamwell’s objective is to examine art as inspired by mathematics and not the inverse situation. The “cultural” settings within which these creative movements take place is the art scene itself.

Within the immense task of surveying the history of mathematical influence in the conception and creation of the visual arts — painting, sculpture, graphic design, photography… — just where does one begin? Gamwell’s Chapter I: “Arithmetic and Geometry”, seeks to provide an historical basis for mathematics and associate its place in an emerging Greek philosophical scaffolding. Since it is mainly the Western world that the ensuing discussions will concern, the scaffolding is Greek thought, specifically Platonism.

The reader is told that mathematicians are Platonists, theorizing about realities they can never fully understand: “shadows on a wall”. Perhaps. But, on second thought, wouldn’t a consideration of the Greek philosophical dichotomy between “arithmetike”, the theoretical, and “logistike”, the applied, be more fitting in appraising both the development of mathematics and its interactions with the processes of artistic creation?

In the concise survey of the first chapter, the reader traverses eleven millennia of human progress, from the symmetric design of Paleolithic spear points, through the geometry of soaring Gothic Cathedrals, to the cosmic speculations of Johannes Kepler and Isaac Newton: a long span. During this interval, when generalizations converge to focus on particular events, factual errors occur. On page 11, it is noted that during a visit to “peninsular Italy”, Plato sees Dion in Syracuse. Syracuse is in Sicily, an island off the mainland and a kingdom unto itself. On page 32, the Islamic mathematician Al’Khowarizmi (ca. 825) is credited with the introduction of letters for variables in algebraic computation. Not in my readings of his works!

Occasionally, similar errors appear in other sections of the book, for example: in a sidebar on page 124, the British mathematician, John Wallis (1616–1703) is mentioned for introducing the symbol “∞” to represent infinity and naming it “lemniscate”. Yes, he adopted the symbol from existing Roman numerals, but did not name the geometric curve. Jacob Bernoulli did this in 1694. On page 127, the reader is told that “Galileo invented the telescope”, which he did not; and on page 192, the Chinese Communists are credited with initiating the Chinese Revolution of 1911. The Communist Party did not exist as an entity in China until 1921. The Revolution of 1911 was nationalistic in nature and promoted mainly by disenchanted students. With such a great scope of material being considered, however, some errors are likely to occur.

While the narrative of the first chapter may disturb an historian of mathematics, the following chapters pick up intellectual and conceptual momentum. The author is apparently more comfortable discussing post-nineteenth century philosophical trends and artistic developments. Beginning with the second chapter, “Proportions”, the reader is transported to a rich and challenging environment of visual and conceptual information. The proportions considered are based on Western models via Vitruvius, Dürer and Da Vinci. Personally, I would also like to have seen some examination of other non-Western, systems of proportion, for example, those evident in Egyptian, Buddhist and Hindu art. The “Golden Proportion” is debunked as an intuitive component of artistic composition. I was pleased to learn that the controversial Spanish artist Salvador Dali (1904–1989), in his rendering of the painting, “The Last Supper”, was influenced by the fifteenth century works of Luca Pacioli and Leonardo da Vinci. Although I have viewed this particular painting several times, I did not identify the segment of a dodecahedron [the Platonic universe] arching over the subjects. Dali certainly was influenced by mathematics.

Throughout the book, in several instances, the artists, themselves, will credit their techniques to mathematical influences: the Russian formalist Aleksandr Rodchenko (1891–1956) credits Greg Cantor (1845–1918) and his concept of the continuum for inspiring his 1918 series of paintings “Black on White” and in the 1940s, the Swiss formalist painters Paul Lohse (1902–1988) and Max Bill (1908–1994) openly used group theory in their compositions.

The examination of the development of art is definitely stronger for the post-eighteenth era. Chapter 12, “Computers in Mathematics and Art” examines knots, networks and fractal geometry; while the following and last chapter, “Platonism in the Postmodern Era” returns the reader to the issue of philosophical directions established at the beginning of the narrative. Most mathematicians I know just do mathematics: solve the problem, probe the theory; and do not devote much, or any, contemplation as to the existence of their “worlds”.

By the time I was halfway through this book, I had accumulated ten pages of notes. While some of my comments were for this review, the majority, were for my own reference: information to pursue, new wonderful facts I had not previously known: so much information, so many issues to consider or reconsider in light of new revelations and so many inspiring and intriguing images. This is the beauty and power of this book: it is an intellectual tour de force of art history and its interaction with mathematics that will draw most readers, including me, back for further reading and study. The high quality of the illustrations and design, as well as the quantity of information provided, makes this book a true bargain at the stated price. I realize now why it was so heavy.

Frank Swetz, Professor of Mathematics and Education, Emeritus, The Pennsylvania State University, is the author of several books on the history of mathematics. His research interests focus on societal impact on the development, and the teaching and learning, of mathematics.

FOREWORD by Neil deGrasse Tyson IX

1 Arithmetic and Geometry 1
2 Proportion 73
3 Infinity 109
4 Formalism 151
5 Logic 197
6 Intuitionism 225
7 Symmetry 249
8 Utopian Visions after World War I 277
9 The Incompleteness of Mathematics 321
10 Computation 355
11 Geometric Abstraction after World War II 385
12 Computers in Mathematics and Art 455
13 Platonism in the Postmodern Era 499