Geometry and the Visual Arts

Geometry and the Visual Arts is a Dover republication of a book originally published in England by Penguin Books in 1976 and in the United States by St. Martin’s Press in 1978. The book’s original title was Geometry and the Liberal Arts. Neither title is entirely appropriate. The first four chapters bear the titles “Vitruvius,” “Albrecht Dürer,” “Leonardo da Vinci,” and “Form in Architecture.” The bulk of the book, however, including these four chapters, is a presentation of a variety of geometrical facts with some proofs. An exception to this observation is the nice discussion of one-point perspective in the Dürer chapter.

The central chapter is the sixth, “Euclid’s Elements of Geometry.” In a mere 32 pages, Pedoe manages to move from the five postulates through absolute geometry, the fifth postulate, and non-Euclidean geometry in a coherent manner. The unwritten subtext is the central position of Euclidean geometry in Western Civilization: think of Aquinas, Spinoza, and Newton and the explicit modeling of their work on a logico-deductive system. Until recently, we in the West withheld belief in philosophical pronouncements unsupported by an argument. A mere listing of maxims ala oriental mystics just wasn’t our style. This mode of intellectual activity is a direct consequence of the impact of Euclid.

The longest chapter is the penultimate one, entitled “Curves.” Starting with some properties of circles, Pedoe continues with various methods of drawing conic sections and ends with some of the curves one considers in calculus, such as cardioids, limaçons, and spirals. Many of these Pedoe describes first in terms of envelopes, then as point loci, and finally using a method he calls mathematical embroidery.

This is a fun chapter, but, more so than the other parts of the book, raises the question “For whom is this book written?” In the preface, Pedoe makes a passing reference to the “general reader,” but even thirty-five years ago the “general reader” would be thrown by the gratuitous mention of the polar equation of a cardioid (p 222) without any introduction of polar coordinates.

This is only the most obvious of the anomalies throughout the book. Each chapter ends with a modest list of exercises that Pedoe says are there for the reader to derive some additional pleasure. These exercises don’t transform the book into a text. The general reader Pedoe has in mind is someone with a mathematical background beyond high school, say an engineer or scientist or mathematician.

In a few places, having introduced a mathematical object using a set of labels, a subsequent discussion of properties of that object uses the same labels for different points. On page 279 there occurs the only serious error that I noticed. Pedoe gives a formula for the sum of the dimensions of two subspaces (points, lines, planes, or solids) of four-dimensional space that implies 1 + 1 = 4. Maybe there is an interpretation of the formula which I am missing.

If any of the above seems like negative carping, attribute it to the reviewer being a grumpy old guy. When I finished reading ‘Geometry and the Visual Arts’ the overall feeling I had was one of having had some fun. A copy of this book has been on my shelves for over thirty years and I’m happy to have finally read it.

Ronald Infante graduated from Rutgers University and received a PhD from Yale University about fifty years ago. He taught at Seton Hall University for fifteen years. After working in the defense industry for 27 years, he retired 3 years ago. Currently he is attempting to understand Wiles’ proof of Fermat’s Last Theorem (with little success), building furniture for his children, and harboring a fond desire that all calculus reformers disappear from the face of the earth.