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# Geometry Meets Art in an Arcimboldesque Assemblage

The Evil Genius of a King, Giorgio de Chirico, 1914-15. From Manifold Mirrors.
BOOK REVIEW: Manifold Mirrors: The Crossing Paths of the Arts and Mathematics. By Felipe Cucker, Cambridge University Press, New York and Cambridge, UK, 2013, 432 pages (including three indices: names, concepts, symbols), \$29.99.

Flipping through a publisher’s catalog that just arrived, I find a selection of mathematics books described variously as:

• “For the mathematical novice who wants to explore the universe of abstract mathematics.”
• “For teachers and students so that they can access the subject matter.”
• “To activate student learning and comprehension.”
• “To supplement a standard undergraduate course.”
• “To add to the mathematics curriculum of life science students.”

In publicizing their books, authors and publishers try to make clear the purpose of the books and their intended audience. As I read, or more likely skim, a new book, I usually ask myself the same questions: What is the book about and who is it for?

On the back cover of Manifold Mirrors, the author and the publisher assert clearly who and what the book is for:

“The book began life as a liberal arts course and is certainly suitable as a textbook. However, anyone interested in the power and ubiquity of mathematics will enjoy this revealing insight into the relationship between mathematics and the arts.”

In his book, Felipe Cucker, who is Chair Professor of Mathematics at City University of Hong Kong, and whose specialty is foundational aspects of numerical algorithms, has interpreted “arts” broadly. He considers, at the very least, painting, printmaking, photography, design, decoration, the weaving of rugs and carpets, dance, poetry, architecture, and music.

The mathematics discussed is perhaps less sweeping in scope, but the reader will find in Cucker’s book mini or micro lectures on linear transformations, symmetries, groups, tilings, projections, perspective, the seven frieze and the 17 wallpaper design groups, axioms, formal languages, elements of music, and hyperbolic, projective, and spherical geometries. Here and there the reader will find proofs of some of the theorems mentioned; a prefatory user’s manual offers vague suggestions as to how a reader might deal with the proofs. Cucker gives a critique of the axiomatic method as set out in Euclid’s Elements and expounded by Hilbert and later writers, e.g., Marvin J. Greenberg.

Manifold Mirrors is liberally illustrated, with images from the worlds of both geometry and art. Many of the works of art are in color; a number of them were quite unfamiliar to me. As to how works of famous artists are connected to mathematics, Cucker uses as an example the emergence (“eclosion”) of pictorial arrangements in elliptical form  in Renaissance art.

Cucker’s text is liberally strewn (and decorated) with quotations from famous authors. He includes—confesses, in fact, that he cannot resist including—an often quoted mistranslation of Euripides on mathematics and art: “Mighty is geometry; joined with art, resistless.” He quotes Leonard Bernstein on music as mathematics, Charles Dodgson (Lewis Carroll) on existence, and Gödel—in his incompleteness theorem—on consistency. Cucker points out that the “greatest English poets,” among them Shakespeare, Milton, and Wordsworth, preferred iambic pentameter for their poetry.

You can also read here a metrical analysis of the canons of Johann Sebastian Bach, and you will find a detailed explanation of how Poincaré visualized hyperbolic geometry. Cucker quotes Richard Feynman on a particularly beautiful gate in Neiko, Japan:

“. . . when one looks closely he sees that in the elaborate and complex design along one of the pillars, one of the small design elements is carved upside down; otherwise the thing is completely symmetrical. If one asks why this is, the story is that it was carved upside down so that the gods will not be jealous of the perfection of man.”

Mini and micro biographies pepper the text. We meet, among many others, Gauss, Felix Klein, and George Birkhoff, along with J.S. Bach, John Milton and Jorge Borges, Vincent Van Gogh and Maurits C. Escher, art critic and historian Ernst Gombrich, Bohemian music critic Eduard Hanslick.

In short, Cucker has produced a pot au feu, an eclectic catch-all. There is much that can be learned from Cucker’s presentation of the marriage of mathematics and art.

Giuseppe Arcimboldo (1526-1593), an Italian painter to the Hapsburg Court in Prague and Vienna, was famous for creating portraits from assemblages of fruit, vegetables, flowers, grains, animals. I consider Manifold Mirrors Arcimboldesque in that it is an assemblage of many basic mathematical ideas and constructs, adding up to . . . well, to a unique work. Answering the questions I mentioned at the outset of this review, the book is a fine source book for teachers of mathematics who, in laying out a curriculum, want to go beyond the dry sequences of definitions, theorems, proofs and yet who have little interest in standard applications—of analytical mechanics, say. The book is a Newton-free environment.

Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at philip_davis@brown.edu.