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Manifold Mirrors: The Crossing Paths of the Arts and Mathematics

Felipe Cucker
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Hardy Grant
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Felipe Cucker deals admirably with two-thirds of the agenda implicit in his subtitle. He provides an authoritative account of the mathematics that underlies certain forms of artistic creation, and he describes with deep insight a wonderful variety of works of art in those forms. The third aspect of his self-imposed mandate, the “crossing paths” between mathematics and the arts, presents a subtler challenge.

His mathematical chapters concentrate almost wholly on geometry. “Art and geometry”, he says (p. 12), clearly intending here only the visual arts, “are linked in a primordial manner: the way one visually perceives things presupposes a geometry which decodes this perception”. On this clue the discussion proceeds to various geometrical spaces — the Euclidean plane first and foremost. The possible isometries in the plane, and the symmetry groups of planar objects, are described in detail; the payoffs include a full proof that there are exactly seven frieze patterns and a description of the seventeen possible wallpapers. The exposition then supposes a “stretching” of the plane, and duly covers homothecies, similarities, shears, strains, affinities. Later chapters explore more exotic geometries: projective (with a bow to the influence of Renaissance perspective on Desargues), affine, non-Euclidean, spherical, cylindrical — this last in a chapter on cosmology whose interest trumps its seeming lack of relevance.

Cucker’s mathematical emphasis naturally dictates his choice of exemplifying works of art. The predominance that he gives to geometry over arithmetic, to shape over number, entail for example that one finds here no mention of the numerical proportions detectable in Gothic cathedrals (Otto von Simson), or of the “Pythagorean palaces” of Renaissance Italy (George Hersey). Moreover the account of geometry leans heavily, as the author concedes (p. 373), toward two dimensions, with the consequence that architecture gets relatively little attention, and sculpture still less. The great majority of the illustrations are paintings, drawings and designs. Eye-catching friezes, wallpapers and carpets from many times and places are, mathematically speaking, combinations of translations, reflections, rotations and glides. The focus on such transformations extends (under the unfortunate title “Aural wallpaper”) to the book’s only chapter on music, in which the spotlight falls on the quasi-spatial manipulations of themes which generate the canons in Bach’s Musical Offering. Returning to the visual arts, Cucker moves from a good account of Renaissance perspective to the alternatives associated with more recondite geometries, offering (p. 260ff) ”a repertoire of drawing systems”, based on diverse projections, vantage points, locations of the picture plane. Here Asian art dominates his roster of examples, for “in the Far East”, he explains (p. 286), “perspective was rarely used.” Perhaps surprisingly, he says (p. 321) that his chapter on non-Euclidean geometry “bears little on art” – but his final illustrations, the amazing “Circle Limit” woodcuts that M.C. Escher produced after consulting Harold Coxeter, exhibit isometries and tessellations in the hyperbolic plane.

Cucker’s approach to the “crossing paths” of his subtitle is hinted by his book’s announced theme: “the role of mathematics in rule-driven artistic creation” (p. 381). He points out that rules, including the “laws” of the relevant geometry, work in two ways: they specify possibilities, but they also impose constraints. He recognizes, of course, that not all constraints are mathematical. Indeed, it is mostly in this context that the book takes its limited notice of literature — with examples in which the mathematical component of the constraint(s) is either zero (Georges Perec’s novel written entirely without the letter “e”) or trivial (the numerical aspects of poetic metre and sonnet form). Cucker gives full play to the ways in which artists resist constraints: a chapter on “The vicissitudes of perspective” chronicles “Divergences” and even “Abandonment”. In this context he is especially good on artists’ diverse approaches to symmetry. He regards symmetry as particularly valuable in the “interplay” between the “formal” and the “semantic” aspects of works of art (p. 128) — but here too he recognizes the aesthetic appeal of purposeful deviations. In Escher’s magical world departure from symmetry, Cucker says (p. 107), is “no less rigorous” than adherence.

Certain deeper issues about those “crossing paths” remain elusive in these pages — but then they pose questions which lack easy answers. One can ask, for example: does it help a creative artist to know the underlying mathematics? Cucker’s few scattered hints suggest that ignorance need not hamper. He concludes (p. 82) that in the making of “patterns” — wallpapers, tiles, porcelain, … — mathematics merely provides a “catalogue” of possibilities, of which artisans generally remain unaware, without loss. He asserts (p. 240) that most of the early practitioners of perspective “lacked a basic education in mathematics”. He recalls (p. 366n) the remarkable fact (which Kenneth May pointed out long ago) that Escher ”solved” certain problems before the working out of the relevant mathematical theory. And as in the mysterious process of creation, so also in the demanding activities of contemplation and appreciation: how essential, for a viewer or listener, is mathematical understanding? Again Cucker provides (p. 387) a clue: “constraints having a mathematical nature”, he says, “are almost invariably on form”, not on content — and so, one is tempted to add, not in the directions in which works of art speak to us most powerfully. Perhaps then an awareness of those constraints, and of the artist’s skill in coping with them, may fascinate or delight but will not usually tug at deeper emotions or decisively shape judgments of greatness.

The book’s cover blurb says correctly that although it began as a text for a liberal-arts course it should appeal to a wide general audience. Apparently the “target” reader is, mathematically speaking, very unsophisticated but a very quick study: we get Euclid’s Elements and analytic geometry absolutely from scratch — en route to Gödel’s Completeness Theorem and the Poincaré model of hyperbolic geometry, among other esoterica. Not all of the thorny technicalities bear directly on the ostensibly central theme of connections with the arts; surely, for example, the significance of non-Euclidean geometry in that context could have been set out without a rigorous introduction to formal languages and model theory. But Cucker makes at the outset (p. ix) a mathematician’s case for the value of full detail — and then genially allows that readers can skip as they please.

In his acknowledgements (p. 403) the author lauds the “phenomenal” help given by his publisher, but his adjective errs on the side of generosity. The book is marred by a regrettable array of the kind of mistakes in English from which good editors will save an author who is not writing in his first language. But against that flaw must be set the publisher’s painstaking treatment of a manuscript that must have been typographically daunting at many points.   

The merits of this big, ambitious book greatly exceed its deficiencies. The few omissions of ostensibly relevant themes, and the many infelicities in expression, are dwarfed by a richness of scope and detail, on the sides both of mathematics and of its artistic embodiments, that a short review must be content merely to salute. The illustrations make an anthology of marvels, clever and beautiful, delightful and moving, spanning continents and centuries. Felipe Cucker’s immense learning, and his often densely technical presentation of mathematical complexities, are balanced by a pervasive lightness of tone and by a flair for offbeat allusions that range from Euripides to Busby Berkeley. His book is a joy for the eye and a feast for the mind.

Hardy Grant ( retired a number of years ago from the mathematics department at York University, Toronto, where his specialty was an undergraduate Humanities course on the cultural career of (Western) mathematics.

Mathematics: user's manual
1. Space and geometry
2. Motions on the plane
3. The many symmetries of planar objects
4. The many objects with planar symmetries
5. Reflections on the mirror
6. A raw material
7. Stretching the plane
8. Aural wallpaper
9. The dawn of perspective
10. A repertoire of drawing systems
11. The vicissitudes of perspective
12. The vicissitudes of geometry
13. Symmetries in non-Euclidean geometries
14. The shape of the universe
Appendix: rule-driven creation
Index of symbols
Index of names
Index of concepts.