CAN you find your own butterflies, swallowtails and wigwams? They are right there on your body: the geometrical figures that appear when smoothly curved surfaces are viewed from the right angle. Structural engineer Allan McRobie鈥檚 The Seduction of Curves is your guide to these most intimate of mathematical objects.
McRobie鈥檚 underlying point is a good one: why do we teach geometry as a rectilinear discipline when so much of what we encounter in life, from bodies to landscapes to drapery, consists of curves? And not just the familiar parabolas and valleys of quadratic and cubic functions, but the singularities that result from a three-dimensional view of surfaces. McRobie explores these familiar yet unrecognised shapes, navigating with the aid of mathematician Ren茅 Thom鈥檚 catastrophe theory.
Thom was a formidable expert in topology who won the coveted Fields medal in 1958. Yet his catastrophe theory feels now like the mathematical equivalent of flares and patchouli oil: very much of its time, namely the 1960s and 1970s. It was the chaos or complexity theory of its day, allegedly pertaining to pretty much anything from economics to fundamental physics 鈥� without ever quite explaining any of it.
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Thom wrote in a dense, obscure and quasi-mystical style, full of enigmatic definitions that postmodernists like psychoanalyst Jacques Lacan lapped up and mimicked. Jean-Luc Godard made a documentary about him; Salvador Dali painted an homage to him. That鈥檚 the kind of thing that turns you into a cult figure with a finite shelf-life.
Yet McRobie鈥檚 revival of Thom鈥檚 taxonomy of curves is persuasive, showing how they define the landscape of the artistic nude. They progress in a series of increasing complexity from the parabolic fold to the cusp (that point where the two arcs of a loop of wire just start to visually overlap, making a sharp apex), the double-cusped 鈥渟wallowtail鈥�, triple-cusped 鈥渂utterfly鈥� and so on. These curves feature even more prominently in optics: they are the bright bands and cusps of caustics, seen as light passes through a wine glass or a pool.
鈥淲hy do we teach geometry as a rectilinear discipline when so much of the world consists of curves?鈥�
Mathematically, cusps are singularities: places where things get infinite. They are related to critical points in statistical physics 鈥� which is one reason why catastrophe theory wasn鈥檛 much of a revelation to physicists.
But it was useful to structural engineers, since cusps are also related to the point at which stressed surfaces buckle and wobbly structures topple. McRobie ignores, however, the extensive recent literature on buckling and wrinkling, such as the analysis of the folds of drapery and paper, or the dimples and ridges of seed pods and fruits 鈥� a shame, since these issues too beg for aesthetic investigation.
No, it鈥檚 bodies that are the focus here, particularly in relation to art. Was Leonardo da Vinci鈥檚 recommendation that bodies be depicted in twisting posture an implicit claim for the aesthetic appeal of the fleshy cusp? Thom鈥檚 curves involve surfaces hidden but implied behind cusps and folds: a kind of geometry of desire. McRobie has some sly fun with it: 鈥淚 once spent the best part of a pleasant evening trying to find a butterfly [cusp] on my wife鈥檚 hip,鈥� he tells us. (What he found instead was a rare form called a gull.)
In taking on not just the artistic but the evolutionary and sexual aspects of his subject, he could have gone further. His notion of the nude is the idealised form of Velazquez鈥檚 Rokeby Venus or Michelangelo鈥檚 David, and the book hovers on the edge of saying something provocative about whether there鈥檚 a geometry of maximal arousal, both aesthetic and erotic. That would set a cat among the pigeons.
Book information
Princeton University Press
This article appeared in print under the headline 鈥淔etching figures鈥�
