Einstein had remarked that to appreciate the universe one would need to understand curves. Not sure if he was seduced by curves, but it is clear that straight lines and square corners do not determine what we make of the universe. Like human body, the universe is beautiful because of its magical patterns and exquisite shapes. Be it the human cell, the lowly creatures or the distant galaxies, the remarkable theory of curve is at work everywhere. Such is the magic of this theory that the entire universe manifests itself in curves, from the fold to a wigwam.
Reproductive biologist may consider curve as a work of evolution but for a structural engineer curve is what connects mathematics with art. And, within this connection may lie insights on understanding why things are shaped the way they are. While we often take shapes for granted but deep within each must be a cause for the object to take a particular form. University of Cambridge structural engineer Allan McRobie, who previously designed boringly rectilinear bridges and towers, found that the stability of engineering structures is essentially governed by their smoothly curved energy surfaces. This is how an unexpected link between the world of careful engineering calculation and a freer graphical expression emerged, leading to a beautiful language of folds, cusps and swallowtails.
Nothing comes closer than ‘human nude’ to draw a connection between art and mathematics, as a large part of our fascination for curves originates there. Need it be said that evolutionary biology rests on curves, our genes guide us to like the body shapes of our mates. The pervasiveness of these curves is striking because human body is a one-stop object for viewing different type of curves, a perfect justification for a book connecting mathematics with art to have a seductive curve on the cover. The curve resembles a swallowtail, the essence of beauty depicted by two back-to-back cusps connected by folds. Beautiful though it may be, a swallowtail on the waist of a human body is nowhere near as profound as the immense organization required to create all the components of that living, breathing, thinking human. The question worth exploring is how biological shapes emerge to be the shapes they finally become.
In some ways, the beauty in the swallowtail curve is indeed a precarious point of stability which engineers use to explain the catastrophe theory, the sudden change that induces abrupt outputs, developed by noted mathematician Rene Thom. However, the final shape of an object is on account of the energy of a protein that folds itself into any one of a number of possible configurations, indicating which stable pathway will eventually be followed. That is how some cells arrange to become bones and others into hair follicles.
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| The Swallow's Tail |
Spread across nineteen profusely illustrated chapters, the book is about seven curves, four from cuspoid (fold, cusp, swallowtail and butterfly) and three of umbilic family (elliptic, hyperbolic and parabolic), which are the basic building blocks, the fundamental components of curved form. These curves also represent a way by which something can suddenly change. This is the essence of the catastrophe theory, which studies how smooth and subtle changes in a system can result in sudden and abrupt outputs. Collapse of elevated roads or oil rigs are classic examples of such drastic end results. Given the significance of curves, one wonders if as children we should have been taught alphabets of curved geometry beginning with fold, cusp, swallowtail and butterfly in place of straight-line Euclidean geometry sequence of terms like triangle, square, pentagon and so on. Will it begin to change anytime soon?
In many ways this is a remarkable book which has more visuals than text, and it will be a pity if it goes unread. Interestingly, the illustrations and pictures are a study in themselves providing depth, dimension and relevance of curves in our lives, making one wonder if there were no curves in our universe. At some point in reading through the book one begins to view straight lines and angles with disdain, much like well known Brazilian architect Oscar Niemeyer, who was influenced by Le Corbusier but had developed an aversion for straight lines. Niemeyer was attracted to free-flowing, sensual curves, and had remarked ‘I am attracted to curves that I find in the mountains of my country, in the sinuousness of its rivers, in the waves of the ocean, and on the body of the beloved woman’. Several of the famous architectural structures standing today bear testimony to the power of the curved form. The future seems to be in triggering sensuous resonances in the minds of observers.
The book is a bold attempt at evoking multiple feelings towards curves. Allan McRobie deserves praise for sensually drawing parallels between natural and the constructed world.
The Seduction of Curves
by Allan McRobie
Princeton University Press, Oxford
Extent: 159, Price: $35
First published in Current Science, dated Aug 25, 2018.
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