“In the whole pattern of civilisation there have been two tendencies, one toward straight lines and rectangular patterns and one toward circular lines. There are reasons, mechanical and psychological, for both tendencies. Things made with straight lines fit well together and save space. And we can move easily — physically or mentally — around things made with round lines. But we are in a straitjacket, having to accept one or the other, when often some intermediate form would be better.” — Piet Hein [1] Draw a line, any line, and it will have to be either straight or curved. Extend a curved line far enough and it closes to become a circle. Cross two straight lines and you get a corner; straighten the corner and you get a right angle. “Straight vs. curved” is a very natural paradigm that shows up all around us, and from its beginning mathematics has been built on the study of their differences. Euclid’s Elements, the most famous geometry textbook of all time, begins with the assumption that straight lines, right angles, and circles can always be drawn. The idea of translating these geometric shapes and problems into algebra, which was first described by Descartes only about a century after the beginning of algebra in its modern form, unified those two main areas of mathematics. It is the reason we still learn about Cartesian graphs (graphs drawn with the coordinate system that we are all familiar with) in school. On the usual x-y plane, the equation for a straight line is familiar: y = mx + c. Linear equations, where the highest-order term is x or y, always produce straight lines. Quadratic equations, with x2 or y2, will produce one of the curved conic sections – circles, ellipses, parabolas, or hyperbolas [1]. Consider x2/a2 + y2/b2 = 1, the general equation for the ellipse, which becomes the equation for a circle when a = b: the equation x2 + y2 = 1 should look familiar. By 1818, all this was well known to mathematicians. In a geometry book published that year, Gabriel Lamé decided to take things further by thinking about what happens with exponents n other than 2 [2]. The generalized curve xn + yn = 1 comes in many varieties depending on n [3], but the most interesting one for us is what happens when n is a fraction p/ q, with p even, and q odd and greater than 1. These fractions for n include all the even integers 2, 4, 6, 8, … From the circle at n = 2, you can see the shape become progressively closer to a square as n increases through the even integers (Figure 1). By adding absolute values, we allow n to be any number; now the Lamé curve |x|n + |y|n = 1 varies continuously for all n > 2 from a circle at n = 2 to a square in the limit as n goes to infinity. So does the general Lamé curve |x/a|n + |y/b|n = 1, which turns into a rectangle instead when the same limit is taken. If you never knew the equation for a square or rectangle, congratulations! Now you do. Thanks to the visual tool and shorthand provided by algebra, we have an intermediate shape between square and circle that can be described in a simple form. This simplicity is the reason for the sudden revival of this obscure curve in some unusual applications. Here are a few of them. In 1959, Swedish architects wanted to build a roundabout that could fit into a rectangular space between the buildings of downtown Stockholm [4]. Roundabouts tend to be circles, of course, but a circle would leave part of the space unused. An ellipse has pointed ends that would be harder for traffic to navigate. The city planners also tried a combination of eight arcs, but that would create too many points where the curvature would suddenly change. A Danish designer and scientist, Piet Hein, came across this problem when it was announced as a design challenge, and leaned on his mathematical background to find a compromise shape between ellipse and rectangle. He came across the Lamé curve and experimented with the value of n for an ellipse of width 6 and height 5 (i.e. a = 6, b = 5), eventually deciding on n = 5/2 or 2.5 (Figure 2) [5], which he called a “superellipse”: Figure 1 Transition of Lamé curves as n increases (n = 2 and n = 4 from left to right). By Peace Foo 胡適之
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