Alex's Adventures in Numberland - Alex Bellos [87]
In 1818 the French mathematician Gabriel Lamé started to play around with the formula for the circle and ellipse. He wondered what would happen if he started to tweak the exponent, or power, rather than the values of a and b.
The effect of this adjustment was fascinating. For example, consider the equation xn + yn = 1. When n = 2, this creates the unit circle. Here are the curves produced by n = 2, n = 4 and n = 8:
When n is 4, the curve looks like an aerial view of a Babybel cheese squashed in a box. Its sides have become flattened and there are four rounded corners. It’s as if the circle is trying to become a square. When n is 8, the curve is even more like a square.
In fact, the higher you push n, the closer the curve is to a square. In the limit, when x8 + y8 = 1, the equation is a square. (If anything deserves to be called the squaring of the circle, surely this is it.)
The same thing happens to an ellipse. If we take the ellipse described by ( )n + ( )n = 1, then by increasing the values of n, the ellipse will eventually turn into a rectangle.
In downtown Stockholm there is a main public plaza called Sergels Torg. It’s a large rectangular space, with a pedestrian lower level and a traffic circle on top. It’s the place activists choose to hold political rallies, and where sports fans congregate when Sweden’s national teams win a major event. The plaza’s dominant feature is a cetral section with a sturdy 1960s sculpture that locals love to hate – a 37m-high glass and steel obelisk that lights up at night.
During the late 1950s, when city planners were designing Sergels Torg, they encountered a geometrical problem. What, they asked themselves, is the best shape for a roundabout in a rectangular space? They didn’t want to use a circle because that would not use the rectangular space fully. But nor did they want to use an oval or an ellipse – which do fill the space – because the pointed ends of either shape would hinder the smooth flow of traffic. Searching for an answer, the architects on the project looked abroad, and consulted Piet Hein, a man once described as the third-most famous person in Denmark (after the physicist Niels Bohr and the writer Karen Blixen). Piet Hein was the inventor of the grook, a style of short aphoristic poem that he published in Denmark during the Second World War as a form of passive resistance against Nazi occupation. He was also a painter and mathematician, so possessed the right combination of artistic sensibilities, lateral thinking and scientific understanding to give fresh ideas to Scandinavian planning problems.
Piet Hein’s solution was to find a shape that was halfway between an ellipse and a rectangle, using simple mathematics. To achieve this, he used the method described on the previous page. He adjusted the exponent in the equation for the ellipse to get a shape that would fit inside the rectangular plaza at Sergels Torg. In algebraic terms, he did what Lamé did by playing around with the n in the ellipse equation:
As I showed previously, increasing the n from 2 to infinity takes you from a circle to a square, or from an ellipse to a rectangle. Piet Hein judged that the value of n such that the curve was the most aesthetic compromise between round and right-angled was when n = 2.5. He could have called his new shape a ‘squircle’. Instead, he called it a superellipse.
More than just an elegant piece of maths, Piet Hein’s superellipse touched on a deeper human theme – the ever-present conflict in our surroundings between circles and straight lines. As he wrote, ‘In the whole pattern of civilization there have been two tendencies, one toward straight lines and rectangular patterns and one toward circular lines.’ His piece continued, ‘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