Alex's Adventures in Numberland - Alex Bellos [43]
The first published origami proof of the Delian Problem was by a Japanese mathematician in 1980, and angle trisection followed by an American in 1986. The boom of interest stemmed in part from frustration with more than two millennia of Euclidean orthodoxy. The restrictions imposed by Euclid’s limitation to working with only a ruler and compass had narrowed the scope of mathematical enquiry. As it turns out, origami is more versatile than a ruler and compass, for example, in constructing the regular polygons. Euclid was able to draw an equilateral triangle, square, pentagon and hexagon, but recall that the heptagon (which has seven sides) and nonagon (nine) eluded him. Origami can fold heptagons and nonagons relatively easily, although i meets its match with the 11-agon. (Strictly speaking, this is one-fold-at-a-time origami. If multiple folds are allowed any polygon can in theory be constructed, even though a physical demonstration may be so hard as to be impossible.)
Far from being child’s play, origami is now at the cutting edge of maths. Literally. When Erik Demaine was 17 he and his collaborators proved that it is possible to create any shape with straight sides by folding a piece of paper and making just one cut. Once you decide on the shape you want to make, you work out the fold pattern, fold the paper, make the single cut, unfold the paper and the detached shape will fall out. While it might appear that such a result would be of interest only to schoolchildren making increasingly complex Christmas decorations, Demaine’s work has found uses in industry, especially in car airbag design. Origami has connections to protein folding, and now has applications in the most unexpected spheres: in creating arterial stents, robotics, and in the solar panels of satellites.
A guru of modern origami is Robert Lang, who as well as advancing the theory behind paper-folding has turned the pastime into a sculptural art form. A former NASA physicist, Lang has pioneered the use of computers in designing fold patterns to create new and increasingly complex figures. His original figures include bugs, scorpions, dinosaurs and a man playing a grand piano. The fold patterns are almost as beautiful as the finished design.
The United States now has as good a claim as Japan does to being at the forefront of origami, partly because origami is so embedded within Japanese society as an informal pursuit that there is more of a barrier to taking it seriously as a science. The cause is not helped by a Monty Pythonesque factionalization between different organizations, each claiming exclusive access to origami’s soul. I was surprised to hear Kazuo Kobayashi, chairman of the International Origami Association, dismiss the work of Robert Lang as elitist: ‘He is doing it for himself,’ he tut-tutted. ‘My origami is about the rehabilitation of the sick and educating children.’
Nevertheless, there are many Japanese origami enthusiasts doing interesting new work, and I travelled to Tsukuba, a modern university town just north of Tokyo, to meet one of them. Kazuo Haga is a retired entomologist, whose professional expertise is in the embryonic development of insect eggs. His tiny office was stacked with books and display cases of butterflies. Haga, who is aged 74, was wearing large glasses with a thin black rim that framed his face geometrically. I noticed immediately that he is a very shy man, soft and modest – and was rather nervous of being interviewed.
But Haga’s timidity is only social. In origami he is a rebel. Choosing to stay out of the origami mainstream, he has never felt constrained by any conventions. For example, according to the rules of traditional Japanese origami, there are only two ways to make the first fold. Both are folding it in half – either folding along a diagonal, bringing two opposing corners together, or along