Alex’s Adventures in Numberland (17 page)

‘For a long time I wondered if I was doing something completely and utterly ridiculous,’ she told me. ‘But when I had got it done I stood next to it and realized the scale gave it a grandeur. A particularly wonderful thing is that you can stick your head and shoulders into the model and see this amazing figure from a viewpoint that you have never seen before.’ It was endlessly fascinating to look at because the more she zoomed in, the more she saw the patterns repeating themselves. ‘You simply look at it and it doesn’t need to be explained. You can understand it just by looking at it. It is an idea made solid; math made visual.’ The business-card Menger sponge is a beautifully crafted object that creates an emotional and intellectual response. It belongs just as much to geometry as it does to art.

Although origami was originally a Japanese invention, paper-folding techniques also developed independently in other countries. A European pioneer was the German educator Friedrich Fröbel, who used paper-folding in the mid nineteenth century as a way of teaching young children geometry. Origami had the advantage of allowing kindergarten pupils to feel the objects created, rather than just see them in drawings. Fröbel inspired the Indian mathematician T. Sundara Row to publish
Geometric Exercises in Paper Folding
in 1901, in which he argued that origami was a mathematical method that in some cases was more powerful than Euclid’s. He wrote that ‘several important geometric processes…can be effected much more easily than with the compass and ruler’. But even he did not anticipate just how powerful origami can actually be.

In 1936 Margherita P. Beloch, an Italian mathematician at the University of Ferrara, published a paper that proved that starting with a length L on a piece of paper, she could fold a length that was the cube root of L. She might not have realized it at the time, but this meant that origami could solve the problem given to the Greeks at Delos, where the oracle demanded that the Athenians double the volume of a cube. The Delian Problem can be rephrased as the challenge to create a cube with sides that are
– the cube root of two – times the side of a given cube. In origami terms, the challenge is reduced to folding the length
from the length 1. Since we can double 1 to get 2 by folding the length 1 on itself, and we can find the cube root of this new length following Beloch’s steps, the problem was solved. It also followed from Beloch’s proof that any angle could be trisected – which cracked the second great unsolvable problem of antiquity. Beloch’s paper, however, remained in obscurity for decades, until, in the 1970s, the maths world began to take origami seriously.

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 the midline, bringing adjacent corners together. These are known as the ‘primary creases’ – the diagonals and the midlines of the square.

Haga decided to be different. What if he folded a corner on to the midpoint of a side? Ker-azee! He did this for the first time in 1978. This simple fold had the effect of opening the curtains on to a sublime new world. Haga had created three right-angled triangles. Yet these were not any old right-angled triangles. All three were Eygptian, the most histct eggsand iconic triangle of them all.

Haga’s theorem: A, B and C are Egyptian.

Feeling the thrill of discovery, but having no one to share it with, he sent a letter about the fold to Professor Koji Fushimi, a theoretical physicist known to have an interest in origami. ‘I never got a reply,’ said Haga, ‘but then all of a sudden he wrote an article in a magazine called
Mathematics Seminar
referring to Haga’s Theorem. That was the substitute for his reply.’ Since then Haga has given his name to two other origami ‘theorems’, although he says he has another 50. He tells me this not as evidence of arrogance but as a measure of how the area is so rich and untapped.

In Haga’s Theorem a corner is folded on to the midpoint of a side. Haga wondered if anything interesting might be revealed if he folded a corner on to a random point on the side. Deciding to demonstrate this to me, he took a blue piece of square origami paper and marked an arbitrary point on one of the sides with a red pen. He folded one of the opposite corners on to this mark, leaving a crease, and then unfolded it. He then folded the other opposite corner on to the mark to make a second crease, leaving the square now with two separate intersecting lines.

Haga’s other theorem.

Haga showed me that the intersection of the two folds was always on the middle line of the paper, and that the distance from the arbitrary point to the intersection was always equal to the distance from the intersection to the opposite corners. I found Haga’s folds totally mesmerizing. The point had been chosen randomly, and was off-centre. Yet the process of folding behaved like a self-correcting mechanism.

Haga wanted to show me a final pattern. The name he chose for this discovery sounded like a haiku:
an arbitrarily made ‘mother line’ bears eleven wonder babies
.

Mother line showing seven of her eleven wonder babies.