Alex's Adventures in Numberland - Alex Bellos [42]
Thinking inside the box: Jeannine Mosely and her Menger Sponge.
The Menger sponge is a brilliantly paradoxical object. As you continue the iterations of taking out smaller and smaller cubes the volume of the sponge gets smaller and smaller, eventually becoming invisible – as though the woodworm have eaten the whole lot. Yet each iteration of cube removal also makes the surface area of the sponge increase. By taking more and more iterations you can make the surface area larger than any area you want, meaning that as the number of iterations approaches infinity, the surface area of the sponge also approaches infinity. In the limit, the Menger sponge is an object with an infinitely large surface area that is also invisible.
Mosely constructed a level-three Menger sponge – in other words, a sponge after three iterations of cube removal (F). The project took her ten years. She enlisted the help of abut 200 people, and used 66,048 cards. The finished sponge was 4ft 8in high, wide and deep.
‘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