Alex's Adventures in Numberland - Alex Bellos [41]
Another Eastern culture has long embraced the joys of geometric shapes. Origami, the art of paper-folding, evolved from the custom of Japanese farmers thanking the gods at harvest time by making an offering of some of their crops on a piece of paper. Rather than placing the produce on a flat sheet, they would make one diagonal fold in the paper to give the offering a human touch. Origami flourished in Japan over the last few hundred years as an informal pastime, the kind of thing parents did with their children for fun. It fitted in perfectly with the Japanese love of artistic understatement, attention to detail and economy of form.
The Sri Yantra.
Business-card origami sounds like the ultimate Japanese invention, uniting two national passions. In fact, the practice is abhorrent to them. The Japanese see business cards as an extension of the individual, so playing with them is considered grossly offensive, even with origamic intent. When I tried to fold one in a Tokyo restaurant I was almost ejected for my antisocial behaviour. In the rest of the world, however, business-card origami is a modern paper-folding subgenre. It dates back more than a hundred years, to the (now obsolete) practice of visiting-card origami.
A simple example is to fold a business card so that the bottom right corner meets the top left corner, and then fold the overlaps, as shown overleaf. Repeat this with another business card, except this time fold the bottom left to the top right. You now have two pieces that can be slotted together to form a tetrahedron. It is a winning way, so I am told, to hand over your business card during mathematics conferences.
The octahedron can be made from four cards, and an icosahedron with ten of them. It is also easy to make a fourth Platonic solid – the cube. Put two cards on top of each other like a plus sign and fold the flaps as shown above. This creates the shape of a square. Six cards folded in this way slot together to form a cube, although the flaps are on the outside. You need another six cards to slide on to each face in order to make the cube clean.
The Zen master of business-card origami is Jeannine Mosely, a software developer from Massachussetts. A few years ago she found herself with 100,000 cards in her garage – she inherited three batches from her colleagues at work, the first time when the company changed its name, the second when the company moved addresses, and then again when it was discovered that the new cards all had a typo. You can make a lot of tetrahedrons with 100,000 business cards. Yet Mosely had much grander ambitions than the Platonic solids. Why limit herself to the ancient Greeks? Had 2000 years of geometry not produced a more exciting 3-D shape? With her resources Mosely felt ready to tackle the ultimate challenge of her art, the Menger sponge.
Before we get to the Menger sponge, I need to introduce the Sierpinski carpet. The bizarre shape was invented by the Polish mathematician Waclaw Sierpinski in 1916. You start with a black square. Imagine it is made of nine identical subsquares, and remove the central one (figure A). Now for each of the remaining subsquares, repeat the operation – that is, imagine they are made of nine subsquares and remove the central one (B). Repeat this process again (C). The Sierpinski carpet is what you get if you continue this process ad infinitum.
In 1926 the Austrian mathematician Karl Menger came up with the idea of a three-dimensional version of the Sierpinski carpet,