Alex's Adventures in Numberland - Alex Bellos [72]
Pi is a mathematical concept that has been studied for thousands of years, yet it holds many secrets. There have been no great advances in understanding its nature since transcendence was proved almost a century and a half ago.
‘We actually don’t know about most of this stuff,’ said Gregory.
I asked if there would ever be a new advance in understanding pi.
‘Of course, of course,’ said Gregory, ‘There are always advances. Mathematics moves forward.’
‘It will be more miraculous but it won’t be nice,’ David said.
Nineteen sixty-eight was a year of counter-cultural uprisings around the world, and Britain was not immune to such generational upheaval. In May the Treasury announced the introduction of a revolutionary new coin.
The 50p piece was designed to replace the old ten-shilling note as part of the switch from imperial to decimal currency. Yet what set the coin apart was not its denomination, but its unorthodox shape.
‘It’s no ordinary coin,’ exhorted the Daily Mirror. ‘Why, the Decimal Currency Board even go so far as to call it a “multilateral-curve heptagon”.’ Never before had a country introduced a seven-sided coin. And never before had a nation been so outraged over the aesthetics of a geometric shape. Leading the foment was retired army colonel Essex Moorcroft, of Rosset, Derbyshire, who formed the Anti-Heptagonists. ‘We take our motto from Cromwell’s heartfelt cry, “Take away this bauble”. I have founded the society because I believe our Queen is insulted by this heptagonal monstrosity,’ he said. ‘It is an ugly coin and an insult to our Sovereign, whose image it bears.’
Nonetheless, the 50p entered circulation in October 1969, and Colonel Moorcroft did not take to the barricaes. Indeed, by January 1970 The Times reported that ‘the curvaceous heptagon seems to have won for herself some affection’. Today the 50p piece is considered a distinctive and cherished part of British heritage. When a 20p piece was introduced in 1982, it too was heptagonal.
The 50p and 20p pieces are, in fact, design classics. Their seven-sided shape means that they are easily distinguished from circular coins, helping in particular the blind and partially sighted. They are also the most thought-provoking coins in circulation. The circle is not the only interesting round shape in mathematics.
A circle can be defined as a curve for which every point is equidistant from a fixed point, the centre. This property has many practical applications. The wheel – generally trumpeted as humanity’s first great invention – is the most obvious. An axle attached to the centre of a wheel will stay at a fixed point above the ground when the wheel rotates smoothly along a surface, which is why carts, cars and trains run smoothly without bobbing up and down.
For the transportation of very heavy loads, however, an axle might not take the weight. One alternative is to use rollers. A roller is a long tube with a circular cross-section, laid out on the ground. If a heavy object with a flat base (such as a giant cuboidal piece of stone intended for a pyramid) is put on several rollers, then it can be pushed smoothly over the rollers, with new rollers being put in front as it inches forward.
Curves of constant width include the Reuleaux triangle (left) and the multilateral-curve heptagon, better known as the 50p piece (centre).
The crucial feature of a roller is that the distance between the ground and the top of the roller is always the same. This is obviously the case with a circular cross-section, since the width of a circle (the diameter) is always the same.
Do all rollers have to have a circular cross-section? Are there any other shapes that will work? It may appear counter-intuitive,