Alex's Adventures in Numberland - Alex Bellos [61]
The first genius whose extreme passion for discovery about pi does justice to the aspirations of Givenchy’s aftershave is equally the man who took the most famous bath in the history of science. Archimedes slipped into the tub and noticed that the volume of water he displaced was equal to the volume of his own body under the water. He instantly realized that he could therefore find the volume of any object by submersing it, in particular the crown of the King of Syracuse, and so would be able to ascertain if that piece of royal bling was made of pure gold or not by working out its density. (It wasn’t.) As a result, he ran naked into the streets shouting ‘Eureka! [I have found it!]’, thus displaying – for the citizens of Syracuse, at least – the eternal masculine. Archimedes loved to grapple with problems in the real world, unlike Euclid, who dealt uniquely in abstractions. His many inventions were said to have included a giant catapult and a system of huge mirrors that reflected the sun’s rays with such intensity that they set Roman ships ablaze during the Siege of Syracuse. He was also the first person to come up with an apparatus to capture pi.
To do so he first drew a circle, and then he constructed two hexagons – one that he fitted inside the circle, and one that he put outside it, as in the diagrams below. This already tells us that pi must be somewhere between 3 and 3.46, which is determined by calculating the perimeters of the hexagons. If we let the diameter of the circle be 1, then the perimeter of the inner hexagon is 3, which is less than the circumference of the circle, which is pi, which is less than the perimeter of the outer one, which is 3 , or 3.46 to two decimal places. (The way that Archimedes calculated this value was by using a method that was essentially a fiddly precursor to trigonometry, and which is too complicated to go into here.)
So, 3 < pi < 3.46.
Now if you were to repeat the calculation using two regular polygons with more than six sides, you would get a narrower bound for pi. This is because the more sides that the polygons have, the closer their perimeters are to the circumference, as we can see in the diagram above that uses a 12-sided polygon. The polygons act like walls closing in on pi, squeezing it from above and from below, between narrower and narrower limits. Archimedes started with a hexagon and eventually constructed polygons of 96 sides, allowing him to calculate pi as follows: 3 < pi < 3
This translates into 3.14084 < pi < 3.14289, an accuracy of two decimal places.
But pi hunters weren’t about to stop there. In order to get closer to the number’s true value, all that was required was to create polygons with more sides. Liu Hui, in third-century China, employed a similar method, using the area of a polygon of 3072 sides to pin pi to five decimal places: 3.14159. Two centuries later Tsu ChungChih and his son Tsu Keng-Chih went one digit further, to 3.141592, with a polygon of 12,288 sides.
The Greeks and the Chinese were hampered by cumbersome notation. When mathematicians were eventually able to use Arabic numerals, the record tumbled. In 1596 the Dutch fencing master Ludolph van Ceulen used a souped-up polygon of 60 × 229 sides to find pi to 20 decimal places. The pamphlet on which he printed his result ended with the words: ‘whoever wants to, can come closer’, and no one felt the urge as strongly as he did. He went on to calculate pi to 32 and then 35 decimal places – which were engraved on his tombstone. In Germany, die Ludolphsche Zahl, Ludolph’s number, is still understood as a term for pi.
For 2000 years the only way to pinpoint pi with accuracy was by using polygons. But, in the seventeenth century, Gottfried Leibniz and John Gregory ushered in a new age of pi appreciation with the formula:
In other words, a quarter of pi is equal to one minus a third plus a fifth minus a seventh plus a ninth