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January 10, 2006

Fascinating "Pi"

I think the number Pi is known to all of us. It is the all important factor necessary when calculating the area of a circle and circles have been fascinating people for thousands of years.

Ancient civilizations knew two perfect geometric figures: square and circle. Both figures were of mystical importance and a circle in particular because it has no beginning and no end like the sky that encircles the earth (or, as some believed, the disc). Both sun and moon are of circular shape as well and, owing to an apparent whim of the gods, they appear to be of the same size, as the moon will completely cover the sun during an eclipse. The square on the other hand is like a symbol for the four directions North, South, West, and East. With the help of a ruler and a pair of compasses all other geometric figures can be derived from these two: triangles, polygons, trapezoid, rhombus, cube, pyramid, sphere and cone. If you put two squares symmetrically into a circle and the diagonals of the squares equal the diameter of the circle this will result in an octagon. Octagons have quite often been used as a layout for ecclesiastical buildings.

By about 2000 B.C. clever mathematicians could calculate the area of plane geometrical figures and the volume of three dimensional ones like cubes, but the area of a circle eluded them. If the circle wasn't too big they could use a piece of string to measure the circumference, but, its area, was a totally different matter. The first idea was to find a geometrical procedure (only ruler and a pair of compasses allowed!) to turn a circle into a square of the same area - easier said than done! The first approach was to put a square into the circle. Its diagonal equals the diameter of the circle. One glance tells you that the area of the square is smaller than the area of the circle. You can also put a second square around the circle so that the length of the side will equal the diameter of the circle. Now the area of the square is of course much bigger than the that of the circle. Clearly a different approach needs to be taken.

Around the same time mathematicians found out that the ratio between circumference and diameter is constant regardless of how big a circle is. The circumference is always slightly more than three times the diameter of the circle. So Pi (the Greek letter equivalent to the Latin "P" was first used in the 18th century) was slightly larger than 3. "Slightly larger" meant between 1/8 and 1/4 of the diameter resulting in 3,125 < Pi < 3,25. But the question was how to determine exactly where between these points is the value of Pi?

If you cut up a circle into an even number of equal segments you can put the segments together to form a rectangle. You can easily calculate the area of a rectangle. The more segments you use the greater your accuracy will be. This is how the equations for determining the circumference (Pi times the diameter) and the area (Pi times the square of the radius) were originally found. The problem was that this didn't help much as the exact value of Pi was still unknown.

By 1800 BC the Egyptians had come to the conclusion that the area of a square whose side equalled 8/9 of a circle's diameter equalled the area of the circle. Well, it does, but with an error of only 0.6%!

Euclid (340-270 BC) calculated the value of Pi to be smaller than 270/70 (3.1428) but bigger than 270/71 (3.1408) which would yield a mean value of 3.1418.

Archimedes (287-212 BC) devised a method of putting polygons inside a circle to calculate the area. A polygon can be divided into isosceles triangles, each triangle having two sides of equal length. Archimedes did not calculate the area of the triangles, but of the length of the bases. Adding up the sum of the bases of the isosceles triangles inside the circle would lead to the circumference of the circle. In this way he calculated Pi to have the value of 3.1419! No better value for Pi was found for hundreds of years.

With the decline of the Roman Empire at about 400 AD, the leading role of scientists from Greece and Alexandria came to an end. Europe plunged into difficult times of wars and religious conflicts which had no place for science or scientific discovery. Small wonder that from now on scientific progress took place in Arabia, India and China. So our story moves to about 500 AD when the Chinese mathematician Tsu Chung Chi used polygons with 24576 sides and calculated a value for Pi of 3.1415929!

Towards the end of the 12th century the Italian mathematician Fibonacci brought a value for Pi of 3.1418 back to Europe from Arabia, reviving interest in mathematics in Europe, and in particular the calculation of the area of circles.

One of the last "polygon calculators" was Ludolph van Ceulen (1539-1610). In his methodology he used polygons with more than 32 billion (!) sides. Using this laborious methodology he was able to calculate the value ofPito 32 decimal places. This will have taken up so much time that he must have spent all his working life calculating Pi! Advances in mathematics from the end of the 17th century also marked the end of all attempts to calculate Pi with geometrical methods.

From now on analytical methods could be employed to calculate Pi and an ongoing competition started between mathematicians to see who could calculate Pi to the highest number of decimal places. In 1948 1000 decimals of Pi were identified, this was pushed to 2000 in 1950 and this has been pushed to a billion today! Such enormous numbers are of no practical value, of course, but seem to provide mathematicians with a lot of fun.

An interesting fact about Pi is that it has an infinite number of decimal places. Moreover, all numbers between 0 and 9 appear with the same frequency but their sequence is totally accidental. If you pick one decimal the mathematical probability for the following decimal to be any number between 0 and 9 has the same "chance" as you would have gambling with throwing a 10-sided dice. This is, in fact, a definition for a transcendental number. Another example for a transcendental number is "e" the base of the system of natural logarithms. Transcendental numbers are always connected with laws of nature which is probably not accidental.

So how big is the value of the proportion between diameter and circumference? How big is infinity? Fascinating, isn't it?

Posted by Mausi at January 10, 2006 10:00 AM

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Mausi, it's my guess that most peiople now know who the intellectual one of us is! I agree that Pi is a fascinating mathematical little conundrum, but one question remains. How did B S Johnson achieve a circle with a circumference where Pi = 3 in going Postal? And what ramifications does this have for the universe at large?

Posted by: The Gray Monk at January 10, 2006 09:01 AM

I love hearing about pi. When I was a teenager, I had it memorized to the 100th decimal place. Yaaa. I'm a geek. Thanks for all the historical information.

Posted by: vw bug at January 10, 2006 02:28 PM

Sorry, Monk, haven't met B S Johnson yet although Going Postal is already waiting for me on the bookshelf. My guess is that he did it the same way the Romans did it: they generally used PI=3 or even 4 for their purposes. Their military successes and also their buildings seem to prove them right, don't they? I am not sure about the ramifications for the universe at large. Perhaps, that there's a lot out there that we don't understand?

Posted by: Mausi at January 10, 2006 08:41 PM

Are you sure 70727 about this?!?

Posted by: Flots Masriach at October 4, 2006 07:19 AM


Posted by: seema at May 20, 2007 11:42 AM