When Homer Simpson's superhero alter-ego the Pie Man first threw a fruit-filled pastry in anger a few years ago, Springfield's local news anchor was moved to observe: “Pie! Tasty dessert, tricky math thing, and now sword of righteousness.”

He wasn't the first to make the connection between sticky puddings and the most intriguing number in maths - since 1988 mathematicians across the land have been celebrating pi day each year by tucking into a feast of them. The number has obsessed generations of mathematicians for millennia, and not because it's an excuse to eat pudding.

Pi is related to one of the most important geometric objects in nature: the circle. The number defines the ratio of the circumference of a circle to its diameter. So if a circle is one metre across, the number of metres it takes to go round the outside of the circle is 3.14159... and then the numbers spiral off to infinity in a dizzying dance of digits. And it's the digits of pi that define the date for the celebration of pi day. The first digit gives you the month: 3 for March. The next two give you the date: the 14th. The next three give you the time that celebrations kick off, hence 1.59. Most mathematicians interpret the time as pm rather than am (the afternoon is a better time for pie).

Calculating an exact value has obsessed mathematicians since ancient times. The Rhind Papyrus, written by the Egyptian scribe Ahmes in about 1650BC, approximates pi as 256/81 or roughly 3.16. Not bad for a first estimate.

As mathematics developed, so more cultures had a go at trying to capture this important number. The Ancient Greek mathematician Archimedes used a 96-sided shape to estimate that pi lay between 223/71 and 22/7. This is where we get the approximation that most engineers use for pi of 22/7. In fact, engineers celebrate pi day on July 22, but then they never did care quite so much about precision as mathematicians.

It was in Kerala in the south of India in the 15th century that a mathematician called Madhava came up with an exact formula for pi. By successively adding and subtracting different fractions Madhava discovered that he could capture pi precisely. If you start with the number 4 and then take away four thirds then add four fifths then take away four sevenths then add four ninths and you kept on doing this, each time alternating between 4 divided by the next odd number, when you've done this infinitely many times you'll hit pi exactly. The formula, although beautiful, is not very practical. You have to add up a lot of fractions before it starts honing in on pi.

A curious way to calculate pi was discovered in 1777 by the Compte de Buffon, a French naturalist. If you take a needle and a page of lined paper, on which the lines are separated by the same distance as the length of the needle, then if you continually throw the needle on to the page, the proportion of times the needle will cross a line when it lands is 2 divided by pi. This gives you a physical way of calculating pi. Keep chucking down the needle and the more you toss the better your estimate for pi should be.

Mathematical methods and computing power mean that we know pi to a staggering trillion digits. Of course no one needs to know so many values for the practical purpose of calculating circumferences. You need to know only 39 digits to calculate the circumference of a circle the size of the observable universe to the precision comparable to the size of a hydrogen atom.

The use of decimal numbers is anyway a very human construct, dependent on the fact that we have ten fingers. If we had evolved with a different anatomy, say with eight fingers like Homer Simpson, pi would still be the same expression of the unchanging ratio between the circumference and diameter of a circle but using powers of eight rather than powers of ten as our natural base would mean pi began 3.110375...

So while we'll have to wait till Saturday for a mathematical excuse to tuck into tarts, the Pie Man should be celebrating today. But even if he's metaphorically number-munching on a different day, as mathematicians we'll all be agreeing with him when his eyes glaze over and he sighs, “Mmm... pi... .”

Marcus du Sautoy presents Sunday Feature on Radio 3 at 9.30pm on Sunday, on the art and science of the Baroque

Conundrum

A duck sits in the middle of a perfectly circular pond, the radius of which is 40 metres. On the edge of the pond is a hungry fox that can't swim. The fox can run four times faster than the duck can swim. Will the fox always be able to catch the duck the moment it reaches the shore?

Answer: Not if the duck swims just short of 10m from the centre of the pond, and then swims round a circle concentric with the edge of the pond. As its circumference is just less than a quarter of the circumference of the pond, the duck can swim round its circle faster than the fox runs round the edge. The duck then keeps swimming until it is diametrically opposite to the fox, before heading to shore. It must cover just over 30m. The fox in that time can run just over 120m. But to reach the duck it must run a distance of pi x 40m - approximately 125m. So the duck (just) makes it first. Of course, it could have flown.

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