Cornering π: from inscribed polygons to Mādhava's exact series

0:00Draw a circle, and inside it draw a six-sided figure with all its corners on the rim. Every one of its six sides is exactly the radius - not nearly, exactly - because each slice from the centre has two radii and a sixty degree corner, which makes it equal-sided. So the way round that figure is six radii, which is three diameters, with nothing rounded and no constant needed.

0:29And the circle is longer than anything drawn inside it. So the number we are after is bigger than three. Not probably bigger. Bigger, provably, from a picture you can draw in a minute. Which means every civilisation that used three was already using a number known to be too small. The oldest written value we have is about four thousand years old, and it is three and an eighth.

0:59Three point one two five. That is low by about one and seven tenths hundredths - and hold on to that number, because it is going to come back at the worst possible moment. It is not a bound. Nobody could say whether the true value was above it or below it, and nothing about it says how wrong it might be. It is a figure that worked well enough for building.

1:25Turning that into mathematics took one more idea. The idea is Archimedes', and it is this. A figure drawn inside the circle is shorter than the circle. A figure drawn outside it, with every side just touching, is longer. So put both in the same picture and you have not one estimate but two bounds, one from below and one from above, with the answer trapped between them. That is a completely different kind of knowledge.

1:59You no longer have a number you hope is close. You have a pair of numbers, and a guarantee. And every improvement from then on came from the same move: replace the curve by something straight you can actually measure, and squeeze. Now do the outer six-sided figure properly. Drop a perpendicular from the centre to one of its sides. That perpendicular is the radius, one, and it lands in the middle of the side.

2:30For a six-sided figure the distance from the centre out to a corner is the same as the side itself - call it a. Now you have a right-angled triangle with legs a over two and one, and hypotenuse a. Squares on the two legs add to the square on the third, so a squared over four plus one equals a squared, which gives a squared equal to four thirds.

2:56Six of those sides make a way round of four root three, over a diameter of two. So the number is less than two root three, about three point four six. Three below, two root three above, from two hexagons. Now double the corners. Six becomes twelve, twelve becomes twenty-four, and each time both bounds tighten. Archimedes stopped at ninety-six sides on each figure and published three and ten seventy-firsts below, three and a seventh above.

3:32The whole interval is one four-hundred-and-ninety-seventh wide. And here is what that famous computation actually settles: the two ends agree on two decimal places. Three point one four. The third digit is still open. Ninety-six sides sounds like a lot and buys less than people assume. Push the same method harder and it keeps working, and that is the trouble with it. Liu Hui and then Zu Chongzhi drove the side count to twenty-four thousand five hundred and seventy-six, which pins seven decimal places.

4:14But look at the price. Two hundred and fifty-six times as many sides bought a bracket sixty-six thousand times narrower - because the width falls with the SQUARE of the side count, and with nothing better than that. Eleven decimal places would need eighteen doublings and a figure with over a million and a half sides. And no matter how far you push it, every one of those figures is strictly shorter than the circle from the inside and strictly longer from the outside.

4:48Never equal. The method has no last step. Zu Chongzhi left two fractions. One convenient: twenty-two sevenths, high by about thirteen ten-thousandths. One astonishing: three hundred and fifty-five over one hundred and thirteen, high by under three ten-millionths. Take every denominator below fifteen thousand and pick the nearest fraction each time - not one of them beats it, and the only ones that match it are multiples of its own denominator.

5:25There is a wrinkle in the convenient one, though. Twenty-two sevenths is exactly three and a seventh, which is Archimedes' upper bound from seven centuries earlier. It was never a new number. What was new was the machinery that produced the other one beside it. In the year four hundred and ninety-nine, Aryabhata gave the value three point one four one six. It is high by about seven millionths - ten times better than Ptolemy's, three and a half centuries earlier.

6:01But the number is not the interesting part. The interesting part is what they called it. They labelled it asanna: approaching. Not the value. A value that comes close. That single word is doing more work than most of the arithmetic in this story, because it is the difference between having an answer and knowing what kind of thing your answer is. And now the moment this story is usually told without.

6:32In six hundred and twenty-eight, Brahmagupta adopted the square root of ten. It is easy to carry through algebra, which is why they wanted it. It is also high by about two hundredths. Line that up against the value from four thousand years earlier - three and an eighth, low by one and seven tenths hundredths - and the newer value is FARTHER from the truth than the older one.

7:00Worse: it sits above three and a seventh, an upper bound that had been in print for nearly nine hundred years, so it was already known to be out of range when it was taken up. Lay all the named values out by date and ask how often a later one is worse than an earlier one, and it happens in seven of the fifteen pairs. Accuracy was one goal among several, and not always the winning one.

7:30Every advance so far came from the same move, and every one of them hit the same wall: a polygon is not a circle, and no number of sides changes that. Madhava, in Kerala, around fourteen hundred, stopped trying. They did not replace the circle by a straighter shape. They wrote down an unending list of fractions. A quarter of the number equals one, minus a third, plus a fifth, minus a seventh, and on for ever in the odd numbers.

8:04That is not an approximation with an error to be quoted. It is an equation. The number IS what that list adds up to - and this is the first time in the whole story that anyone has been able to write the thing down exactly rather than corner it. So let us actually add it up. One term gives four. Two terms give two point six seven. Three give three point four seven, four give two point nine zero, five give three point three four.

8:39They jump over the answer and back, closing in from alternate sides and never landing on it. How fast? After n terms the gap left is close to half the next term - fifty two hundredths at ten terms, and a half by a hundred. So the gap shrinks like one over two n, and to reach eleven decimal places you would need four hundred thousand million terms. Exact and unusable, at the same time.

9:11Which tells you something about the eleven places Madhava is credited with. Three point one four one five nine two six five three five eight. Nobody added four hundred thousand million fractions. There were correction terms, and faster relatives of the same series, and that is what the following centuries were about - Nilakantha, then Machin, then Ramanujan, then machines running the Chudnovskys' work out to hundreds of trillions of digits.

9:45Exactness and speed are different virtues, and the series won the first one outright. As for the letter: William Jones picked it in seventeen hundred and six, because perimetros, the Greek for the way round, starts with it. Euler made it stick. Four thousand years to corner a number, and then a formula that names it exactly - and it is still shorter to write than the thing it means.