Of course, six is a ridiculously small number of steps, and the resulting hexagon is a crude caricature of a circle, but Archimedes was just getting started. Once he figured out what the hexagon was telling him, he shortened the steps and took twice as many of them. Then he kept doing that, over and over again.
A man obsessed, he went from 6 steps to 12, then 24, 48 and ultimately 96 steps, using standard geometry to work out the ever-shrinking lengths of the steps to migraine-inducing precision. By using a 96-sided polygon inside the circle, and also a 96-sided polygon outside the circle, he ultimately proved that pi is greater than 3 + 10/71 and less than 3 + 10/70.
Take a moment to savor the result visually:
3 + 10/71 < π < 3 + 10/70.
The unknown value of pi is being trapped in a numerical vise, squeezed between two numbers that look almost identical, except the first has a denominator of 71 and the last has a denominator of 70. By considering polygons with even more sides, later mathematicians tightened the vise even further. Around 1,600 years ago, the Chinese geometer Zu Chongzhi pondered polygons having an incredible 24,576 sides to squeeze pi out to eight digits:
3.1415926 < π < 3.1415927.
By allowing the number of sides in the polygons to increase indefinitely, all the way out to infinity, we can generate as many digits of pi as we like, at least in principle.
In taming infinity, Archimedes paved the way for the invention of calculus 2,000 years later. And calculus, in turn, helped make the world modern. Archimedes’s mathematical strategy is used in computer-generated movies, approximating Shrek’s smooth belly and trumpet-like ears with millions of tiny polygons. The smooth glide of an Ella Fitzgerald song is digitally represented in streaming audio by an enormous number of bits.
In every field of human endeavor, from reconstructive facial surgery to the simulation of air flowing past a jet’s wing, billions of tiny, discrete elements stand in for an inherently smooth and analog reality. It all began with the computation of pi. Pi represents a mathematical limit: an aspiration toward the perfect curve, steady progress toward the unreachable star. It exists, clear as night, with no end in sight.