Here's how the ancient Greeks found the first few digits of pi.
We start with a circle of diameter equal to one, so that, by definition, its circumference will equal π. Using some basic geometry and trigonometry, Archimedes observed that the length of each of the sides of the inscribed blue hexagon would be 1/2, and the lengths of the sides of the circumscribed red hexagon would be 1/√3.
The perimeter of the inscribed blue hexagon has to be smaller than the circumference of the circle, since the hexagon fits entirely inside the circle. The six sides of the hexagon all have length 1/2, so this perimeter is 6 × 1/2=3. Similarly, the circumference of the circle has to be less than the perimeter of the circumscribed red hexagon, and this perimeter is 6 × 1/√3, which is about 3.46.Fractals are math's trippiest concept, and they get even weirder when used to solve a puzzle involving the British coast
Archimedes, through some further clever geometry, figured out how to estimate the perimeters for polygons with twice as many sides. He went from a 6-sided polygon, to a 12-sided polygon, to a 24-sided polygon, to a 48-sided polygon, and ended up with a 96-sided polygon. This final estimate gave a range for π between 3.1408 and 3.1428, which is accurate to two places.
Archimedes' method of approximating π with polygons, and similar techniques developed in China and India, would be the dominant way mathematicians would approach the calculation of the digits π for centuries.from calculus, and our ever-faster computers allow us to find trillions of digits of π.
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