Happy Pi Day

“Probably no symbol in mathematics has evoked as much mystery, romanticism, misconception and human interest as the number pi.” – William L. Schaaf

The ancient Egyptians started the hunt for the mysterious number 4,000 years ago. In his book The History of Pi (1971), Petr Beckman speculates that the ancient Egyptians drew a circle, and then measured the 1e06c2ec-48f2-4e90-a3bb-2c2c2f63c863circumference and diameter with rope. They determined that pi was a sliver greater than three, and came up with the value 3 1/8 or 3.125.

But the ancient Egyptians didn’t stop at rope measurements. According to the Rhind Papyrus, which was written by an Egyptian scribe named Ahmes around 1650 BCE, he claimed: “Cut off 1/9 of a diameter and construct a square upon the remainder; this has the same area as the circle.” For us non math geeks out there, Ahmes basically said pi = 4(8/9)2 = 3.16049, which was pretty accurate for a mathematician three thousand years ago.

The ancient Greeks, like most things, built upon what the ancient Egyptian mathematicians had done and made two revolutionary leaps forward. Antiphon and Bryson (who both hailed from the city-state Heraclea) thought of the clever idea to inscribe a polygon inside a circle, find its area, and then double the sides repetitively.

This leads us to the main who most people call the father of pi, Archimedes (from the city-state, Syracuse). archimedes-1-sizedWhere Antiphon and Bryson failed, Archimedes succeeded.  Archimedes focused on the polygons’ perimeters as opposed to their areas, so that he approximated the circle’s circumference instead of the area. He started with an inscribed and a circumscribed hexagon then doubled the sides four times to finish with two 96-sided polygons.

To quote Archimedes himself in his work entitled, Measurement of a Circle: Given a circle with radius, r = 1, circumscribe a regular polygon A with K = 3(2n-1 sides and semi perimeter and inscribe a regular polygon B with K = 3(2n-1 sides and semi perimeter bn. This result in a decreasing sequence a1, a2, a3… and an increasing sequence b1, b2, b3… with each sequence approaching pi. We can use trigonometric notation (which Archimedes did not have) to find the two semi perimeters, which are: an = K tan ((/K) and bn = K sin ((/K). Also: an+1 = 2K tan ((/2K) and bn+1 = 2K si n ((/2K). Archimedes began with a1 = 3 tan ((/3) = 3(3 and b1 = 3 sin ((/3) = 3(3/2 and used 265/153 < (3 < 1351/780. He calculated up to a6 and b6 and finally reached the conclusion that 3 10/71 < b6 < pi < a6 < 3 1/7.

Well, I didn’t get any of that. But, for the next few hundred years most people accepted Archimedes’ James_Gregorycalculations. That was until Archimedes’ calculations were refined by James Gregory in 1672 and Gottfried Leibniz in 1685. By 1750, mathematicians could express pi in an infinite series.

Now we have computers to do the work for us, but it all comes back to what the ancient Egyptians started with a piece of rope and drawing a circle in the sand.

Sources:

Archimedes. Measurement of a Circle. From Pi: A Source Book.

Beckman, Petr. The History of Pi. The Golem Press. Boulder, Colorado, 1971.

Wilson, David. History of Mathematics, Rutgers, Spring 2000

About Joshua G. Silverman