Numbers & Geometry

Puzzle Inspired Story


π — the ratio of a circle’s circumference to its diameter — is the most famous of the numbers. Despite its ubiquity, devising a method for calculating π’s value is not for the fainthearted. The ancient mathematical genius Archimedes of Syracuse used a regular polygon with 96 sides inscribed in a circle to approximate the circle’s circumference. The method yielded the following estimate

223/71 < π < 22/7

which is good to only two decimal digits: π ≈3.14. Working with polygonal perimeters is difficult, as their lengths converge to the length of the circumference very slowly. Also, despite the final approximations appearing deceptively simple, the intermediate calculations require the ability to handle very large numbers. Some historians believe that when Archimedes learned that another great geometer, Apollonius of Perga, found a better approximation of π—implying the latter’s greater mastery with large numbers—the famous Syracusian devised a challenge known as the “Archimedes’ Cattle” problem. The challenge had two parts, and was to demonstrate that Archimedes, its creator, was still unbeaten as far as large numbers were concerned. The elementary problem — still too elaborate to be presented here — asked in a poetically written epigram to “Compute the number of oxen of the Sun, giving thy mind thereto, if thou hast a share of wisdom ...”. It is not surprising that for 22 centuries, no one was able to give a solution to Archimedes’ challenge as the sought number of oxen was shown in modern times to have 206,545 digits.

Unfortunately, sources do not specify that better approximation Appolonius calculated which so enraged Archimedes. It could have been the innocent looking, but very efficient fraction 355/113, which gives π to 6 decimal digits

355/113 = 3.141592...

This remarkable fraction is believed to have been first discovered by Chinese mathematician and astronomer Tsu Ch'ung-Chih in the fifth century A.D. Incredibly, Ch'ung-Chih used a polygon with 12,288 sides to obtain his value.

Further improvements were possible only with the development of modern mathematics, which built on the findings of luminaries of the Renaissance and Enlightenment. Yet even today, with all that knowledge available at one’s fingertips, a full understanding of what goes into calculating, say, 1000 digits of π, requires advanced, graduate-level mathematics.

That said, in October, 2011, two amateurs, Shigeru Kondo, a Japanese hardware engineer, and U.S. student Alexander Yee calculated 10 trillion digits of π on a home PC assembled by Kondo. The feat took them 371 days, 180 of which were spent troubleshooting hardware failures and other problems. The PC generated a lot of heat, raising the temperature of the room it was housed in to 104°F (40°C). The computer required 48 terabytes of hard drive space, and taxed the pair $390 a month in electricity bills.



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