An Example


Let's try Archimedes' algorithm

starting with inscribed and circumscribed squares


Here are the results of 12 iterations of the algorithm.

2 2.8284271247461901 4
3 3.0614674589207182 3.3137084989847604
4 3.1214451522580523 3.1825978780745281
5 3.1365484905459393 3.1517249074292561
6 3.1403311569547529 3.1441183852459043
7 3.1412772509327729 3.1422236299424568
8 3.1415138011443011 3.1417503691689665
9 3.1415729403670914 3.1416320807031818
10 3.1415877252771597 3.1416025102568089
11 3.1415914215112 3.1415951177495891
12 3.1415923455701177 3.1415932696293073
13 3.1415925765848727 3.1415928075996446
14 3.141592634338563 3.1415926920922544

These computations were done using a spreadsheet with only limited accuracy (ostensibly, 15 decimal places). Nevertheless, notice that either of the last two entries agree with the actual value of to at least 6 places. Another half dozen iterations would yield to 9 places. Not bad!

A better starting estimate would obviously have helped our situation here. Archimedes started with regular hexagons (an inscribed perimeter of 3 and a circumscribed perimeter of ).

Archimedes' algorithm falls is one of a larger class of related algorithms.


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Neal Carothers - carother@bgnet.bgsu.edu