starting with inscribed and circumscribed squares
Here are the results of 12 iterations of the algorithm.
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.
Neal Carothers - email@example.com