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.
Neal Carothers - carother@bgnet.bgsu.edu