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**.

[Main Index] | [Pi Index]

Neal Carothers - carother@bgnet.bgsu.edu