Okay, last one for tonight… since I upset √2 with the previous formula, I thought I’d have a go using square roots, but didn’t find anything yet. Maybe another day.

But… I did find something using cube roots… in fact, the cube roots of 2, 3, 5, and something derived from 7 …. and the accuracy is the best yet, even if the formula is horribly complex. Beauty and the beast, I suppose.

I suppose we could write 10 as cube root of 1000 as well.

The comparison table now looks as follows:

Method | Value | Difference from π/6 | Abs difference |
---|---|---|---|

π/6 | 0.52359877559830 | 0.00000000000000 | 0.00000000000000 |

φ²/5 | 0.52360679774998 | 0.00000802215168 | 0.00000802215168 |

Ave Year | 0.52360779708785 | 0.00000902148956 | 0.00000902148956 |

π – φ² | 0.52355866483991 | -0.00004011075839 | 0.00004011075839 |

10²/φ²π²ⅇ² | 0.52376444099003 | 0.00016566539172 | 0.00016566539172 |

cube roots | 0.52360035017194 | 0.00000157457365 | 0.00000157457365 |