Mathematics

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | C | D | E | F | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 24 | 23 | 24 | 18 | 17 | 22 | 15 | 18 | 22 | 24 | 18 | 21 | 23 | 15 | 24 | 14 |

1 | 20 | 23 | 15 | 19 | 21 | 17 | 20 | 17 | 21 | 30 | 20 | 22 | 18 | 24 | 24 | 22 |

2 | 18 | 28 | 22 | 24 | 21 | 17 | 19 | 11 | 23 | 25 | 23 | 18 | 9 | 18 | 28 | 22 |

3 | 16 | 18 | 25 | 19 | 20 | 16 | 23 | 23 | 20 | 17 | 18 | 20 | 19 | 17 | 25 | 20 |

4 | 22 | 22 | 21 | 15 | 16 | 24 | 20 | 22 | 17 | 15 | 19 | 16 | 18 | 12 | 16 | 15 |

5 | 24 | 20 | 25 | 18 | 22 | 24 | 22 | 26 | 22 | 16 | 16 | 16 | 19 | 21 | 23 | 20 |

6 | 31 | 20 | 23 | 18 | 16 | 20 | 24 | 14 | 17 | 21 | 22 | 20 | 28 | 8 | 22 | 22 |

7 | 19 | 24 | 20 | 21 | 19 | 20 | 14 | 21 | 13 | 24 | 16 | 20 | 13 | 15 | 20 | 16 |

8 | 16 | 24 | 16 | 17 | 23 | 19 | 17 | 17 | 22 | 19 | 16 | 20 | 15 | 18 | 23 | 23 |

9 | 14 | 20 | 17 | 20 | 25 | 19 | 25 | 16 | 19 | 25 | 21 | 29 | 22 | 14 | 17 | 21 |

A | 23 | 24 | 15 | 26 | 17 | 26 | 17 | 19 | 13 | 19 | 18 | 20 | 16 | 27 | 19 | 21 |

B | 17 | 17 | 20 | 26 | 14 | 31 | 27 | 17 | 22 | 23 | 21 | 15 | 19 | 20 | 11 | 16 |

C | 17 | 16 | 18 | 21 | 16 | 22 | 22 | 24 | 20 | 18 | 22 | 12 | 21 | 10 | 13 | 16 |

D | 20 | 15 | 21 | 17 | 9 | 19 | 20 | 11 | 13 | 19 | 19 | 23 | 14 | 14 | 13 | 23 |

E | 22 | 18 | 18 | 15 | 22 | 19 | 20 | 19 | 24 | 19 | 27 | 18 | 20 | 17 | 22 | 22 |

F | 19 | 21 | 26 | 22 | 12 | 19 | 19 | 20 | 17 | 10 | 24 | 26 | 14 | 20 | 24 | 18 |

This shows the first 5000 hexadecimal digits of pi. The rows represent values of d(n), the columns represent values of d(n+1). Each entry in the table shows the frequency of events which turn a given d(n) to a given d(n+1). This convention is adopted throughout these webpages.

The c++ program, pifreq.cpp (6K) can be used to generate phase spaces for other regions of pi.

Pseudo-phase spaces are used to search for patterns within number streams. At the moment, I doubt whether the above phase space could be distinguished from the phase space of a random number generator. I intend to include a Kolmogorov-Smirnoff test at some point. As a rough estimate, I think it would need at least 16M digits to produce any useable data.

{number of digits N=16M,

random walk error E = sqr(N) = 4K,

expected number per interval EXP(n) = N/256 = 64K}

Calculating the digits takes time, an obvious next step would be to save the results so that new digits could be added to the phase space.

That's enough for now.

Links at other sites...

Created 27/9/99

Modified 6/10/99