Human ecology
Mathematics

A pseudo-phase-space of pi

0123456789ABCD EF
0 242324181722151822241821 23152414
1 202315192117201721302022 18242422
2 182822242117191123252318 9182822
3 161825192016232320171820 19172520
4 222221151624202217151916 18121615
5 242025182224222622161616 19212320
6 312023181620241417212220 2882222
7 192420211920142113241620 13152016
8 162416172319171722191620 15182323
9 142017202519251619252129 22141721
A 232415261726171913191820 16271921
B 171720261431271722232115 19201116
C 171618211622222420182212 21101316
D 20152117919201113191923 14141323
E 221818152219201924192718 20172222
F 192126221219192017102426 14202418

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.



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Created 27/9/99
Modified 6/10/99