Juggling Pi as a siteswap.—strach
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Re: Pi as a siteswap.
12 Apr 2005 10:07:29 GMT
peterb...@hotmail.com.nospam (Peter Bone)

The longest in the first 1000,000 digits is period 17 - there’s only 1.

  f(17) = 343721. This is a 4 ball siteswap 05268611514962570.

If anyone can get hold of more than 1000,000 digits then you can put them in a text file and use my program to find more: <atlas.walagata.com>—Peter

I’m currently in the process of doing just that. I’ve found a period 20 ss somewhere around the 8000000 mark, but since I’ve got 100,000,000 digits to play with, I would expect to find more. When I’ve got all of the results I’ll post them up here, along with the location fo the source file (which I can’t currently remember since I had to reboot into Windows to use the program). It has the first 4.2 billion digits if anyone has a really fast computer, fast connection and lots of time on their hands... Guy—Guy

I’ve written my own program now and it’s currently searching for all of the siteswaps between periods 14 and 30 in the first 100 million digits of pi. When it’s done, it’ll spit out a file with every ss between these periods, in order of where it occurs, with the number of balls. I can’t imagine it’ll take less than a few hours to process it all though.   Stay tuned. Guy—Guy

f(18) = 1928549 This is a 5 ball siteswap 538936820638552953.

I don’t know why I’m waisting my time on this really. I’ve updated my program to search a range of periods and it’s now much faster. <atlas.walagata.com>

I still can’t find more than 4 million digits though. I found the site with 4.2 billion but I think my firewall doesn’t like the ftp or something. It’s also stored as 2 digits per byte, which means I’d have to write another program to convert it to 1 digit per byte.—Peter
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

The theoretical prediction for that would be n!/(n^n) * 100000. It seems to match quite closely with the few I tested. —Peter

I haven’t done the others yet, but I might get round to it. An interesting one is the number of n period siteswaps in a given number of places for pi. "What would that look like?" I hear you ask. Well, in the first 100,000 digits of pi, here are the frequencies of the different siteswap periods:

  n frequency  1 100000  2 48910  3 22496  4 9328  5 3842  6 1673  7 642  8 260  9 95  10 37  11 6  12 4  13 5

A graph of which can be found here: <ntlworld.com> You’ll notice that the log graph is very linear (except at one end, but that’s because the frequencies get smaller - this would be fixed by examining more digits).—Guy

Yes, another power law. And with a gradient apparently close to -8/9 ... Does the gradient converge towards some value as we do more digits ?

I wonder how this depends on the 0..9 range of our digits ? Pi is presumably working as just a good source of random digits here.—Peter