Pi as a siteswap.
11 Apr 2005 11:17:06 GMT
kuba.straszew...@poczta.fm.nospam (strach)

Juggling Pi as a siteswap. since pi is of course not a correct infinite siteswap it can’t be done. but pi contains proper siteswaps.

there are two questions that come into my mind.

wheater there are siteswaps of any length contained in Pi? if so then how far they are?

let f(n) = position in pi which starts the first n-length siteswap. what is the asympthotical complexity of f?

i just started thinking about it, and don’t have any proofs yet.—strach

f(5) is 1 for a start. 31416 is jugglable... tis fun too! (Of course, this way of writing whatever it is you’re considering writing seems to assume you’ve ignored the decimal point... If you want to juggle fractional objects, it’s easiest to use rings, cut out the required proportion (for which purpose, knowing pi is quite useful!) and then juggle with the decimal number rounded down worth of extra objects. Of course, you almost certainly didn’t wish to know that, and I can’t for the life of me think why I bothered to write it...)

Tom - definitely in procrastination mode!—Schwolop

i’ve examined the first 10000 digits of pi. the longest siteswap in it is 13 digits long—strach

i checked the existence of siteswaps up to 50 length. i doubt if there are any longer ones in first 1000 digits. i wonder how far is the first 100 digits long siteswap in Pi.

f(n) looks as O(some very high exponent)—strach

Since you can’t have siteswaps with two digits n places apart where the first digit is the second digit plus n[1], it should be easy to find a lot of such pairs in whatever section of pi you’re looking at, so limiting the maximum length of a siteswap to the largest distance between these pairs. Actually it would be from the digit after the first digit in the first pair to the digit before the second digit in the second pair.

For example in 3.1415926535897932384 (twenty digits) there can be no                       ^^ ^^ siteswap of period > 8 because it would have to include 65 or 32, which aren’t allowed.—Rory

f(5) is not 1 unfortunately   pi is 3.14159...   f(1) = 1. This is a 3 ball siteswap 3.  f(2) = 1. This is a 2 ball siteswap 31.  f(3) = 2. This is a 2 ball siteswap 141.  f(4) = 13. This is a 7 ball siteswap 9793.  f(5) = 9. This is a 6 ball siteswap 53589.  f(6) = 247. This is a 4 ball siteswap 190914.  f(7) = 120. This is a 5 ball siteswap 4709384.  f(8) = 461. This is a 6 ball siteswap 96274956.  f(9) = 3890. This is a 4 ball siteswap 783123552.  f(10) = 3889. This is a 4 ball siteswap 4783123552.  f(11) = 6048. This is a 5 ball siteswap 53779914037.  f(12) = 2599. This is a 5 ball siteswap 858900971490.  f(13) = 119. This is a 5 ball siteswap 6470938446095.

and of course I ignore the decimal point. i don’t what the 3.14 as real number has to do with siteswap? I will download more digits of pi, and try to find possibly longest siteswap there. Anyone else to try?—strach


I’d call it 0, but then I’m prone to zero-based indexing. I’d also call it wrong because 6 only appears there if you round up. That digit should be a 5.

  f(1) = 0 (3; 3 props)  f(2) = 0 (31; 2 props)  f(3) = 1 (141; 2 props)  f(4) = 12 (9793; 7 props)  f(5) = 8 (53589; 6 props)  f(6) > 50, I can’t be bothered to find it. I might explore this programmatically in the future.—Rory

Well, one of the main conjectures with Pi is that it is a random number. This means that every sequence of digits that you can imagine repeats infinitely often in the digits of Pi. If this conjecture is true, then any proper siteswap of any length appears in Pi infinitely often. Now, that doesn’t answer your question regarding the position of the first n-length siteswap, but I hope it helps otherwise.—RPN

I’ve heard that said but I don’t know of any proof. Since it appears that we don’t even know whether pi is a normal number[1]--or even which of the digits appear infinitely often in the decimal expansion--we can not know that every possible digit block occurs in it.—Rory

Thats a guess not a fact.

And as for pictures of things we do run into some philosophical questions. Lets keep out mothers out of it and instead talk about a picture of me juggling a 5 club cascade. You may find a picture (or even a video) that looks a lot like me juggling 5 clubs, but it isn’t really since no such picture exists. The same could of course be said about a picture of me juggling 3 clubs. This actually raises a lot of questions about what constitutes a picture and I think I rather juggle than spend time on them right now. :)—KlasA

You’re confusing "picture" with "photograph"

I’ve probably never met you, have no idea what you look like and yet I could quite happily draw a picture of you juggling 5 clubs. The picture would exist, and it would be of you juggling 5 clubs. [1]

I could even use a drawing package on my pc to draw a picture of you juggling 5 clubs and save it as a jpg. It would be a picture of you, juggling 5 clubs, in jpg format, and may appear somewhere in the depths of pi.

OK, so it might not be a very accurate picture of you, but it would still exist. pictures don’t have to depict reality - look at the work of M.C. Escher or Salvador Dali. I doubt very much that any of the subjects that they painted exist in reality - but the pictures still exist.

You might think you were on safer ground with "video" - but not so!

There exists video of a grown man wearing spandex swinging around a city attached to a large spider-web. To my knowlegde, this has never actually happened, and is a complete work of fiction. However, the video does exist

I know. How? I paid far too much money to watch that pile of toss in the cinema[2] that’s how.

-Paul I guess it’s perfectly possible that this post might appear somewhere in the depths of pi, but it’s equally possible that there also exists a spellchecked and proofread version.

[1] it wouldn’t be a very good picture as I can’t draw for shit! [2] I still can’t quite believe I went back to see if the sequal was any better - it wasn’t.—Little


Whenever I hear a statement like "given an infinite number if digits - it will appear an infinite number of times" I always feel like I should add ", or not at all". Is that unreasonable?—Struggler