Re: [math] labeled ball juggling

Thanks Miika for all that great info. I hadn't seen a lot of that
terminology.




Here's another interesting thought I had, unrelated to labeled-ball
juggling. I'd love to bounce it off you all.

I'm not sure if we ever write the juggling state for a synchronous
pattern, like (10)(11)(00)(01). Has anyone ever played around with this?

I made a 4-ball synchronous transition matrix using this notation, up to a
throw height of 6. I noticed there were essentially 3 ground states (if I
can use that language) listed on my matrix. The state (11)(11)(00) can be
re-entered indefinitely with a (4,4) or (4x,4x) throw, of course. But the
state (11)(10)(10) can also be re-entered indefinitely with a (6,2) or
(2x, 6x) throw. Likewise, the state (11)(01)(01) can be re-entered
indefinitely with a (2,6) or (6x, 2x) throw. I'm interested in what the
implications are for having multiple ground states.

I'd love to hear your thoughts.

John









Miika wrote:
Here's a few definitions that I found useful for discussing some of
these ideas.


A proper siteswap is not a repetition of any shorter siteswap.

A stacked siteswap has at least one orbit with more than one ball.

An unstacked siteswap has as many orbits as there are balls; that is,
each ball has its own orbit.

An unstacked repetition of a siteswap is the first repetition of a
siteswap such that it is unstacked. The length of the unstacked repetition
is the labelled period of the siteswap. This in turn is the lowest common
multiple of the sums of the throw heights in each orbit.

A labrime (short for labelled prime) siteswap is an unstacked siteswap
which corresponds to a cycle in a labelled state graph.


Examples.
441 is proper and stacked. Its unstacked repetition is 441441441, which
of course is not proper.
55050 is a proper, unstacked and labrime.
(423)^2 = 423423 is unstacked and labrime.
(423)^4 is unstacked. It's not proper or labrime.
(64)^6 is unstacked and labrime but not proper. This is the unstacked
repetition of 64 and 6464 and 646464.
64744 is proper and stacked. (64744)^6 is its unstacked repetition, which
is labrime.
(8884885)^12 is unstacked and labrime.


When converting stack notation into siteswap, the result will always be
an unstacked siteswap (before looking for repetitions).

An unstacked repetition of any (normally) prime siteswap is a labrime
siteswap. The converse is not true; 423423 is labrime but it is a
repetition of 423, which is not prime. Are all non-proper unstacked
siteswaps labrime siteswaps? No, since (7446474455)^6 is not proper or
labrime. Is this the shortest counterexample?

Any closed walk in a labelled state graph corresponds uniquely to an
unstacked siteswap. Unfortunately the opposite situation is not as simple.
An unstacked siteswap with b balls corresponds to b! walks in a labelled
state graph, one for each arrangament of the balls. These can even be
disjoint, as with the siteswap 4440. This shows how the structure of a
labelled state graph is needlessly complex.

Rather than dealing with labelled state graphs, it's probably better
just to stick with analyzing normal state graphs and maybe add some notion
of arrangement of the balls. One way to do this is for each throw observe
how many different balls will be thrown before that same ball is thrown
again. Attach this number k < #balls to the throw in the stategraph. Using
these values it's possible to keep track of the order of the balls in a
siteswap.

A few simple results: For all throws out of ground state k=b-1. For any
state there is at least one throw for which k=b-1. Within any siteswap
there is at least one throw for which k=b-1.


That's about what I had in mind for now.


-Miika


--

(btw, tämä oli kahdeskymmenes viestini)

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