Re: juggling's combinatorics problem with siteswap
From: Xah Lee (xah_at_xahlee.org)
Date: 02/13/05
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Date: 13 Feb 2005 14:06:01 -0800
btw, if you are not a juggler, but find it fascinating, there's a
software called JugglingLab that can simulate by a visual display any
juggling siteswap pattern. From a math perspective, it's more
fascinating than watching people juggling. The software can be
downloaded at
http://jugglinglab.sourceforge.net/
some more thoughts on orbits...
a pattern can have several orbits, but two orbits may be the same. For
example, one hand juggling two balls. Now, both hand does that. That's
a pattern with 2 orbits, but same. They are mirror images. Mirror image
orbits are not necessarily shows only in patterns where each hand minds
its own business.
suppose one hand juggles 2 balls in a cycle. (i.e. shower). That's one
orbit.
suppose one hand juggles 3 balls in a cycle. (i.e. shower). That's one
orbit. Now suppose one throws a ball really high, then do two small
throws of the other two balls, then the 1st ball came down and throw it
high again, then two small throws with the other 2 balls... this
pattern would be 2 orbits. Now a similar pattern with high and low
throws with 3 balls one hand can be made so that it has just one orbit,
such a ball of any color will go thru the same pattern.
in pattern 64645 (5 balls), there are 3 orbits, 2 distinct ones.
Essentially, each hand independently juggles 2 balls, and another ball
goes from hand to hand by itself.
One may ask the question, given a specific orbit, what patterns are
possible using that orbit with n balls.
another concept is that some orbits are entirely juggled by a single
hand, as in many of the above examples. In network theory jargon, we
can call it a loop, as it connects only to itself.
sorry for the rather incoherent ramble... will have to think about
these more coherently and perhaps put together a nice account in the
future.
Xah
xah@xahlee.org
http://xahlee.org/PageTwo_dir/more.html
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