Re: juggling's combinatorics problem with siteswap
From: Xah Lee (xah_at_xahlee.org)
Date: 02/14/05
- Next message: David C. Ullrich: "Re: S. Petersburg Paradox"
- Previous message: examachine_at_gmail.com: "Re: does sqrt(2) exist in CM?"
- In reply to: owenjonesuk: "Re: juggling's combinatorics problem with siteswap"
- Next in thread: Rory Parle: "Re: juggling's combinatorics problem with siteswap"
- Messages sorted by: [ date ] [ thread ]
Date: 14 Feb 2005 04:08:26 -0800
here's a more organized account with some corrections.
-----------
Some definitions
Some formal definitions, to be refined.
* siteswap notation is finite sequence of positive integers, as in
{5,3,4,4,4} (4 balls tennis pattern) that are constrained by these
rules: (to do). each number is single digits, then it can be
abbreviated to 53444. And siteswap always mean canonical siteswap ...
(to do)
* length of sitewap is the total numbers in the siteswap. i.e.
length[53444]==5.
* bs = number of balls. Number of balls is the arithmetic mean of the
siteswap.
* cp = a colored period is the number of throws all colored balls
return to their position. For example, for pattern 53444, the cp is 30.
(to do: need math definition here).
* ucp = uncolored period. The number of throws all uncolored balls
return to their position. In 53444, ucp = 10. In 3-balls cascade, ucp
is 2. (to do: need math definition here).
* rucp = reduced uncolored period. Like ucp, but mirror cycles are
excluded. That is, ucp is a multiple of rucp. For example, rucp for
4-balls tennis 53444 is 5. rucp for 3-balls cascade is 1. (rucp can be
computed as the total numbers in siteswap sequence.).
* obs = the number of orbits in a pattern. (a orbit is the path a ball
travels.) In 53444, obs = 2. (note: each orbit can have its own cp,
ucp, rucp and siteswap notation.)
* amb = true if ambidextrous, ortherwise false. A pattern is
ambidextrous if left and right hand play the same role. (if length of
siteswap is odd, then ambi is true for sure. 8448 is a counter example)
... the above needs to be refined and adding more concepts.
Some Siteswap Examples
* single handed 3 balls: 60. obs=1.
* single handed 3 balls: 8040. obs=2.
* single handed 3 balls: 804060. obs=3.
* 3-balls snake: 55050.
* 3-balls snake: 7070700.
* 3-balls flash 55500. obs=1.
* 3-balls flash 7770000. obs=1.
* 4 balls tennis: 53444. cp=30, ucp=10, rucp=5, amb=true,
orbits={44403,50000}. 53444=44403+50000 = 3*400040004000300 + 50000.
* 2 balls shower: 31
* 3 balls shower: 51
* 4 balls shower: 71
* 4 balls cascade: 55550
* 5 balls cascade: 55555
* 3-balls shower on one hand, 2-balls shower on the other: 64
* each hand doing single handed 3-balls 8040: 8844. 8844=8040+8040.
Some problems to do
* Given a siteswap, decompose the siteswap into siteswaps of its
orbits.
* some orbit's siteswap can be reduced in height. (need to define this
more rigorously) For example, 55050 and 7070700 are both 3-balls
snakes. Find a way to define the notion of heigh or equivalence.
* investigate composition of siteswaps. That is, given two siteswaps,
find another pattern that contains these two as orbits.
* decompose a orbit's siteswap into the siteswap of each balls. For
example, 3-balls cascade {3}, there is a single orbit that's the
siteswap itself. Each ball's siteswap is 300. In 4-balls tennis 53444,
one of the ball's siteswap is 400040004000300.
* When studying handed juggling of 3-balls, there are 60 (obs=1), 8040
(obs=2), and 804060 (obs=3). Note the 0 there. Perhaps there needs to
be a notation with respect to a single node. The siteswap system is
based on 2 nodes. One needs to investigate how to extend it to n nodes.
* note in 8448 where each hand is doing single handed 3-balls 8040,
there's a notion of phase shift.
* 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. Here's a notion where how many nodes
a ball visits. Given a single ball siteswap, if there are odd number in
the sequence, then the ball visit both hands.
I've archived the above at
http://xahlee.org/Periodic_dosage_dir/t2/siteswap.html
Please post corrections and suggestions. Thanks.
Xah
xah@xahlee.org
http://xahlee.org/PageTwo_dir/more.html
- Next message: David C. Ullrich: "Re: S. Petersburg Paradox"
- Previous message: examachine_at_gmail.com: "Re: does sqrt(2) exist in CM?"
- In reply to: owenjonesuk: "Re: juggling's combinatorics problem with siteswap"
- Next in thread: Rory Parle: "Re: juggling's combinatorics problem with siteswap"
- Messages sorted by: [ date ] [ thread ]
Relevant Pages
|
