siteswap maths essay

[kylie]

For the third year of my maths degree I have to write an essay and was
thinking about siteswap as a subject. Is there enough maths in
siteswap for this to be possible? And does anyone know of anywhere I
can find details of any maths behind siteswap, from simple maths to as
complicated as possible. Thanks.

[Scott Kurland]

Yes.

www.jugglingdb.com, start with Beever's stuff.

[Gunnar Andersson]

And have a look at
"Juggling Drops and Descents"
Joe Buhler, David Eisenbud, Ron Graham, Colin Wright
American Mathematical Monthly
January 1994

/ Gunnar

[parit]

i had a similar idea about a year ago, and managed to gather a fair amount
of material on siteswaps. i can't remember all the sources, but I got a
lot of material from http://www.juggling.org/help/siteswap/ I do remember
making a start on the project but postponed it indefinitely for reasons i
can't remember... Will post more sites if i manage to find them.

good luck with yr essay.
ps are u the kylie i know???
parit

----== posted via www.jugglingdb.com ==----

[Alex Gillis]

could also try looking at other juggling maths

[Jack Boyce]

freaky...@talk21.com (kylie) wrote in message news:<21be2369.02082...@posting.google.com>...

I have a couple of articles on the web. If you're interested in
juggling math, the second is probably more interesting:

http://members.directvinternet.com/~jackboyce/juggling/jkbnum/jkbnum.html
http://members.directvinternet.com/~jackboyce/juggling/simple/simple.html

(The "simple patterns" in the second paper are called "prime patterns"
nearly everywhere else. I thought I had good reasons back when I
named them, but thankfully everyone ignored my suggestion.)

Most siteswap math boils down to combinatorics and/or graph theory; if
either of those strikes your fancy, you'll find all kinds of
interesting and unsolved problems. And then you find interesting
connections: when I was looking into long simple (again, read
"prime") patterns, I discovered that some of the most relevant prior
work was done by electrical engineers trying to design shift registers
that could generate long pseudo-random bit sequences. (Of course,
many people think that siteswaps look like random sequences anyway, so
the analogy may be apt.)

Jack

[Aidan]

Here are a couple of meaty maths problems relating to siteswaps:

The number of legitimate site swaps that are n digits long using b (or
fewer) balls is exactly b raised to the n power. Despite its simplicity,
the formula was surprisingly difficult to prove.
source: http://www2.bc.edu/~lewbel/jugweb/science-1.html

for all sequences of n numbers x1, x2, .. xn
whose sum is a multiple of n,
there is a permutation of these numbers
that is a valid siteswap.

example: given 4 3 1 1 1 (whose sum is 10 which is a multiple of 5),
a valid permutation is 3 1 4 1 1.

can you prove this?

A proof is given in "A combinatorial problem on abelian groups"
> by Marshall Hall, Jr., Proceedings of the American Mathematical Society,
> vol. 3 (1952), pp. 584-587.
source:
http://www.jugglingdb.com/news/article.php?id=%3C982v3p$ao6$1...@woodrow.ucdavis.edu%3E

Also once I was told that NASA are interested in siteswaps, something to
do with communicating with satellites. Don't know how much truth there is
in that.

I've posted some stuff about siteswap here:
http://www.geocities.com/aidanjburns/contents
but I don't think the mathematical content is sufficient for an essay in
the third year of a degree!
Aidan.

----== posted via www.jugglingdb.com ==----

[Dan Scoffings]

You could also try 'Juggling Polynomials' by Lute Kamstra at:

http://db.cwi.nl/rapporten/abstract.php?abstractnr=952

Dan

[Lloyd Ramsey]

Fair warning:

Do not read "Just say NO to Site Swap" recently posted to this webgroup.

It is not about juggling numbers, so you wouldn't be interested.

[Jack Boyce]

> The number of legitimate site swaps that are n digits long using b (or
> fewer) balls is exactly b raised to the n power. Despite its simplicity,
> the formula was surprisingly difficult to prove.
> source: http://www2.bc.edu/~lewbel/jugweb/science-1.html

It is actually quite a bit simpler to prove this than the method in
the Graham et al paper. (They prove other interesting results as
well, just more than they need to get to the b^n formula.) See the
attached.

Jack

----------------------------------------

From: kle...@math.berkeley.edu (Michael Kleber)
Date: Wed, 18 Feb 1998 13:21:30 -0800 (PST)
To: jbo...@socrates.berkeley.edu
Subject: Re: b^n proof
Cc: kle...@math.berkeley.edu, tc...@math.berkeley.edu,
wmu...@math.berkeley.edu

Hi all...

Here's the counting argument I mentioned to Will, that shows the
number
of site-swap patterns of period n with exactly b balls is b^n -
(b-1)^n.
I think the idea of the proof is best attributed to Bill Thurston in
'93,
though this exact version is probably the work of me and Jeremy Kahn
and
earlier versions of the same idea came from Walter Stromquist and Adam
Chalcraft the previous year. The agent of cross-pollination that let
all
these ideas spread around was Joe Buhler, of course.

I'll present this in pretty much the same way I'd give a talk on it
(which I've done a few times)...

The basic concept that this proof is built on is the "in-the-air
stack".
At any instant in a n-ball juggling pattern, you can order the balls
from
lasnding soonest (the one you'll catch less that a beat later) to
landing
latest (the one that will take the longest before it next touches your
hand). The position of a ball in this stack just depends on when the
*next* time you're going to touch that ball is, not on how you throw
it
in the future.

Example: take the pattern 441; let's figure out the in-the-air stacks
after each throw. Say the balls are red, green and blue. First let's
work out which ball gets thrown at which beat:

toss: 4 4 1 4 4 1 4 4 1 . . .
ball: R G B B R G G B R . . .

Now it's easy to work out the in-the-air stack: *after* each toss,
just
scan forward through the bottom line and write down in what order you
next encounter the balls:

3rd: R G G B R R G B B
2nd: B R R G B B R G G
1st: G B B R G G B R R
toss: 4 4 1 4 4 1 4 4 1 . . .
ball: R G B B R G G B R . . .

Excercise: make sure you can go back and forth between the site-swap
and
the in-the-air stach for a pattern. Make sure you can do it for 345
or
51414 or something.

Now, the key step is to note what happens to the in-the-air stack when
each toss happens. The bottom thing on the stack is the ball that you
throw, and it gets inserted into some higher position in the stack --
each thing below the place it gets inserted moves down one position,
and
each one above the place you insert the toss stays in place. In other
words, you k-cycle the bottom k elements of the stack, for some k.

Aha! Now all we have to do is write down that value of k that
describes
what happens to the in-the-air stack for each toss! In 441, for
example,
you'll easily notice that when you do a 4 toss you 3-cycle the (bottom
three elements of) the stack; when you do a 1 toss you 1-cycle the
bottom
1 element, which of course does nothing. If we write this down in the
obvious way, we assign the string "331" to the site-swap "441".

Exercise: do this for 345 or 51414 or something.

Certainly, the cycling string we just produced contains enough
information to go back the other way: it's trivial to reconstruct the
in-the-air stack, and from there to get back the original site-swap.

Now here's the beautiful part: every cycling string gives rise to an
in-the-air stack and so to a juggling pattern, with the largest number
in
the string = the number of balls.

Exercise: try this with a random string of digits. I dunno, do 314159
or
something. For a talk, solicit one from the audience. It's really
easy
once you've done it a few times.

Anyway, counting strings is trivial, and that's it, we're done!

(Oh -- my counting assumes there aren't any 0's in this pattern. If
there are, the bijection still works -- an empty hand gives you a 0 in
the cycling string just like in the site-swap -- and the number of
patterns is (b+1)^n - b^n. The in-the-air stack isn't as nice, since
it's hard to tell the difference between 1-cycling the bottom 1 and
0-cycling the bottom 0 elements. I generally just ignore empty hands
for
this talk, though.)

Write back for any clarification.

--Michael Kleber I don't have an overactive
imagination...
kle...@math.berkeley.edu I have an underactive reality...
--EG

[Jack Boyce]

Here's a siteswap math problem I came up with a while back:

Let S be the set of all valid siteswap patterns. (Any length, any
number of balls.) Select a pattern P randomly from S. What is the
probability that P has no throws that are evenly divisible by the
pattern length?

Jack