Siteswaps - tell me something I don't know

[Miika]

Hello everybody :-)

I like juggling. I also like thinking about juggling. In particular I
like thinking about siteswap theory. Stuff like the notation, states,
transitions, transformations, relationships between siteswaps, and some
other mathematical aspects are really interesting to me. But after playing
around with these concepts for close to ten years now, it's hard to come
up with new stuff that still excites me and makes me spend hours trying to
wrap my head around how and why it works. Yet I know there is lots more to
figure out, if we just find the right approach.

My question today is, what else is there? Do you know some weird aspect
of siteswaps that I haven't encountered before? Have you had any ideas
like this that you thought were cool but haven't really shared with
anyone? Do you know someone that has? What mathematical results haven't
been published yet?

Even if it's some small thing, I'd love to hear it. Or if you have some
confusion about some aspect of siteswaps, perhaps sharing it here would
help figure it out. Or if you can think of an unanswered question about
something related to siteswaps, we could try that too.


Swap away,

-Miika

--

Siteswaps of the day: pear , apple , mango

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

[Jason Lu]

There are loads of strange bijections from one set of siteswaps to other
sets of siteswaps.
One nice one is the bijection from any async siteswap to shower patterns.
Map n to ([2n-2]x,2x) or 2n-1 1
Then you get a tonne of shower siteswaps.
e.g. 64 -> d171
or
73 -> d151

Like 73, d151 is excited state (relative to ground state 5 ball shower). A
6 is required to excite to 73 and similarly b1 is required fro d151.

There are also some mathsy bits and bobs which I've thought of, but aren't
very structured. One of which:
Any excited state requires more than 1 throw to return back to its state.
This leads to an alternative definition for ground state. The proof is
really easy if you guys want to give it a go.

I just thought that the above rule doesn't apply to multiplexes, e.g.
splits [32].

J

[Emman]

Siteswap difficulty isn't as objective as one might think.
;)

[David Cherepov]

Jason Lu wrote:
>
> One nice one is the bijection from any async siteswap to shower patterns.
> Map n to ([2n-2]x,2x) or 2n-1 1
> Then you get a tonne of shower siteswaps.
> e.g. 64 -> d171

b171 not d171 arithmetic FAIL but nice observation, I like math, but I
wasn't previously very interested to look at mathy siteswap things, but
now I'll try to do that more

> or
> 73 -> d151
>
> Like 73, d151 is excited state (relative to ground state 5 ball shower). A
> 6 is required to excite to 73 and similarly b1 is required fro d151.
>
> There are also some mathsy bits and bobs which I've thought of, but aren't
> very structured. One of which:
> Any excited state requires more than 1 throw to return back to its state.

No, it doesn't matter how many throws to return to a lower excitation
level. Of course it has to have more than 0 throws, but that's the only
limitation

667713 6677162 66771661 5741571424444....

[Jason Lu]

David Cherepov wrote:
>
> Jason Lu wrote:
> >
> > One nice one is the bijection from any async siteswap to shower patterns.
> > Map n to ([2n-2]x,2x) or 2n-1 1
> > Then you get a tonne of shower siteswaps.
> > e.g. 64 -> d171
>
> b171 not d171 arithmetic FAIL but nice observation, I like math, but I
> wasn't previously very interested to look at mathy siteswap things, but
> now I'll try to do that more

oops!

>
> > or
> > 73 -> d151
> >
> > Like 73, d151 is excited state (relative to ground state 5 ball shower). A
> > 6 is required to excite to 73 and similarly b1 is required fro d151.
> >
> > There are also some mathsy bits and bobs which I've thought of, but aren't
> > very structured. One of which:
> > Any excited state requires more than 1 throw to return back to its state.
>
> No, it doesn't matter how many throws to return to a lower excitation
> level. Of course it has to have more than 0 throws, but that's the only
> limitation
>
> 667713 6677162 66771661 5741571424444....
>
> > This leads to an alternative definition for ground state. The proof is
> > really easy if you guys want to give it a go.
> >
> > I just thought that the above rule doesn't apply to multiplexes, e.g.
> > splits [32].
> >
> > J
> >
>
>
>

I am not sure if I misunderstand what you've written or you've
misunderstood what I've written. I didn't mention anything about a lower
excitation level, whatever that is, just that an excited state requires
more than 1 throw to return to that SAME excited state, i.e. any siteswap
on that excited state is of period >1.

[Will S]

Having two balls/objects going into the same hand at the same time. I
think beatmap deals with this a little bit, but I was never too interested
in the notation - same to a certain extent with trad siteswap. I think
two balls going in and out of the same hand together is quite interesting
- I do it a little bit, but there's no way I could write it (nor do I have
much interest in writing it).
Will

[Joost Dessing]

try letting go of the idea that you are required to throw at the same
frequencies for both hands.
This was discussed here at length before; you may be aware of this.

http://www.jugglingdb.com/news/thread.php?group=1&id=130563

Next step is of course letting go of fixed frequencies altogether. I
wonder whether it makes a lot of sense to write a notation for that (even
though it is possible of course).

cheers,
Joost

[ohioohio]

I always read your posts with interest because i do like siteswaps
notation (theory, application and juggling) and I think you added some
interesting details about it, during your studies.
I guess there are a lot of things to understand about this notation but
actually I wouldn't know in what it is lack. I should tell that it can't
describe how a throw is made o how an object can be catched but this is
also known by all!
Maybe the notation could describe patterns that include a ball that bounce
on the arm, a ball stopped on the head or on the foot. But this won't be
nothing too much new.

However, I think the future studies can be done about passing patterns.
Actually the prechac notation transforms a valid siteswap pattern in a
prechac passing pattern for "n" jugglers, in order to the objects and the
period.
I think new notations can be found (and some jugglers are trying to find
them) to describe passing patterns that include takeaway, weaves and all
the crazy and fun replacing, placing, movements that are in the passing
patterns.

I hope a day I will read an other article of you about it on rec.juggling!

Maurizio

[Jason Lu]

[Daniel S.]

Jason Lu wrote:

> There are also some mathsy bits and bobs which I've thought of, but aren't
> very structured. One of which:
> Any excited state requires more than 1 throw to return back to its state.
> This leads to an alternative definition for ground state. The proof is
> really easy if you guys want to give it a go.

This is really cool :).
The other day I made another siteswap game. I think understanding more in
depth on how what you mentioned works might give a little more strategy to
the game. I'll explain first how the game works.

I'm calling it 'Siteswaptimus Prime' but yeah... I just had to give it a
name :D

Components:
The main board for SP is the state wheel, but you don't necessarily need
it. It just makes it easy to visualize. (state wheel video:
http://www.youtube.com/watch?v=uBL7EIAfEIg )
Tons of little cubes and something graph paperish to set them on.

Alternate Components:
Pencil and paper (preferably graph paper)

Rules:

Setup:
Select object/max height parameters. Example: 5 objects / max height 9
Start in ground state.

On each turn a player makes a 'throw'. If after your turn a state is
repeated then you win.
Here's an example game between player A and Player B. This is an actual
game which was played today by myself and a juggler named Perry at the
local vegan burger joint :D

a: 9 xxxxx
b: 4 xxxx----x
a: 5 xxxx---x
b: 8 xxx-x-x
a: 9 xx-x-x-x
b: 5 x-x-x-x-x
a: 0 -x-xxx-x
b: 9 x-xxx-x
a: 0 -xxx-x--x
b: 3 xxx-x--x
a: 9 xxxx--x
b: 9 xxx--x--x
a: 5 xx--x--xx
b: 2 x--xx-xx
a: 0 -xxx-xx
b: 3 (for the win!) xxx-xx
repeated state! xxxxx
siteswap created: 9458950903995203

In the game above we managed to create a prime ground state siteswap, but
it doesn't have to end up in ground state. Below is an example game from
�ine and I where the winner ended up creating an excited state siteswap.

a: 9 xxxxx
b: 7 xxxx----x
a: 9 xxx---xx
b: 2 xx---xx-x
a: 2 xx--xx-x
b: 7 xx-xx-x
a: 9 x-xx-xx
b: 0 -xx-xx--x
a: 6 xx-xx--x
rep: x-xx-xx
siteswap created: 906

By leaving your opponent with an empty slot they are forced to play a 0
which basically means that you get to move twice, but be careful that they
don't complete the siteswap with a 0! This happened once.. I won't say to
whom.

How does this relate to what Jason was saying?
It might help to know how many moves minimum it would take to repeat a
specific existing excited state if I wanted to add more rules to the game
or make it in any way more interesting.

This is also in response to the original post. I'm not sure if this is
along the lines of what you're looking for Miika, but I'll take a look in
my notes and see if I've found out anything else new about siteswaps.
Most of my finds are game related or graphic representations of siteswaps.

I'm curious if anyone has any comments or additions to the game. I
suppose there's no reason why 3 people couldn't also play. I imagine the
strategy would get more and more random as you add players though.

Daniel

[Poor-]

That's crazy! Any vids of multi-frequency juggling?

[Adrian G]

OK, I have a question that I've been wondering about, more to do with
stack notation but anyway...

In all the descriptions of stack notation I've read it says that it can do
anything that siteswap can, but I've naver seen any synchronous examples.

Standard stack notation wouldn't work, but has anyone made any extensions
to it to support synch?

The best I could think of would be having two 'stacks' and if there is an
x after the number then it would cross to the other stack to normal (e.g.
a 4x would cross but a 5x would go straight, like siteswap). so (4,2x)*
(box) would end up something like (2,1)* because you'd count the number of
balls thrown from each hand does that make sense?

Does anyone have any other ideas?

Also on the topic of stack notation, as far as I can tell, multiplexes are
impossible, which I found quite interesting. This is because a multiplex
pattern will always contain a squeeze catch (e.g. there is a 2 and a 3
that are caught at the same time in [43]23 ). Stack notation doesn't allow
two balls to be caught at once in the same hand as the ball is just
'inserted' into the stack (which is why all stack notation patterns are
valid). Therefore it is impossible to notate multiplex patterns, is that
correct?

Adrian

[Boppo]

Regarding the six ball trick, 999441:

Interestingly, the throws are all perfect squares, and there are n
repetitions of each value of n^2. (So there are three 9s, which is
3^2 = 9, two 4s, which is 2^2=4, etc.) This trick is the third in the
sequence 1, 441, 999441. All continuations of this series are valid
vanilla siteswaps: 16 16 16 16 9 9 9 4 4 1 is valid trick, with 10
balls. (Oh, sorry, that's gggg999441 to y'all.)

The fact that all these tricks are valid proves an interesting
mathematical relationship. Consider the series of perfect cubes, 1, 8,
27, 64 ... and the series of triangular numbers, 1,3,6,10,15 (which
are all partial sums of all the counting numbers, 1, 1+2, 1+2+3,
1+2+3+4 ...) The relationship is, that the sum of the first n cubes
(1+8+27) = the nth triangular number squared (6^2). And this is
*because* this trick and its extensions are all valid vanilla
siteswaps. It's straightforward to prove by induction, and the
relationship to the triangular numbers is how many balls each
different trick has; the square of that number is the total value of
all the throws. They're triangular because each next square has one
more iteration than the last - one 1, two 4s, three 9s, etc. So the
six ball trick has 6 (balls) * 6 (period) = 36 total throw value.
Which is the sum of the first three cubes, 1+8+27.

Did you know that? (Is it the sort of thing you are looking for?)

-boppo

[Q Juggler]

Hi Miika

I invented the juggling number system long ago and I have not seen
much new sense then, so I am also interested. I haven't read all of
the stuff out there so I'm not totally sure if there really is
anything new but I imagine so. But the newest thing for me is that the
original siteswaps is explained on the wolfram math website. I have
been wondering about it for the past 25 years. Boppo's last post is
something to me also. I have a few things that I haven't heard or read
about but I'll tell you later.

Paul

[Jason Lu]

That's a nice game, I like the idea of that! How do you keep track of all
the states that's occurred before? Also, why start from ground state? :)

The only application I can think of for my thing is that:
a) You can define ground state as the state that can return to itself with
one throw.
b) If you want to engineer siteswaps, e.g. make 75 symmetric by adding a
6, the best you can do to excited state siteswaps is something with 3
beats, because 1 beat isn't possible.


Does anyone have a nice proof on why one siteswap is only valid on 1
specific state? Clearly the states have to come back to themselves
eventually for the siteswap to loop, but what's not to say that it cycles
through multiple states.
I don't mean each throw of 723 sends
***-*
**-*--*
***--*
***-*
etc,
but that the whole siteswap (723) cycles through different states.
I can show 744 (not 447 or 474) is only valid on ground state by doing it
exhaustively just by tracing back the throwsm but that is hard to
generalise.

J

[Jason Lu]

This is a nice result! How do you show that 1+2+...+n is the number of
balls too? I understand that it is the period but why the number of balls?

[Boppo]

That's because the trick is freezeframe-like in that (1) it's ground-
state, and (2) blocks of throws land later than previous blocks of
lower throws. (In the freezeframe, each throw individually lands
after all previous throws. But for these, this is only true between
blocks, but within each block the first throw lands first.) If all
the throws land after the 1 does, and all the high throws land after
all the throws lower than it, then each ball can only be thrown only
once. Therefore, you don't reuse balls during the trick, so the
number of balls has to equal the period.

This is basically the same reason as for the freezeframes, for which
it might be more obvious that the number of balls has to equal the
period. These tricks are sort of like "triangular-block
freezeframes," or maybe just "triangular freezeframes." In the
freezeframes, the throws heights are all the odds, up to a value; in
the triangular freezeframe the throws heights are all the squares, up
to a value. [1]

-boppo

[1] Of course, you can always add or subtract a constant from every
throw and get a valid trick so long as the throws remain positive, so
8642, for which the throws are even, is 1 + 7531, and 888330 is 999441
- 1, for which the throws are, obviously, n^2 - 1. But those aren't
the base tricks about which the argument is being made. In
particular, when you do a proof by induction, after showing truth for
the n=1 case, you assume validity of case n and then show that case n
+1 is also valid. Let's be sure to pick the proper trick, e.g. 441
and not 330 nor 552 nor any other displacement, for case n. Case n
has to apply to the case of n=1 for the proof to be valid - it has to
end in 1, for one thing. It also has to be made up of consecutive
squares, so for example 771 (not made of squares) and 16 16 1 (not
made of consecutive squares) would be other examples of incorrect
choices for case n. You have to make your case n be correctly
triangular-in-the-squares.

[Boppo]

Incidentally, having mentioned the freezeframes, what is the sum of
the first n consecutive odd numbers?

1 = 1
1+3 = 4
1+3+5 = 9
1+3+5+7 = 16

Yep, that's right, it's the squares. A freezeframe of n balls takes n
throws, each with an average of n, so the total combined throw value
is n*n=n^2. That's the sum of the first n odd numbers, the sum of all
the throws in a freezeframe.

Suppose you had a freezeframe of n throws, and you wanted to add one
ball and throw the new balls such that it lands last. What throw
value would it need? [Left as an exercise.] There's your proof by
induction.

Who knows, maybe there's a proof of the Riemann Zeta hypothesis
lurking in siteswaps. (Although I kind of doubt it.) Fermat's last
theorem, though. Maybe.

Ok, maybe not.

-boppo

[Boppo]

> -boppo- Hide quoted text -
>
> - Show quoted text -

Typo:

Suppose you had a freezeframe of n throws, and you wanted to add one

ball and throw the new ball [singular] such that it land[ed] last.

Sorry about that.

-boppo

[Jason Lu]

:)
Reminds me of the story of some maths lecture, when a lecturer finished a
proof with "something something" is obvious. A student asked if it really
was obvious and the lecturer came back after 30 minutes and said "yes!"

[capricornwhite]

Might be an idea to look into social siteswaps. There's a wealth of new
stuff when it comes to that and i haven't even begun to grasp all the
theory behind it. Most of the normal siteswap rules work with it, but
then you have the prechac patterns, you can go with 2 jugglers doing the
same thing at the same time, the same thing at different times (staggered)
or different things at the sime time so long as the period is the same
with jugglers.

I particularly like working with ultimates but i'd like to work out how to
change odd numbers patterns from one juggler juggling the even numbers to
the odd numbers a transition between the 2 without any self throws so for
7 balls <4p|3p> to <3p|4p> or <7p|4p><5p|5p><3p|1p><1p|2p> to
<3p|5p><4p|6p><2p|2p><3p|3p> for examples. You can do staggered patterns
in ultimates as well the 4 ball splits pattern [43],2,3 works in a
staggered 10 ball pattern with 2 jugglers, dunno what the actual siteswap
notation of that would be though, but i do know it runs.

Have fun :)

[David Cain]

Hmmm. Something you might not know about siteswaps. Audiences don't
care about them one little bit! Sad, perhaps, but true.

[TABjuggler]

I just made this post in another thread. Some of the info might be
relevant here:

"Important Observations:
*Even numbers never cross but throw to the same hand they threw from

*Odd numbers always cross

*If you add up the total of a siteswap and takes its average, if the
average is a whole number, than that tells you the number of balls
required to perform the siteswap (ex: 531 = (5+3+1)/3 = 9/3 = 3 balls) and
(ex: 7441 = (7+4+4+1)/4 = 16/4 = 4 balls)

*The order of a siteswap's numbers do not matter, but are usually written
with the largest number first (ex: 423 = 234 = 342)

*Just because the average is a whole number does not mean that the
siteswap is valid. However, you can always take a permutation of the
numbers and get a valid siteswap. (ex. 432 = (4+3+2)/3 = 9/3 = 3 Balls,
but the siteswap is not actually possible. So you have to look at all the
combinations of those numbers (423, 432, 234, 243, 342, 324). Due to the
above observation (423 = 234 = 342 and 432 = 324 = 243). That means that
you only really have two possible permutations of the siteswap numbers and
one doesn't work. That means that 423 = 234 = 342 are all valid
siteswaps.

*Another cool observation of the above is that all two period siteswaps
that have whole number averages are valid siteswaps. (ex: 51 can only
permutate to 15, which is just 51 in a different order -> 51 = 15)

*Any siteswap that is written as descending sequential numbers is invalid.
(ex: 432, 765431, and 987 are all invalid). There is another fact about
that that is best explained with an example (If you take any of the above
examples of invalid siteswaps and change out any of the numbers, the
pattern is still invalid if at least 2 of the numbers would be sequential
and descending..... ex: (765431, 7x5xxx, 7xx4xx, and 7xxxxx1 are all
invalid no matter what numbers x is equal to in those examples)

Hopefully that isn't too much info, but its most of the relevant parts of
siteswap that will actually help you.

If all of this wasn't too bad I'll give you the most useful formula out of
Mathematics of Juggling that I have found so far. You just had to take it
at face value that 432 and all descending siteswaps are invalid and that
all two period swaps with correct averages are valid. This formula lets
you do the math for valid vs invalid in your head without too much
difficulty. "

[Miika]

Yeah, those bijections are nice. They are usually derivative from
combinations of simpler transformations. Not all the names are in common
use, but here's quick list of some:

cyclic shift: write the pattern starting from a different place
writing hands differently: example for synch throws switch the (6,4) to
(4,6)
global shift: add/subtract constant to every throw
local shift: add/subtract the period from a single throw
time reversing: like 51234 to 52413
site swap: you know this, like 555 changes to 645 or 500001 to 300201 :-)
site sliding: like 8646 to (8,6)(4,6) but can also be done a throw at a
time
take a detour: depending on the state, append another siteswap, like 42 to
423
zero extension: add a 0 somewhere and adjust other throws' heights
accordingly
adding dummy hands: zero extensions at certain intervals like 31 to 6020

So your example of shower siteswaps can be thought of as adding one dummy
hand, site swapping the throws with the zeros, and possibly site slid to
get the synch version. Like 423 -> 080406 -> 713151 ->
(6x,2x)(2x,2x)(4x,2x) . The use of this breakdown is that it's easier to
prove that the result is still a siteswap.

I like that you shared your idea about the ground state. There was a
discussion somewhere about synch state graphs having multiple ground
states because there are several states with loops, like (x,x)(x,x)(x,-)
and (x,x)(x,-)(x,-)(x,-) for five balls. For small ideas like this, it's
hard to know if anyone else has thought of them without discussing it.

Thanks for your time,

-Miika

--

Siteswaps of the day: rat , hen , fox

[Miika]

Emman wrote:
> Siteswap difficulty isn't as objective as one might think.
> ;)

But how does DB55555 factor in difficulty between (?) DB97531 and D666666
?

-Miika

--

Siteswaps of the day: spy , ski , sex

[Miika]

Will S wrote:

> Miika wrote:
> > My question today is, what else is there? Do you know some weird aspect
> > of siteswaps that I haven't encountered before? Have you had any ideas
> > like this that you thought were cool but haven't really shared with
> > anyone? Do you know someone that has? What mathematical results haven't
> > been published yet?
> >
> > Even if it's some small thing, I'd love to hear it. Or if you have some
> > confusion about some aspect of siteswaps, perhaps sharing it here would
> > help figure it out. Or if you can think of an unanswered question about
> > something related to siteswaps, we could try that too.
> >
>

> Having two balls/objects going into the same hand at the same time. I
> think beatmap deals with this a little bit, but I was never too interested
> in the notation - same to a certain extent with trad siteswap. I think
> two balls going in and out of the same hand together is quite interesting
> - I do it a little bit, but there's no way I could write it (nor do I have
> much interest in writing it).
>

Just use multiplex siteswaps. Though the notation doesn't exactly require
you to catch at the exact same time, you can just say "do [53]31 with
squeeze catches". It's generally more work to look for multiplex patterns
that don't require simultaneous catches but have holds in the appropriate
places :-)

-Miika

--

Siteswaps of the day: cat , hog , boa

[Miika]

Joost Dessing wrote:
> try letting go of the idea that you are required to throw at the same
> frequencies for both hands.
> This was discussed here at length before; you may be aware of this.
>
> http://www.jugglingdb.com/news/thread.php?group=1&id=130563
>
> Next step is of course letting go of fixed frequencies altogether. I
> wonder whether it makes a lot of sense to write a notation for that (even
> though it is possible of course).
>

Yes, I've heard about these somewhere :-)
http://www.jugglingdb.com/compendium/geek/notation/siteswap/tweaked.html#polyrhythms
And then there was
http://www.siteswapgeneration.com/vanilla-1.1.8/comments.php?DiscussionID=14&page=1#Item_0

Multi-frequency stuff is a neat concept and even in practice the basics
are quite fun to try out. The 3:2 rhythm in doubles:singles is one of the
very few four club tricks I've bothered learning (not just multifrequency
tricks, but any tricks with four clubs).

Of course the notation of these patterns in siteswap isn't as elegant as
the idea of juggling at different frequencies, more so if we let go of
fixed frequencies altogether. At some point it's just best to use ladder
diagrams and forget about the numerical code :-)

Do you have a list of good patterns to try? I could generate a lot of
them, but the harder part is always picking out the ones that actually are
decent to juggle.

How much multi-frequency passing has been explored in practice?

-Miika

--

Siteswaps of the day: flea , lamb , snail

[Peter Bone]

David Cain wrote:
>
> Hmmm. Something you might not know about siteswaps. Audiences don't
> care about them one little bit! Sad, perhaps, but true.

Who mentioned audiences?

[Miika]

ohioohio wrote:
> I always read your posts with interest because i do like siteswaps
> notation (theory, application and juggling) and I think you added some
> interesting details about it, during your studies.
> I guess there are a lot of things to understand about this notation but
> actually I wouldn't know in what it is lack. I should tell that it can't
> describe how a throw is made o how an object can be catched but this is
> also known by all!
> Maybe the notation could describe patterns that include a ball that bounce
> on the arm, a ball stopped on the head or on the foot. But this won't be
> nothing too much new.
>
> However, I think the future studies can be done about passing patterns.
> Actually the prechac notation transforms a valid siteswap pattern in a
> prechac passing pattern for "n" jugglers, in order to the objects and the
> period.
> I think new notations can be found (and some jugglers are trying to find
> them) to describe passing patterns that include takeaway, weaves and all
> the crazy and fun replacing, placing, movements that are in the passing
> patterns.
>
> I hope a day I will read an other article of you about it on rec.juggling!
>

The easiest way to incorporate bounces into siteswap notation is to say
the underlying siteswap pattern and separately express which throws are
bounces. Like "run a cascade and do a round of 531 with the 5 bounced off
your head" :-) You can also think of the pattern as a three handed pattern
and usually have a 0 on the third hand except when there is the bounce, so
the above could be something like 0!33...0!2h32h!1...0!33 which I'm not
even going to bother justifying!

Yeah, some of the crazy stuff you can do with takeaways and the like are
practically useless to actually notate in text. I guess beatmap can do it
if you want, but many times just a video of the trick will be much better.
If we're just talking about the order of the objects and the hands that
juggle them, there's pretty much not anything that can't be expressed in
siteswap. But still this is an area for me where I don't have any
practical juggling experience, so coming up with meaningful new ideas is
difficult. Like even just notating the number of spins for clubs in
passing patterns doesn't seem to be something that interests jugglers who
actually pass!

-Miika

--

Siteswaps of the day: asp , gnu , orang

[Miika]

> Áine and I where the winner ended up creating an excited state siteswap.

Nice :-) A gaming aspect was one of the approaches I haven't much thought
about. This one seems to work well and has a very mathematical feel to it.
It can be solved without too much effort (meaning to find a winning
strategy for one of the players), but it's flexible enough to not matter,
because you can always choose the starting parameters so that the players
haven't memorized the solution.

For example, if playing with 4 balls and maximum height 6, the second
player can always win by aiming for 44, 53, 6622, 6631, or 666600,
depending on how the first player plays.

As for a variant, you could allow adding or dropping balls. Perhaps that
gets too complex, so maybe just gradually allow for higher throws to be
used. Like for every 0 your opponent makes you throw, the maximum height
you (personally) can throw to grows by one.

You can also allow multiplexes, but I'd suggest not having one person
'throw' both/all the balls in a multiplex. Like if you're in the state
[xx]x-x one persons move would be 5 to get [-x]x-x-x and the other person
then plays 6 to get x-x-xx. The first person shouldn't win even he repeats
a (virtual) state after the first throw, but only once the whole multiplex
is dealt with can someone win.

As for strategy, you should avoid leaving your opponent with a state
starting with x- and definitely not x-x-. Like in your first example game,
after five throws in state x-x-x-x-x player B could have forced a win with
70701 back to ground state.

I just know thinking about this game will just take up too much of my
time. So thanks! In the meanwhile, keep looking through those notes.

-Miika

--

Siteswaps of the day: rocket , caught , zipper

[Jason Lu]

As is the case with geometric transformations, it might be good to try and
think of some basic transformations, where all other transformations are
compositions of the basic transformations.

I'm wondering if a bijection is the right word for this thing. It clearly
is a bijection, but it is more interesting than just a map from A to B. It
satisfies some other conditions which could make an isomorphism if you
came up with an appropriate binary operation on siteswaps.

Instead of transforming throws, you can also think of transformation on
states. E.g. I think of shower patterns as
*-*-*...
where as ground state is
***...
without the gaps.
So the transformation from all siteswaps to shower can be done by just
adding a gap in between every beat of the state.

J

[Boppo]

On Nov 2, 1:07 am, tab.jugg...@gmail.com.nospam.com (TABjuggler)
wrote:

> *The order of a siteswap's numbers do not matter, but are usually written
> with the largest number first (ex: 423 = 234 = 342)
>
> *Just because the average is a whole number does not mean that the
> siteswap is valid. However, you can always take a permutation of the
> numbers and get a valid siteswap. (ex. 432 = (4+3+2)/3 = 9/3 = 3 Balls,
> but the siteswap is not actually possible. So you have to look at all the
> combinations of those numbers (423, 432, 234, 243, 342, 324). Due to the
> above observation (423 = 234 = 342 and 432 = 324 = 243). That means that
> you only really have two possible permutations of the siteswap numbers and
> one doesn't work. That means that 423 = 234 = 342 are all valid
> siteswaps.

These two observations conflict; the first is overstated.

I would say it as: the rotation of a trick doesn't matter (all that
changes is the level of excitation) but the order of the throws does
matter.

423 and 234 are different rotations of the same trick, but 234 and 432
have the throws in a different order and the latter is invalid.

-boppo

[David Cherepov]

Jason Lu wrote:
>
> David Cherepov wrote:

> >
> > Jason Lu wrote:
> > >
> > > One nice one is the bijection from any async siteswap to shower patterns.
> > > Map n to ([2n-2]x,2x) or 2n-1 1
> > > Then you get a tonne of shower siteswaps.
> > > e.g. 64 -> d171
> >

> > b171 not d171 arithmetic FAIL but nice observation, I like math, but I
> > wasn't previously very interested to look at mathy siteswap things, but
> > now I'll try to do that more
>
> oops!

>
> >
> > > or
> > > 73 -> d151
> > >
> > > Like 73, d151 is excited state (relative to ground state 5 ball shower).
A
> > > 6 is required to excite to 73 and similarly b1 is required fro d151.
> > >

> > > There are also some mathsy bits and bobs which I've thought of, but
aren't
> > > very structured. One of which:
> > > Any excited state requires more than 1 throw to return back to its state.
> >

> > No, it doesn't matter how many throws to return to a lower excitation
> > level. Of course it has to have more than 0 throws, but that's the only
> > limitation
> >
> > 667713 6677162 66771661 5741571424444....

> >
> > > This leads to an alternative definition for ground state. The proof is
> > > really easy if you guys want to give it a go.
> > >

> > > I just thought that the above rule doesn't apply to multiplexes, e.g.
> > > splits [32].
> > >

> > > J
> > >
> >
> >
> >
>
> I am not sure if I misunderstand what you've written or you've
> misunderstood what I've written. I didn't mention anything about a lower
> excitation level, whatever that is, just that an excited state requires
> more than 1 throw to return to that SAME excited state, i.e. any siteswap
> on that excited state is of period >1.
>
Yeah, I misunderstood what you were saying.

[David Cherepov]

> Having two balls/objects going into the same hand at the same time. I
> think beatmap deals with this a little bit, but I was never too interested
> in the notation - same to a certain extent with trad siteswap. I think
> two balls going in and out of the same hand together is quite interesting
> - I do it a little bit, but there's no way I could write it (nor do I have
> much interest in writing it).

> Will
>
>

Squeeze catches!

[David Cherepov]

I can't really say that you conclusively prove that your conjecture is
true in all cases. But then again, it would be pretty easy to prove with
sequences.

[Joost Dessing]

Hey Miika,
yes, I had actually seen that site before. I've been absent from
rec.juggling for a while and just had forgotten about it (and that it was
you who wrote it :))

Indeed, the difficult thing with these multi-frequency/polyrhythmic
patterns is figuring out which ones are fun to do. As mentioned in the
original post I created a list of patterns at some point. I printed out a
30 page sample of these and just randomly picked patterns with any number
of balls that seemed to be doable (in terms of number of balls and throw
heights) and started working on it. For me, the fun in that was in
translating the numbers into a juggling pattern, not coming up with
variations on the spot. I remember spending quite some time just on 2-ball
patterns :).
From what I understand, you have also created your own list/program. Let
me know if you want me to send you my lists anyways (I only created
beatmap and sync siteswap lists and I think I deleted anything with more
than 4 balls; and any pattern with more than 5 throws for a hand in a
cycle [that is, up to 5:4]).

At some point I had stable-ish 8:3 and 8:5 pattern with 4 balls, but
nowhere near the stability needed for inserting swaps. But I was working
on 3:2, 4:3, 5:2, 5:3, 5:4 swaps.
One interesting aspect of these patterns (except maybe 3:2) from a
juggler's point of view is that you can definitely 'feel' when you capture
the pattern, while at that point, from a spectator's point of view it
still looks like a mess.

I think in the original post many people were asking about passing
variations, but I don't know how much of this has actually been explored.

greetings,
Joost

[Chris Bendall]

Miika wrote:
>
> Hello everybody :-)
>
> I like juggling. I also like thinking about juggling. In particular I
> like thinking about siteswap theory. Stuff like the notation, states,
> transitions, transformations, relationships between siteswaps, and some
> other mathematical aspects are really interesting to me. But after playing
> around with these concepts for close to ten years now, it's hard to come
> up with new stuff that still excites me and makes me spend hours trying to
> wrap my head around how and why it works. Yet I know there is lots more to
> figure out, if we just find the right approach.
>

> My question today is, what else is there? Do you know some weird aspect
> of siteswaps that I haven't encountered before? Have you had any ideas
> like this that you thought were cool but haven't really shared with
> anyone? Do you know someone that has? What mathematical results haven't
> been published yet?
>
> Even if it's some small thing, I'd love to hear it. Or if you have some
> confusion about some aspect of siteswaps, perhaps sharing it here would
> help figure it out. Or if you can think of an unanswered question about
> something related to siteswaps, we could try that too.
>
>

> Swap away,
>
> -Miika
>
> --
>

> Siteswaps of the day: pear , apple , mango
>

Hey Miika,

This is something extrapolated through a good friend and myself (with help
from TAB as well).

After reading Burkard Polster's "The Mathematics of Juggling" we came
across the notion of other means of checking an average valid siteswap (as
explained by TAB).

441 ---> 4+4+1=9
p=period = 3
9/p = 3
Therefor it is a 3 ball siteswap

As TAB was saying some average valid siteswaps do not work

432 ---> 4+3+2=9
9/p=3
BUT...
if a 4 and 3 are thrown in that order they will land at the same time
(meaning it doesn't "work")

The suggested method for checking from Polster's book states

take the s.s.
say 7531
add 0->(p-1)
p=4


7531
0123
----
7654

now mod out p

7654
4444
----
3210

as long as you get 0->(p-1) after the mod you have a valid s.s. (they
don't have to be in order, it just worked out as such because of the s.s.)

Here's where the extrapolation came in...

Reverse it!
Determine the period first then
write 0->(p-1) in any order
say...
(p=5)
40312

now subtract 0->(p-1) in order

40312
01234
-----
4(-1)1(-2)(-2)

normally negatives are a bad thing in s.s. but we can deal with them here

decide the number of objects you wish to juggle
say 6

you have a couple options here
1) add 6 to the numbers and you have a valid s.s.

4(-1)1(-2)(-2)
6 6 6 6 6
--------------
a5744

30/5=6

Tada!

or

2)divide the number of objects up in to pieces (the same number as p-1),
then multiply p (or a multiple of p) to each of them, then add them to the
set of numbers

6=1+2+0+1+2
12012

5*1=5 5*2=10 5*0=0 5*1=5 5*2=10
5a01a

so...
4(-1)1(-2)(-2)
5 a 0 5 a
--------------
99138

30/5=6
Yea!

That is just the tip of the iceberg...


lets say we do something along the lines of
p=3
o=3

(subtract)
021
012
----
01(-1)

(disperse and multiply)
3=2+1+0

(3) 210
630

(add)
01(-1)
63 0
------
64(-1)

GASP!!! there is a negative in the s.s.

not to worry...

just add the period (or a multiple of the period depending on how large
the negative is) to the negative number

64(-1)
3
------
642

6+4+2=12
12/3=4
Tada!


Extra Notes...

adding p to the s.s. is valid even if there are no negative numbers
take 531

it can become 561 or 831 or 534 or even 864!

you can also subtract
7531

3531 (just 531 with an extra 3 - to make it one sided)

or mix it up
7531

3935
it still works...

just remember adding or subtracting p (or multiples of p) will change the
number of balls in the s.s. (that can be proven by the average theorem).



Hope this sorta makes sense, I know it took me a minute or two to grasp
what my friend was saying on some parts.

I hope to make another post on this soon about matrices and finding s.s.
with a similar theme in mind.

If anyone finds this helpful or would like to point out any errors I look
forward to hearing from you and learning.

-Chris

[Boppo]

On Nov 2, 7:05 am, alia...@yahoo.com.nospam.com (David Cherepov)
wrote:

What do you find lacking in the proof by induction? Specifically, if
you prepend one more block of throws to the trick, having length one
more than the former first block, what must the throw heights be?

Case for n=1 is true: 1^3 = Triangle(1)^2 = 1^2 = 1

Suppose case n is true: n^2 ... n^2 (n-1)^2 ... (n-1)^2 .....999441
is valid, and equals sum of cubes up to that level.

Want to prepend (n+1) throws up front, and keep the trick valid (and
ground state). What must the throw height be? The trick used to
start with throw height n^2, and had n of them. So from the former
first throw, the last-to-land lands n^2+n later. (Example: in 999441,
n=3, and there are three 9s, so the first 9 lands 9 after the first
throw (obviously... what it means to be a 9), which is to say in the
10th position counting the 9 as position #1, but there are two more
nines, so the last 9 lands twelve after the first throw, 12 = 9+3 =
n^2+n for n=3.)

Now, there are (n+1) additional throws, so they have to land all the
previous time (= throw height) later, plus 1 at the end (so it doesn't
squeeze with the former last landing), plus n more at the front
because of the needed delay getting all n+1 throws up in the air
before you even start the former trick. (It's only n more, and not n
+1 more, because after the new first throw, there are only n remaining
new balls you have to throw, and not n+1 more throws).

The needed throw height, then = n^2+n (former length) + 1 (after
former last landing) + n (more remaining of the n+1 after making first
throw) = n^2+n+1+n = n^2 + 2n + 1 = (n+1)^2. So the throw height needs
to be the square of the next higher number. This establishes the sum-
of-triangles-squared part. So what remains is the sum-of-cubes part,
which was already established above but can be repeated here: if you
have (n+1) iterations of a throw whose value is (n+1)^2, then the
total combined added throw height is (n+1)(n+1)^2 = (n+1)^3.

QED.

-boppo

[Brook Roberts]

Multi-frequency is in some sense regularly done - whenever a passer says
to do a hurry, it means to do a throw much earlier than you'd want to -
breaking the rhythm and can (if it is not a two-sided pattern) result in
one hand having more beats where it throws.

[Jason Lu]

This is a nice method of checking validity of a siteswap!
I understand it as: it checks that the balls you throw don't end up
landing in the same beat and instead land in an async rhythm.

I haven't tested it, but I would guess that this doesn't work for excited
state siteswaps, like 723 for example. It checks that none of the balls
land in the same beat, but it won't yield the same result of getting 0 to
p-1 in some unspecified order.

[Peter Bone]

This method of generating siteswaps is given in Ben Beever's book here
(Permutations).
http://www.jugglingdb.com/compendium/geek/notation/siteswap/bensguide.html?page=4
However, he doesn't mention the part you gave in 2) for dividing up the
number of objects and multiplying by the period for each.

[5am]

David Cain wrote:
>
> Hmmm. Something you might not know about siteswaps. Audiences don't
> care about them one little bit! Sad, perhaps, but true.
>
>

That depends purely of the audience. Maybe you just have wrong kind of
audiences.

[Boppo]

Whether or not the observation applies to all audiences, I am happy to
say that an increasing number of jugglers disregard it altogether.

-boppo

[Daniel S.]

> Regarding the six ball trick, 999441:
>
> Interestingly, the throws are all perfect squares, and there are n
> repetitions of each value of n^2. (So there are three 9s, which is
> 3^2 = 9, two 4s, which is 2^2=4, etc.) This trick is the third in the
> sequence 1, 441, 999441. All continuations of this series are valid
> vanilla siteswaps: 16 16 16 16 9 9 9 4 4 1 is valid trick, with 10
> balls. (Oh, sorry, that's gggg999441 to y'all.)
>
> The fact that all these tricks are valid proves an interesting
> mathematical relationship. Consider the series of perfect cubes, 1, 8,
> 27, 64 ... and the series of triangular numbers, 1,3,6,10,15 (which
> are all partial sums of all the counting numbers, 1, 1+2, 1+2+3,
> 1+2+3+4 ...) The relationship is, that the sum of the first n cubes
> (1+8+27) = the nth triangular number squared (6^2). And this is
> *because* this trick and its extensions are all valid vanilla
> siteswaps. It's straightforward to prove by induction, and the
> relationship to the triangular numbers is how many balls each
> different trick has; the square of that number is the total value of
> all the throws. They're triangular because each next square has one
> more iteration than the last - one 1, two 4s, three 9s, etc. So the
> six ball trick has 6 (balls) * 6 (period) = 36 total throw value.
> Which is the sum of the first three cubes, 1+8+27.
>
> Did you know that? (Is it the sort of thing you are looking for?)
>
> -boppo
>
>

Pretty neat. Seems also interesting that 1 , 441 , 999441 , and 16 16 16
16 999441 all have only one orbit. Will this be the case infinitely?

[Chris Bendall]

Miika wrote:
>
> Hello everybody :-)
>
> I like juggling. I also like thinking about juggling. In particular I
> like thinking about siteswap theory. Stuff like the notation, states,
> transitions, transformations, relationships between siteswaps, and some
> other mathematical aspects are really interesting to me. But after playing
> around with these concepts for close to ten years now, it's hard to come
> up with new stuff that still excites me and makes me spend hours trying to
> wrap my head around how and why it works. Yet I know there is lots more to
> figure out, if we just find the right approach.
>
> My question today is, what else is there? Do you know some weird aspect
> of siteswaps that I haven't encountered before? Have you had any ideas
> like this that you thought were cool but haven't really shared with
> anyone? Do you know someone that has? What mathematical results haven't
> been published yet?
>
> Even if it's some small thing, I'd love to hear it. Or if you have some
> confusion about some aspect of siteswaps, perhaps sharing it here would
> help figure it out. Or if you can think of an unanswered question about
> something related to siteswaps, we could try that too.
>
>
> Swap away,
>
> -Miika
>
> --
>
> Siteswaps of the day: pear , apple , mango
>

Another interesting topic I have stumbled upon is negatives in siteswap
(but leaving them there and learning how to juggle them).

http://www.youtube.com/watch?v=_5Q4UhyajC4

that is all I know, but you asked for something different so why not time
travel...

-Chris

[Chris Bendall]

Miika wrote:
>
> Hello everybody :-)
>
> I like juggling. I also like thinking about juggling. In particular I
> like thinking about siteswap theory. Stuff like the notation, states,
> transitions, transformations, relationships between siteswaps, and some
> other mathematical aspects are really interesting to me. But after playing
> around with these concepts for close to ten years now, it's hard to come
> up with new stuff that still excites me and makes me spend hours trying to
> wrap my head around how and why it works. Yet I know there is lots more to
> figure out, if we just find the right approach.
>
> My question today is, what else is there? Do you know some weird aspect
> of siteswaps that I haven't encountered before? Have you had any ideas
> like this that you thought were cool but haven't really shared with
> anyone? Do you know someone that has? What mathematical results haven't
> been published yet?
>
> Even if it's some small thing, I'd love to hear it. Or if you have some
> confusion about some aspect of siteswaps, perhaps sharing it here would
> help figure it out. Or if you can think of an unanswered question about
> something related to siteswaps, we could try that too.
>
>
> Swap away,
>
> -Miika
>
> --
>
> Siteswaps of the day: pear , apple , mango
>

Another topic from "The Mathematics of Juggling" to discuss and expand
upon...

He suggest to generate a random valid s.s. create a matrix with the number
of desired juggling objects in the upper left-hand corner. From the make
a matrix of equal length and height decreasing number as read Left to
Right and increasing going from Top to Bottom.

e.g.

3 2 1
4 3 2
5 4 3

now select a number from the first column (say 5)

moving across the matrix choose a number from every column so that is does
not line up with a number from the same row.

so you can choose 531 or 522 - that's it

you could also go 333 342 423 441
(yes there is a repeat but it is the same thing just transposed)

now you can increase this to whatever dimension/ number of objects you like
e.g.

5 4 3 2 1 0 (-1)
6 5 4 3 2 1 0
7 6 5 4 3 2 1
8 7 6 5 4 3 2
9 8 7 6 5 4 3
a 9 8 7 6 5 4
b a 9 8 7 6 5

this will generate any 5 ball s.s. (up to a period of 7), you could also
take the top 3 x 3 matrix and have any 5 ball 3 period s.s.

5 4 3|2 1
6 5 4|3 2
7 6 5|4 3
------
8 7 6 5 4
9 8 7 6 5

It's all part of the same machine.

Now for the expanding...

take a smaller set of data

3 2 1 0 (-1)
4 3 2 1 0
5 4 3 2 1
6 5 4 3 2
7 6 5 4 3

you can use that (-1)

let's say 7531(-1) [it takes a number from every row and column]

while this is normally invalid all we need to do is add yet again - the
period to the negative number (if you read my previous post you'd
understand the irony).

so take 7531(-1) and add 5 to the (-1)...

you get 7531[4]
which if you think about it, you're just making a one sided 4 ball trick
two sided.

If you added a multiple of 5 (say 10) then you get 75319, the same as
97531!

While I failed to state this in my other post I will say it here and now...

I don't know if this is specifically for ground state s.s. or what, but if
someone can show me something new/an error I would love to learn.

[Chris Bendall]

Not entirely sure about excited, I still need to woodshed that one...

-Chris

[Chris Bendall]

Thanks for the link, gotta check it out in more detail soon...

-Chris

[Adrian G]

Actually, this method works for any sort of siteswap (it can even easily
be extended for synch and multiplex siteswaps)

take 723 for example

723
012 Add 0,1,...(p-1)
---
735

735
333 Work out each throw mod p
102

102 has no repetitions so 723 is valid

Adrian

[TABjuggler]

I think you slightly misunderstood me. I'm just not sure how to say (423
= 234 = 342) in more correct terms. I'm not talking about changing the
order of the throws. Changing the order of the throws makes the siteswap
invalid potentially.

I didn't consider excitation though. In my example all 3 of those
siteswaps should be ground state, so they don't need to be excited.

If the average works out for a siteswap, but it isn't valid, you can
change the order of the throws to some permutation of those numbers that
IS valid. That was my point.

What is an example of a siteswap that isn't period 2, which might be a
good siteswap to think about the difference in rotations of the same
siteswap?

[TABjuggler]

Sweet I was really feeling lazy and didn't want to type most of this out.
Well done :P

[Adrian G]

Chris Bendall wrote:
>
>
> Another interesting topic I have stumbled upon is negatives in siteswap
> (but leaving them there and learning how to juggle them).
>
> http://www.youtube.com/watch?v=_5Q4UhyajC4
>
> that is all I know, but you asked for something different so why not time
> travel...
>
> -Chris
>

On the topic of negative siteswaps, has anyone thought about trying to put
them into state diagrams?

I had a go but wasn't having much luck.

Adrian

[Adrian G]

> While I failed to state this in my other post I will say it here and now...
>
> I don't know if this is specifically for ground state s.s. or what, but if
> someone can show me something new/an error I would love to learn.
>
> -Chris
>

As far as I can tell, yes that is only for ground state, JugglingLab give
me the following for 5 ball period 3 max-throw 7 ('*'s on each side
represent excited state patterns):

555
645
663
744
753
* 672 *
* 771 *

The ground state patterns can all be gotten through that method but the
others can't

However, if we give 771 the entrance and exit it needs to be ground state,
we get 667713 which can be found in your grid.

I'm not sure but maybe that we can find the pattern *with* the entrance
and exit would allow us to work out a way we can find the original pattern.

This is my idea:

Let's say we want to find period 3, 5 ball siteswaps excited by a 7.

first we take a smaller grid like this, we use the 7 in the first column
and work out the exit, which is 44.

5 4 3
6 5 4
7 6 5

Now, we know we're starting with a 7, so we do this as before but only
finding three numbers before using the 44.

5 4 3 2 1 0

6 5 4 3 2 1

(7)6 5 4 3 2
8 7 6 5 (4)3
9 8 7 6 5 (4)

a 9 8 7 6 5

So we have:
447 (744)
483
537
582
933
942

Anyway, you get the idea

Adrian

[Adrian G]

Um, gggg999441 has three orbits:

gggg999441....mod p
6666999441

if we now look at the landings, we get:
6666999441
1449996666
We can see that a ball thrown as the last 9 would shift backwards before
becoming a 6, a 1 and then back to a 9.

This doesn't involve the fours. the middle two sixes become fours and then
immediately become sixes again.

Adrian

[Daniel S.]

Doh! Miscount.

Thanks,
Daniel

[Miika]

Adrian G wrote:
> OK, I have a question that I've been wondering about, more to do with
> stack notation but anyway...
>
> In all the descriptions of stack notation I've read it says that it can do
> anything that siteswap can, but I've naver seen any synchronous examples.
>
> Standard stack notation wouldn't work, but has anyone made any extensions
> to it to support synch?
>
> The best I could think of would be having two 'stacks' and if there is an
> x after the number then it would cross to the other stack to normal (e.g.
> a 4x would cross but a 5x would go straight, like siteswap). so (4,2x)*
> (box) would end up something like (2,1)* because you'd count the number of
> balls thrown from each hand does that make sense?
>
> Does anyone have any other ideas?
>
> Also on the topic of stack notation, as far as I can tell, multiplexes are
> impossible, which I found quite interesting. This is because a multiplex
> pattern will always contain a squeeze catch (e.g. there is a 2 and a 3
> that are caught at the same time in [43]23 ). Stack notation doesn't allow
> two balls to be caught at once in the same hand as the ball is just
> 'inserted' into the stack (which is why all stack notation patterns are
> valid). Therefore it is impossible to notate multiplex patterns, is that
> correct?
>

A nice question, and the answer is actually simpler than you think. Well,
at least I thought that when starting to write this :-)

As the basic stack notation doesn't care about the times when the balls
are juggled, the decent way to use multiplexes and synch patterns is the
same format we use for siteswaps to determine the times of the throws and
the hands used, while letting the numbers deal with the intertwining of
the order. So to turn [43]23 into a stack sequence, we first look at how
the orbits form. Here we have our first problem, does the multiplexed 3
stay the same ball throughout the pattern or is it part of the orbit with
the 4 and 2? My preference is to look at when the balls about to be
multiplexed were last thrown. One was previously a 3 and the other was a
2, so I'd assign the higher throw to the first throw in the multiplex,
which is the 4, leaving the 2 to become the 3. So in fact the 3 is not its
own orbit (that's the pattern I'd call [34]23 if it's required to
distuinguish the two). So the orbits of [43]23 form in the same way as in
6334, if there was no multiplex. We convert this to the stack sequence
$4334 and just put the first two throws in brackets to communicate the
timing $[43]34. For the other pattern, we get
[34]34 -> 4534 -> $4434 -> $[44]34. So that's one answer.

But the problem isn't completely dealt with yet. What if we had chosen
[43]23 to have the three separate orbits? We'd have [43]23 -> 6424 ->
$4324 -> $[43]24, and [34]23 -> 5524 -> $4424 -> $[44]24. In a
siteswap-free world, who's to say how the patterns $[43]34 and $[44]24, or
the patterns $[44]34 and $[43]24, could be distinguished as being the
same? I don't have an immediate solution, so I'd just say that there's two
ways of writing the same pattern in stack notation, and allow it to
complicate things even worse for patterns with more and larger multiplexes.

Then for the synchronous stacks. Here too we could just apply the
(R,L)(R,L)* format in the same way as above for [RR]LR, but now I'm
discouraged that it would lead to worse complications than above, so let's
not get into it. At the very least it wouldn't allow for the nice rule of
writing the mirror image of a pattern by just switching the throws, like
in siteswap we can say that (8,6)(4,6) is the mirror image of (6,8)(6,4).
That property is needed to be able to use * as a shorthand for symmetric
patterns. Let's try the approach of using two stacks instead.

So each hand throws to its own stack, unless there is an x attached to
the throw. This should work, but what ambiguities does it create? For
anything without crossing throws, there is no problem, like (6,4) =
$(3,2). Or anything without throws that are simultaneously thrown to the
same stack, like (8,4)(4x,4x) = $(3,2)(2x,2x). In both these cases I'm
thinking that the bottom ball in each stack is removed (if there is no 0)
before inserting them back into place. But then there are patterns like
(8x,4)(4x,8)(8,4x)(0,4x). Since we know the siteswap, we can work out how
we'd like the stacks to look after each pair of throws and then determine
how to use stack notation to record these changes. So, using letters for
the names of the balls, the stacks should be (ABC,DE) -> (BC,EDA) ->
(C,DBAE) -> (DC,BAE) -> (DBC,AE) and then it repeats with the A and D
switched.

So here's where we have to do some thinking. Once we remove the balls
from the bottom of the stacks, the order we put them back in can now
matter since they now go in the same stack. I was first thinking we should
always put the crossing throw in first, or the one that's written first. I
now feel the best way is to put in the smaller one and then the larger one
(and neither "sees" the other ball where it was before this throw). Using
this rule the whole pattern is $(3x,2)(2x,4)(2,1x)(0,2x). This system
actually works much better than I anticipated! For example, the * can be
used for mirrored patterns, like the box becomes (4,2x)* = (4,2x)(2x,4) =
$(2,1x)(1x,2) = $(2,1x)*.

Only small gripes remain, such as two simultaneous throws to the same
position shouldn't be allowed. Like using (3x,3) never creates a pattern
that couldn't be written with either (2x,3) or (3,2x), but only causes
confusion. Also determining the number of balls in this type of
synchronous stack sequence isn't as straightforward as with normal stack
patterns.

Now that I worked that out, I want to revisit the multiplexes. Sorry for
the confusion of presenting another method of doing that in the same post
:-) This time for the multiplex, let's remove all the balls in the
multiplex from the stack before returning them in increasing order of
their values. That sounds like it should work! The states for the two
orbit [43]23 are [xx]xx, xxxx and x[xx]x. The full cycle of repeating
these with "colored balls" the is [AB]CD -> CDBA -> D[BC]A -> [BC]AD ->
ADCB -> D[CA]B -> [CA]BD -> BDAC -> D[AB]C -> [AB]CD. From that we can
read the stacks by ignoring the brackets. We now deduce that a suitable
stack sequence for these changes is $[43]34 by working out each change at
a time. Similarly for the other pattern [34]23 we derive the form $[34]34,
which is pleasantly similar to the first one. Even though we still need to
disallow same values within a multiplex, like [33], I find this method
superior to the one described above. At least now it seems that each
pattern only has one way to write it!

Mixing it together: (6x,[4x4])(2,[4x4x])* = $(4x,[2x3])(2,[3x4x])* :-)

So what do you think? Does that work? I realize I was working parts of
this out as I was typing, so do ask for clarification where needed. Now
let's see someone count the crossing in the braids for these patterns!
(That's what vanilla stack sequences can be used for.)

-Miika

--

Siteswaps of the day: skull , toast , prank

[Chris Bendall]

Miika wrote:
>
> Hello everybody :-)
>
> I like juggling. I also like thinking about juggling. In particular I
> like thinking about siteswap theory. Stuff like the notation, states,
> transitions, transformations, relationships between siteswaps, and some
> other mathematical aspects are really interesting to me. But after playing
> around with these concepts for close to ten years now, it's hard to come
> up with new stuff that still excites me and makes me spend hours trying to
> wrap my head around how and why it works. Yet I know there is lots more to
> figure out, if we just find the right approach.
>
> My question today is, what else is there? Do you know some weird aspect
> of siteswaps that I haven't encountered before? Have you had any ideas
> like this that you thought were cool but haven't really shared with
> anyone? Do you know someone that has? What mathematical results haven't
> been published yet?
>
> Even if it's some small thing, I'd love to hear it. Or if you have some
> confusion about some aspect of siteswaps, perhaps sharing it here would
> help figure it out. Or if you can think of an unanswered question about
> something related to siteswaps, we could try that too.
>
>
> Swap away,
>

> -Miika
>
> --
>

> Siteswaps of the day: pear , apple , mango
>

Something to think about for you and whoever else decides to read this...

I've been working on 4 clubs and the transitions that come with them, I am
curious as to what the s.s. would be for a 3 club cascade with a kick-up
into 4.

I'm throwing a 5 out of my left hand then a 4 out of my right, from there
a kick-up from my left foot (hopefully a double spin) then a 4 from my
left hand to make room for the kick-up (the right hand is still waiting
for the first 4 to land). After that it's all 4's from there...

Unfortunately I don't have a video up but I'm sure I have seen it in
numerous performances.

I hope to get a video up so a better idea can be given, but any help is
appreciated...

-Chris

[Boppo]

On Nov 3, 2:20 pm, gossey2...@cs.com.nospam.com (Chris Bendall) wrote:

> I'm throwing a 5 out of my left hand then a 4 out of my right,

These two throws would collide if that's really what you're doing.
Since whatever you're doing is working, I don't think that's what
you're doing.

-boppo

[Emman]

Miika wrote:
>
> Emman wrote:
> > Siteswap difficulty isn't as objective as one might think.
> > ;)
>
> But how does DB55555 factor in difficulty between (?) DB97531 and D666666
> ?
>
> -Miika
>
> --
>
> Siteswaps of the day: spy , ski , sex
>

It sounds like an absolute bitch.

I like those siteswaps of the day...

[brembl]

Emman wrote:
>
> Miika wrote:
> >
> > Emman wrote:
> > > Siteswap difficulty isn't as objective as one might think.
> > > ;)
> >
> > But how does DB55555 factor in difficulty between (?) DB97531 and D666666
> > ?
> >
> > -Miika
> >
> > --
> >
> > Siteswaps of the day: spy , ski , sex
> >
>
> It sounds like an absolute bitch.
>
> I like those siteswaps of the day...
>
>
>

How about: goaly, sewer, newer, sinew, diner, piety, joust, rinse, gammas,
mammas, astray, payday, [favoritewordhere]

[Emman]

I'd say the only one I like more then Miika's first two is diner.
Also, [Icanttellyoumyfavoriteword].

[Chris Bendall]

Thanks Boppo - I even posted that in my previous replies... (I tend to
forget things like that).

It's doing a double spin but I could just be a 3 that I'm throwing as a
double (just very low).

I really hope to have a video up so someone can show me the proper way...

[Adrian G]

I'm not sure about 'proper' as there are many ways to do it, but here's
the way I do it.

From a 3 club cascade, a straight double (siteswap 4) from my left hand.
right after that a single spin kickup to my right hand. Then straight into
four clubs.

It seems pretty similar to what you're doing but just with a single kickup
instead (as far as I can tell your 'siteswap 5' at the start seems to be
low enough to almost count as a 3, though maybe I've just misunderstood
what you wrote)

As for notating that, I think beatmap would be the way to go.
If I had to do it with siteswap I would do something like n333<3-club
stuff>33344444... where n is the action of putting the club on your
foot/the ground and the numerical value of n is whatever it needs to be to
land in the correct place (which will of course change depending on your 3
club stuff.

Adrian

[Adrian G]

Miika wrote:
>
> A nice question, and the answer is actually simpler than you think. Well,
> at least I thought that when starting to write this :-)
>
> As the basic stack notation doesn't care about the times when the balls
> are juggled, the decent way to use multiplexes and synch patterns is the
> same format we use for siteswaps to determine the times of the throws and
> the hands used, while letting the numbers deal with the intertwining of
> the order. So to turn [43]23 into a stack sequence, we first look at how
> the orbits form. Here we have our first problem, does the multiplexed 3
> stay the same ball throughout the pattern or is it part of the orbit with
> the 4 and 2? My preference is to look at when the balls about to be
> multiplexed were last thrown. One was previously a 3 and the other was a
> 2, so I'd assign the higher throw to the first throw in the multiplex,
> which is the 4, leaving the 2 to become the 3. So in fact the 3 is not its
> own orbit (that's the pattern I'd call [34]23 if it's required to
> distuinguish the two).

Agree with all that so far.

> So the orbits of [43]23 form in the same way as in
> 6334,

Just wondering, how did you work out that it was the same orbits?

> if there was no multiplex. We convert this to the stack sequence
> $4334 and just put the first two throws in brackets to communicate the
> timing $[43]34. For the other pattern, we get
> [34]34 -> 4534 -> $4434 -> $[44]34. So that's one answer.

I assume you mean [34]23 not [34]34 :)

> But the problem isn't completely dealt with yet. What if we had chosen
> [43]23 to have the three separate orbits? We'd have [43]23 -> 6424 ->
> $4324 -> $[43]24, and [34]23 -> 5524 -> $4424 -> $[44]24. In a
> siteswap-free world, who's to say how the patterns $[43]34 and $[44]24, or
> the patterns $[44]34 and $[43]24, could be distinguished as being the
> same? I don't have an immediate solution, so I'd just say that there's two
> ways of writing the same pattern in stack notation, and allow it to
> complicate things even worse for patterns with more and larger multiplexes.

Not quite sure whether I get that part. As far as I can tell, there is
only two forms of [43]23. One with three orbits ([34]23) and one with two
orbits ([43]23), so I don't quite get what you were saying.

> Then for the synchronous stacks. Here too we could just apply the
> (R,L)(R,L)* format in the same way as above for [RR]LR, but now I'm
> discouraged that it would lead to worse complications than above, so let's
> not get into it. At the very least it wouldn't allow for the nice rule of
> writing the mirror image of a pattern by just switching the throws, like
> in siteswap we can say that (8,6)(4,6) is the mirror image of (6,8)(6,4).
> That property is needed to be able to use * as a shorthand for symmetric
> patterns. Let's try the approach of using two stacks instead.
>
> So each hand throws to its own stack, unless there is an x attached to
> the throw. This should work, but what ambiguities does it create? For
> anything without crossing throws, there is no problem, like (6,4) =
> $(3,2). Or anything without throws that are simultaneously thrown to the
> same stack, like (8,4)(4x,4x) = $(3,2)(2x,2x). In both these cases I'm
> thinking that the bottom ball in each stack is removed (if there is no 0)
> before inserting them back into place. But then there are patterns like
> (8x,4)(4x,8)(8,4x)(0,4x). Since we know the siteswap, we can work out how
> we'd like the stacks to look after each pair of throws and then determine
> how to use stack notation to record these changes. So, using letters for
> the names of the balls, the stacks should be (ABC,DE) -> (BC,EDA) ->
> (C,DBAE) -> (DC,BAE) -> (DBC,AE) and then it repeats with the A and D
> switched.
>
> So here's where we have to do some thinking. Once we remove the balls
> from the bottom of the stacks, the order we put them back in can now
> matter since they now go in the same stack.

I didn't even think of that problem.

> I was first thinking we should
> always put the crossing throw in first, or the one that's written first. I
> now feel the best way is to put in the smaller one and then the larger one
> (and neither "sees" the other ball where it was before this throw). Using
> this rule the whole pattern is $(3x,2)(2x,4)(2,1x)(0,2x). This system
> actually works much better than I anticipated! For example, the * can be
> used for mirrored patterns, like the box becomes (4,2x)* = (4,2x)(2x,4) =
> $(2,1x)(1x,2) = $(2,1x)*.

That looks pretty good, but I don't know what you mean by:

> (and neither "sees" the other ball where it was before this throw)

> Only small gripes remain, such as two simultaneous throws to the same
> position shouldn't be allowed. Like using (3x,3) never creates a pattern
> that couldn't be written with either (2x,3) or (3,2x), but only causes
> confusion. Also determining the number of balls in this type of
> synchronous stack sequence isn't as straightforward as with normal stack
> patterns.

I get the feeling that there may be an easy way to do this, but I can't
think of it at the moment. I was thinking the highest total for the pair
but that doesn't quite work.

> Now that I worked that out, I want to revisit the multiplexes. Sorry for
> the confusion of presenting another method of doing that in the same post
> :-) This time for the multiplex, let's remove all the balls in the
> multiplex from the stack before returning them in increasing order of
> their values. That sounds like it should work! The states for the two
> orbit [43]23 are [xx]xx, xxxx and x[xx]x. The full cycle of repeating
> these with "colored balls" the is [AB]CD -> CDBA -> D[BC]A -> [BC]AD ->
> ADCB -> D[CA]B -> [CA]BD -> BDAC -> D[AB]C -> [AB]CD. From that we can
> read the stacks by ignoring the brackets. We now deduce that a suitable
> stack sequence for these changes is $[43]34 by working out each change at
> a time. Similarly for the other pattern [34]23 we derive the form $[34]34,
> which is pleasantly similar to the first one. Even though we still need to
> disallow same values within a multiplex, like [33], I find this method
> superior to the one described above. At least now it seems that each
> pattern only has one way to write it!

Yes, that definitely seems like a better way. It still seems weird though
that some throws just 'happen' to not be inserted and instead fall down at
the same time.

> Mixing it together: (6x,[4x4])(2,[4x4x])* = $(4x,[2x3])(2,[3x4x])* :-)

How long did *that* one take you to work out? :P

> So what do you think? Does that work? I realize I was working parts of
> this out as I was typing, so do ask for clarification where needed. Now
> let's see someone count the crossing in the braids for these patterns!
> (That's what vanilla stack sequences can be used for.)

Yeah, that seems pretty good overall.

Damn, now I don't have an excuse not to put the conversions for synch and
multiplex into my simulator :)

Also, is the '$' symbol to represent stack notation something you've
thought up or is it a widely used syntax? I've never seen it before but it
makes sense to have something to distinguish between them properly.

Adrian

[Miika]

Boppo wrote:
> Miika wrote:
> >  Hello everybody :-)
> >
> >  I like juggling. I also like thinking about juggling. In particular I
> > like thinking about siteswap theory. Stuff like the notation, states,
> > transitions, transformations, relationships between siteswaps, and some
> > other mathematical aspects are really interesting to me. But after playing
> > around with these concepts for close to ten years now, it's hard to come
> > up with new stuff that still excites me and makes me spend hours trying to
> > wrap my head around how and why it works. Yet I know there is lots more to
> > figure out, if we just find the right approach.
> >
> >  My question today is, what else is there? Do you know some weird aspect
> > of siteswaps that I haven't encountered before? Have you had any ideas
> > like this that you thought were cool but haven't really shared with
> > anyone? Do you know someone that has? What mathematical results haven't
> > been published yet?
> >
> >  Even if it's some small thing, I'd love to hear it. Or if you have some
> > confusion about some aspect of siteswaps, perhaps sharing it here would
> > help figure it out. Or if you can think of an unanswered question about
> > something related to siteswaps, we could try that too.
> >
> >  Swap away,
> >
> > -Miika
>

> Regarding the six ball trick, 999441:
>
> Interestingly, the throws are all perfect squares, and there are n
> repetitions of each value of n^2. (So there are three 9s, which is
> 3^2 = 9, two 4s, which is 2^2=4, etc.) This trick is the third in the
> sequence 1, 441, 999441. All continuations of this series are valid
> vanilla siteswaps: 16 16 16 16 9 9 9 4 4 1 is valid trick, with 10
> balls. (Oh, sorry, that's gggg999441 to y'all.)
>
> The fact that all these tricks are valid proves an interesting
> mathematical relationship. Consider the series of perfect cubes, 1, 8,
> 27, 64 ... and the series of triangular numbers, 1,3,6,10,15 (which
> are all partial sums of all the counting numbers, 1, 1+2, 1+2+3,
> 1+2+3+4 ...) The relationship is, that the sum of the first n cubes
> (1+8+27) = the nth triangular number squared (6^2). And this is
> *because* this trick and its extensions are all valid vanilla
> siteswaps. It's straightforward to prove by induction, and the
> relationship to the triangular numbers is how many balls each
> different trick has; the square of that number is the total value of
> all the throws. They're triangular because each next square has one
> more iteration than the last - one 1, two 4s, three 9s, etc. So the
> six ball trick has 6 (balls) * 6 (period) = 36 total throw value.
> Which is the sum of the first three cubes, 1+8+27.
>
> Did you know that? (Is it the sort of thing you are looking for?)
>

Great! I did not know that. And it is one of the sorts of thing I was
hoping for. I'm sure someone will use it in some sort of mathematical
publication at some point. I had noticed the freezeframes and squares
relationship, but this is much nicer. Did you stumble upon it by noticing
the pattern in 999441 and wondering if it continues with more balls? It's
hard to say what type of other relationships between siteswaps and
specific sets of numbers are lurking to be found.

I liked how you refered to the blocks of throws as working like the
individual throws in freezeframes. Some time ago I was working on defining
a form of isomorphisms for siteswaps, and using that terminology we could
say 999441 is isomorphic to 531 when choosing the blocks of throws to map
to each single throw. (I also allowed siteswaps within a block of throws,
so 999531 is similarly isomorphic to 531.) There were still some aspects
of the idea I didn't flesh out completely, but it seems that the concept
is useful in highlighting some similarities in the structure between
different patterns.

Now go make some whiteboard videos!

-Miika

--

Siteswaps of the day: idiom , proof , theorem

[Miika]

Q Juggler wrote:
> Hi Miika
>
> I invented the juggling number system long ago and I have not seen
> much new sense then, so I am also interested. I haven't read all of
> the stuff out there so I'm not totally sure if there really is
> anything new but I imagine so. But the newest thing for me is that the
> original siteswaps is explained on the wolfram math website. I have
> been wondering about it for the past 25 years. Boppo's last post is
> something to me also. I have a few things that I haven't heard or read
> about but I'll tell you later.
>
> Paul

Not much new in almost thirty years? :-) Hopefully the next few decades
will bring at least as much to the table as what we currently have. Of
course there is some limit to what you need siteswaps to do for you, but
it won't stop us from trying to apply it in some new fun way. You know,
like we do with real juggling! It's a great system you guys concocted! We
should really compile a list, or a database even, of the aspects of
siteswaps (and mathematically related things) that have so far been
discovered. That would be quite the project!

Too bad the MathWorld entry is a bit awful. It describes the sequence of
states for a pattern as being the siteswap and misses the fact that it's
rather the sequence of throws. Of course both describe a juggling pattern,
but still...

Anticipating your future reveals,

-Miika


P.S. Hopefully nobody from the Cambridge group will also reply, because
then I'd just pass out. Perhaps even the guy who invented siteswap states
could cause that to happen at this point :-)

--

Quantum juggling patterns of the day: ninth , yard , push

[Q Juggler]

Hi Miika

Thanks for the reply. I'm not much of a math whiz so I can't pretend
I understand everything that is written and I too feel a little woozy
when Boppo lets loose. One thing I would like make clear is that the
ground state is just the cascade or the fountain and excited states
are all the others. So for a given number of objects there is only one
ground state and many excited states. But what I think is interesting
is that there are different levels of excited states. The states that
some people call ground states are what I call level zero excited
states. The states that people refer to as excited are excited states
but are of a higher level or energy. It is hard to change the current
lingo so it is a little difficult for me to figure out what people
mean if they say “ground state” without any numbers to back it up. For
example, a 4b ground state is just 4. Level zero excited states are
53, 552, 633, 642, 5551... Level one excites states are 62, 561,
741... Second level excited states are 71, 822, 831... So that means
level zero excited states do not require transitions. It is a mouthful
to say ground state level excited state, so level zero or L0 seems
like a good alternative if you so choose to except it. If you want to
specify the levels, L0, L1, L2, L3... works. I think you all realize
that the transitional states are just some state that is split in two
parts. The four ball 53 state is used for many transitions to level
zero excited states. The 5 is the excitation and the 3 is the decay.
The 53 itself is a L0 short lived excited state and 56262623 is a long
lived L0 state and just the inner 62 is a L1 state. Sooo L0 requires
no transition and L1 requires at least one small excitation to get
into it. In the four ball 567123 pattern the 6712 within it is a L1
state and the 71 is a L2 state. Now for something completely
different. Catching and throwing an object is what I call a
manipulation. If you catch an incoming 5 and throw a 7 with the same
hand, it's a 5-7 manipulation. Example, in the three ball 441 pattern,
it has three manipulations. 1-4, 4-4, 4-1. I think this is interesting
because this is what makes each pattern unique and some combos of
manipulations are harder than others. Example, 633 is kind of hard
because the manipulations are 6-6, 3-3, 3-3. The vertical displacement
of 6-6 is a lot larger than the 3-3 manipulations. If the
manipulations are the same size like in 97531 it makes things easier.
The manips are 1-9, 3-7, 5-5, 7-3, 9-1. Notice that they all average
out to 5. In the new version of my program (coming out soon), I made a
graphic that shows all of the unique manipulations of a pattern. If
you limit the numbers between 1 min and 9 max, there are 9*9 or 81
unique manipulations. So for all you math whizzes, I have a question.
What is the shortest pattern that utilizes all the manipulations
(because it would make a good practice pattern). I know it must be
more than 81 and if you require both left and right to take part, it
is at least 162. First try limiting it to lets say 3 min and 7 max,
with five objects. Now try not to pass out. More later.

Paul

ps Miika, thanks for using the word quantum juggling.

[Chris Bendall]

I must apologize for my earlier statement...

There is a 5 thrown but it is 522 (not really sure how I missed that), it
goes along the lines of 52[2<kickup - double spin>]4444...

maybe that will shed some new light and solve this conundrum.

-Chris

[Adrian G]

Chris Bendall wrote:
>
> I must apologize for my earlier statement...
>
> There is a 5 thrown but it is 522 (not really sure how I missed that), it
> goes along the lines of 52[2<kickup - double spin>]4444...
>
> maybe that will shed some new light and solve this conundrum.
>
> -Chris
>
>

OK, that makes more sense now :)

Also remember that as 522 is ground state, it's not going to be very
different to 333 , except maybe a bit easier in this case.

But if you were to change it to 333 the only difference between my version
and your version is that I throw a four and then kickup as a single, you
kickup a double and then throw a four. Not positive, but I think this is
approximately equivalent to a 53 instead of 44 or something of the sort.

Adrian

[Daniel S.]

Miika wrote:
>
> Hello everybody :-)
>
> I like juggling. I also like thinking about juggling. In particular I
> like thinking about siteswap theory. Stuff like the notation, states,
> transitions, transformations, relationships between siteswaps, and some
> other mathematical aspects are really interesting to me. But after playing
> around with these concepts for close to ten years now, it's hard to come
> up with new stuff that still excites me and makes me spend hours trying to
> wrap my head around how and why it works. Yet I know there is lots more to
> figure out, if we just find the right approach.
>
> My question today is, what else is there? Do you know some weird aspect
> of siteswaps that I haven't encountered before? Have you had any ideas
> like this that you thought were cool but haven't really shared with
> anyone? Do you know someone that has? What mathematical results haven't
> been published yet?
>
> Even if it's some small thing, I'd love to hear it. Or if you have some
> confusion about some aspect of siteswaps, perhaps sharing it here would
> help figure it out. Or if you can think of an unanswered question about
> something related to siteswaps, we could try that too.
>
>
> Swap away,
>
> -Miika
>

> --
>

> Siteswaps of the day: pear , apple , mango
>

Hey Miika,

For what it's worth I have another siteswap puzzle idea :P

Imagine each line below is a layer of a cake with nine sides plus a top
and bottom. So… a disk with nine sides around the edges. There are 9
different disks each representing the numbers 1-9. There is an X on one
of the sides marking the throw and a V on another side marking the
re-throw.
Grab 9 random disks which have values that when added together are
divisible by 9. If for example we took the numbers 111233457 their sum is
27 which goes into 9 3 times. Because of the same proof [1] that makes
colorswap work (another siteswap puzzle) one will always have at least one
solution to the puzzle. The solution being a valid siteswap.

Rules:
The object of the puzzle is to rotate each disk so that there is one X and
one V in every column.

X V _ _ _ _ _ _ _ 1
X V _ _ _ _ _ _ _ 1
X V _ _ _ _ _ _ _ 1
X _ V _ _ _ _ _ _ 2
X _ _ V _ _ _ _ _ 3
X _ _ V _ _ _ _ _ 3
X _ _ _ V _ _ _ _ 4
X _ _ _ _ V _ _ _ 5
X _ _ _ _ _ _ V _ 7

One solution to this puzzle would be

3 4 5 1 7 1 2 3 1

At this point the puzzle would look like the figure below.

_ _ _ X V _ _ _ _
_ _ _ _ _ X V _ _
V _ _ _ _ _ _ _ X
_ _ _ _ _ _ X _ V
X _ _ V _ _ _ _ _
_ V _ _ _ _ _ X _
_ X _ _ _ V _ _ _
_ _ X _ _ _ _ V _
_ _ V _ X _ _ _ _

Depending on how you read it you end up with some different siteswaps

left to right:
3 4 5 1 7 1 2 3 1
3 objects

right to left:
8 6 7 8 2 8 4 5 6
6 objects

top to bottom:
4 5 1 7 7 2 8 1 1
4 objects

bottom to top:
8 8 1 7 2 2 8 4 5

Probably better would be to add the rule that you suggested for colorswap
which was to have the 9th piece move freely so that one wouldn't need to
do averaging. ie Imagine that the top piece has a fixed X on it, but it's
V could be determined after solving the rest of the puzzle.

[1] Does someone have a link to this proof? And if anyone wants to
explain how it works as if teaching to someone who has jr high math skills
(some place just past learning cool things like greater than and less than
symbols :D ) I'd be all eyes. If you're afraid it would bore the pants
off of the readers here then email me… uh… cool patient mystery person.

Now I gotta make one... 8 sides would be easier to make. Hope it's not
too easy to solve.

Daniel

[Aidan]

a quick search on rec found the proof!
http://groups.google.com/group/rec.juggling/msg/96fbd7662e7ac477
Aidan.

[5am]

Sorry for not giving much but Paul, thank you for your post. I really
enjoyed and agreed with it. It makes lot of sense. Thank you

-S

Bad programming is just a bad excuse not to use numbers. Quantum forever.

[Boppo]

On Nov 4, 2:43 pm, Q Juggler <paulkli...@att.net> wrote:

> Thanks for the reply. I'm not much of a math whiz so I can't pretend
> I understand everything that is written and I too feel a little woozy
> when Boppo lets loose.

Now that makes me upset. It makes me think that those really
expensive stupid pills I got off eBay and have been taking to interact
more smoothly with the rest of the world are all just a big ripoff.
Damn it!

> One thing I would like make clear is that the
> ground state is just the cascade or the fountain and excited states
> are all the others. So for a given number of objects there is only one
> ground state and many excited states.

This usage isn't far from what is commonly understood, inasmuch as a
given state is either ground or it isn't, but see below. You use the
word state differently from everyone else.

> The states that
> some people call ground states are what I call level zero excited
> states. The states that people refer to as excited are excited states
> but are of a higher level or energy. It is hard to change the current
> lingo so it is a little difficult for me to figure out what people
> mean if they say “ground state” without any numbers to back it up.

What you call "states," here, everyone else calls "tricks."[1] As
commonly used, a state is: at any instant, what is the landing
schedule for the balls as just thrown? (Do they land consecutively
with no gaps, or will be be gaps and if so, where and how many?)
Throws transition between states, and a trick is a sequence of states,
that is, a sequence of throws, that returns to the starting state. A
"composite" trick revisits an already-visited state at least once
within the trick, e.g. 4413. The word "transition" is slightly
ambiguous, in that within a trick one can transition between states,
or between tricks one can transition from one level of excitation to
another. Usually context is enough to distinguish these two
meanings.

Continuing with the current lingo, the "ground state" is when all the
balls land with no gaps, and a "ground state trick" is one in which
this ground state is one of the states visited during the trick. e.g.
441. An "excited state trick" is one in which the ground state is
never visited, e.g. 51. There is a conflict between what you call
"ground state" and what everyone else means by it, which I would like
to attempt to resolve by suggesting that the word "trick" already
carries a certain connotation of excitation, compared with the "basic
pattern," namely the fountain or cascade, which we all agree are
"ground state," and, or maybe but, "not a trick". So your "level zero
excited state" is what other people would call a "ground state
trick".

There remains a similar tension in what is meant by "excited state."
Are we talking about just one state by itself, which is not the ground
state, or are we talking about an excited state trick? Normally, when
one talks about "states" plural, at most one of them is the ground
state, and so excitation is already implied to all the others.
Therefore, usually "excited state" is used to speak about a trick (or
a series of tricks) in which the ground state is never visited.

> For
> example, a 4b ground state is just 4. Level zero excited states are
> 53, 552, 633, 642, 5551...  Level one excites states are 62, 561,
> 741... Second level excited states are 71, 822, 831...

Here it is clear that you are describing entire tricks by the word
"states."

Your usage also conflicts with the way quantum mechanicians use the
term; an atom in a certain state does not cycle through a series of
internal conditions, the exception being a "dressed state" or
"superposition of states" which are their own special case, and in
which the latter expression implies a constancy of condition within a
single state. Instead, an atom in a certain state remains in that
state until some event transpires that puts it in a different state,
such as decay or interaction with a passing atom or photon. The
analogy to juggling would be that a throw/catch exchange, your
"manipulation," is the necessary event that can change states. So I
think the common usage for the word "state" is preferable to your own
usage, even within your own context of quantum juggling.

-boppo

[1] Or "siteswap tricks" (as opposed to other kinds of tricks, e.g.
backcrosses) or just "siteswaps."

[ChaseMartin]

My latest thing is finding siteswaps in which certain balls obey a given
rule over and over again. You can then juggle different colored balls to
make this obvious to any audience. A simple one would be 642 (the 64 can
be white then black then white etc.)

For example, one of my favorites is a 88441753. Make the 8's one color and
the rest another, and they get locked in place with all the low/crossing
throws white and all the 8's black.

I've also played with mills mess color coded siteswaps, including
transitions to make the location of each color change. These transitions
are very difficult to work out, but I have some specific ones if any of
you want to give it a try.

I was kind of considering making an entire routine off this idea and
competing with it, but then I remembered that I'm a teacher and have no
free time lol.

Miika wrote:
>
> Hello everybody :-)
>
> I like juggling. I also like thinking about juggling. In particular I
> like thinking about siteswap theory. Stuff like the notation, states,
> transitions, transformations, relationships between siteswaps, and some
> other mathematical aspects are really interesting to me. But after playing
> around with these concepts for close to ten years now, it's hard to come
> up with new stuff that still excites me and makes me spend hours trying to
> wrap my head around how and why it works. Yet I know there is lots more to
> figure out, if we just find the right approach.
>
> My question today is, what else is there? Do you know some weird aspect
> of siteswaps that I haven't encountered before? Have you had any ideas
> like this that you thought were cool but haven't really shared with
> anyone? Do you know someone that has? What mathematical results haven't
> been published yet?
>
> Even if it's some small thing, I'd love to hear it. Or if you have some
> confusion about some aspect of siteswaps, perhaps sharing it here would
> help figure it out. Or if you can think of an unanswered question about
> something related to siteswaps, we could try that too.
>
>
> Swap away,
>
> -Miika
>
> --
>
> Siteswaps of the day: pear , apple , mango
>

[Q Juggler]

Hi Boppo

I'm trying to follow quantum mechanics lingo as close as possible, so
I appreciate your input. I brought up “ground state” subject because I
felt like people were not using it quite right.
Ground state is defined as the lowest, most stable energy state. Do
you agree that the four ball “Ground state” is just 4 and nothing
else? For a pattern like 7333, “ground state trick” makes sense but
it's not “the ground state”. The word “states” can have multiple
meanings but I don't see how my definition conflicts with the way
quantum mechanics uses the term Can you please run that by me again?
(keep it simple). To me “state” either means “ground” or “excited” or
a set of numbers like 5551 or 71.

Paul

[Miika]

capricornwhite wrote:
> Might be an idea to look into social siteswaps. There's a wealth of new
> stuff when it comes to that and i haven't even begun to grasp all the
> theory behind it. Most of the normal siteswap rules work with it, but
> then you have the prechac patterns, you can go with 2 jugglers doing the
> same thing at the same time, the same thing at different times (staggered)
> or different things at the sime time so long as the period is the same
> with jugglers.
>
> I particularly like working with ultimates but i'd like to work out how to
> change odd numbers patterns from one juggler juggling the even numbers to
> the odd numbers a transition between the 2 without any self throws so for
> 7 balls <4p|3p> to <3p|4p> or <7p|4p><5p|5p><3p|1p><1p|2p> to
> <3p|5p><4p|6p><2p|2p><3p|3p> for examples. You can do staggered patterns
> in ultimates as well the 4 ball splits pattern [43],2,3 works in a
> staggered 10 ball pattern with 2 jugglers, dunno what the actual siteswap
> notation of that would be though, but i do know it runs.
>
> Have fun :)
>

The easiest way to get from <4p|3p> to <3p|4p> in all passes is to look
at the pattern from the other side, or rename the jugglers. :-) Otherwise
it's a bit complicated. Assuming both jugglers throw as often and without
zeros, it's not even possible. With those restrictions, and even overall
the shortest transition, would be to use one self from the person throwing
the 4p. Or if you want to give your partner some time to react, the person
with the 3p can throw 5p4 to which his partner responds with a couple
holds (so it's <4p4p...4p22...3p3p | 3p3p...5p44p...4p4p> ).
Animation link for those two: http://tinyurl.com/sstrans-3p-4p

To avoid the self throw, you can use a zero, something like
<4p4p...4p4p4p0...3p3p | 3p3p...4p4p4p4p...4p4p>. Note that this way the
jugglers switch who's throwing straight and who's throwing across. If you
wish to avoid this, you should rather transition into <3xp|4xp>. (Which
way do you usually juggle <4p|3p>? Do the 4p's go straight from left hand
to right hand, or not? I also assume you juggle it as <doubles|singles> to
distinguish it more from 3.5p ultimates in singles, right?)
Normally a transition between <4p|3p> and <4xp|3xp> either requires a
hurry or some slightly higher throws to fit in a 0, to give the jugglers a
chance to switch which hands are throwing at the same times. (Either like
<..RLRL..|..RLRL..> or <..RLRL|..LRLR..>.) Since the 4p4p4p0 transition
above already has a 0, this is a good place to switch the hands and allow
the jugglers to keep passing in the same directions as before. (Though I'm
not really sure if this a desirable goal in practice for lack of
experience.)

Combining these transitions into one sequence:
<4p|3p>...<4p4p4p0|4p4p4p4p>...<3p|4p>
<3p|4p>...<3p4p4p4p0*|5xp5xp5xp5xp0>...<3xp|4xp>
<3xp|4xp>...<4p4p4p4p|4xp4xp4xp0*>...<4p|3p>
Animation link: http://tinyurl.com/sstrans-3xp-4xp
(I'm not sure why I bothered with that one, because it's too confusing to
actually see what's happening in the simulation. It would be more helpful
if there was a way to dynamically change the colors of the balls during a
sequence, so each repeatable pattern could be distinguished from the
transitions by color changes. Now I didn't even bother with getting the
straight passes thrown on the outside to make it look some what decent.)

I guess you could also gradually change the pattern from <4p|3p> into
<3.5p|3.5p> and then onto <3p|4p>, much like a rhythm change between just
4 and (4,4), by slight variations in the dwell times. Can anyone do this
in real life?

As for the transitions between <7p5p3p1p|4p5p1p2p> and
<3p4p2p3p|5p6p2p3p>, would you be willing to settle on doing some selfs?
Using <75p3p1p|4p5p1p2p> and <7p5p3p1p|35p1p2p> is quite close to how the
first pattern looks, though just <3|4p> and <4p|3> would be shorter.
Animation link: http://tinyurl.com/sspassing01
It sounds more like a theoretical question, so I didn't work out other
options like on the first one, but would an explanation of how to go about
finding them yourself be useful? Or can you do that by just knowing what
to look for?

The "staggered 4 ball splits pattern with 10 ball pattern" sounds like
[5.5p 4.5p]23 or ([6xp 6p],2)(4x,2)*.
Animation link: http://tinyurl.com/sspassing02

You're probably right in saying that there's more stuff to look into in
passing patterns. I'm not quite sure what it is, but for now we can have
fun with the stuff we do know :-)

Juggle on!

-Miika

--

Siteswaps of the day: boy , dad , mommy

[Adrian G]

Q Juggler wrote:
>
> Hi Boppo
>
> I'm trying to follow quantum mechanics lingo as close as possible, so

I don't know any quantum mechanics, but this is my interpretation.

> I appreciate your input. I brought up “ground state” subject because I
> felt like people were not using it quite right.
> Ground state is defined as the lowest, most stable energy state. Do
> you agree that the four ball “Ground state” is just 4 and nothing
> else?

I'd actually say that the four ball 'ground state' is 111100..., I'd say
that the siteswap necessary to stay in the ground state is 4.

> For a pattern like 7333, “ground state trick” makes sense but
> it's not “the ground state”. The word “states” can have multiple
> meanings but I don't see how my definition conflicts with the way
> quantum mechanics uses the term Can you please run that by me again?
> (keep it simple).
>

I agree on the usage of 'ground state trick', it is certainly less
ambiguous. However, since 7333 is a trick and we all know that, I think it
is acceptable to say that 7333 is 'ground state'.

However, when a trick is a 'ground state trick' it more means that it can
be accessed directly from a standard pattern. e.g. I consider 441 to be a
ground state trick but 414 to be an excited state trick. In other words, I
only look at the state that you need to *enter* the trick rather than the
states at each point in the trick as I think you do unless I misunderstood
your description.

> To me “state” either means “ground” or “excited” or
> a set of numbers like 5551 or 71.

I think that may be where some of the confusion is coming from, I'd say
that 5551 isn't a state. I'd say that is *has* a state which it needs to
start (in this case 1111...) however it is not a state in itself.

For me a state is a series of numbers, where the number represents that
amount of balls coming down at that particular time. e.g. 1111... the 4
ball 'ground state' has a ball coming down at each of the next four beats
(well, 2 to be perfectly precise as the first two are already caught I
suppose). 11011... would have two ball coming down, one each for the next
two beats (or in the hands if you want to think of it that way), a gap of
a beat and then two more balls landing on each over the next two beats.

A 'ground state' is simply a state with no zeros, e.g. 111... 11111...
etc, an excited state is any other state.

Re-reading through this you probably know it all but it might help work
out the differences in the ways we describe the same thing :-)

Adrian

[Miika]

David Cain wrote:
> Hmmm. Something you might not know about siteswaps. Audiences don't
> care about them one little bit! Sad, perhaps, but true.
>

Ah, but I kinda did know that. I could say I don't care, but it wouldn't
really be true. I bet someone doesn't, though. I do find myself doing more
siteswap tricks in performances where I don't get paid anything :-)

God bless,

-Miika

--

Siteswaps of the day: ten , big , wet

[Adrian G]

Q Juggler wrote:
>
> Hi Miika
>

With your L0,L1,etc notation, I like it but have a couple of questions:

111 is L0
1101 is L1
1011 is L2
0111 is L3
that much makes sense, now what about these?

- From what you've written, 1101 is L1, but what about 11001? is it still
L1?
- Is the number after the L simply the minimum number of throws needed to
get there?
e.g. you need one throw (from 111) to get to 1101, 11001, 110001 and so
on, so these would all be 'L1' states?

> Now for something completely
> different. Catching and throwing an object is what I call a
> manipulation. If you catch an incoming 5 and throw a 7 with the same
> hand, it's a 5-7 manipulation. Example, in the three ball 441 pattern,
> it has three manipulations. 1-4, 4-4, 4-1. I think this is interesting
> because this is what makes each pattern unique and some combos of
> manipulations are harder than others. Example, 633 is kind of hard
> because the manipulations are 6-6, 3-3, 3-3. The vertical displacement
> of 6-6 is a lot larger than the 3-3 manipulations. If the
> manipulations are the same size like in 97531 it makes things easier.
> The manips are 1-9, 3-7, 5-5, 7-3, 9-1. Notice that they all average
> out to 5. In the new version of my program (coming out soon), I made a
> graphic that shows all of the unique manipulations of a pattern. If
> you limit the numbers between 1 min and 9 max, there are 9*9 or 81
> unique manipulations. So for all you math whizzes, I have a question.
> What is the shortest pattern that utilizes all the manipulations
> (because it would make a good practice pattern). I know it must be
> more than 81 and if you require both left and right to take part, it
> is at least 162. First try limiting it to lets say 3 min and 7 max,
> with five objects. Now try not to pass out. More later.

That sounds pretty cool, have you looked at how to work out which
combinations are harder? I think that might be useful for a difficulty
algorithm[1]

Not sure about the smallest one that will pass through all combinations
though.

Adrian

[1] - I've been compiling a list of all the difficulty algorithms I can
find and comparing them in the hope to make a somewhat reliable one (I'll
do a post on it here soon), so anything else I can try out for it is great
:)

[Boppo]

On Nov 5, 6:58 pm, Q Juggler <paulkli...@att.net> wrote:
> Hi Boppo
>
>  I'm trying to follow quantum mechanics lingo as close as possible, so
> I appreciate your input. I brought up “ground state” subject because I
> felt like people were not using it quite right.
> Ground state is defined as the lowest, most stable energy state. Do
> you agree that the four ball “Ground state” is just 4 and nothing
> else?

The question is whether state applies to a trick or to a sequence of
landings. You apply the term to mean a trick, but the common usage
applies it to a sequence of landings. In the four fountain, if you
were to suddenly stop juggling, the four balls would land 1-2-3-4 with
no gaps, left-right-left-right - that's the "ground state". In a 4
fountain, of course the balls *always* land that way (or in the mirror-
image way), so it's certainly a ground-state trick, but it's not the
only one. For example, if with four balls you throw 6666, the balls
eventually land that way, but with two gaps first: pause-pause-left-
right-left-right. If you throw 666600, the two zeros take up the time
for the two pauses, and then the balls land left-right-left-right, so
the trick 666600 is a "ground-state trick" because the state left-
right-left-right is experienced during the trick. In contrast, in the
trick 71 the balls never, ever land left-right-left-right so 71 is an
"excited-state trick." So, in short, no I do not agree that 4 and
nothing else is the "ground state:" "left-right-left-right" is the
ground state. The 4 fountain involves the ground state but so does
5551; 71 does not involve the ground state.

> For a pattern like 7333,  “ground state trick” makes sense but
> it's not “the ground state”.

No, states apply to landing schedules, of which a four-period trick
(such as 7333) has up to four of. In the case of 7333, one of those
states is the ground state, but the other three are not.

> The word  “states” can have multiple
> meanings but I don't see how my definition conflicts with the way
> quantum mechanics uses the term  Can you please run that by me again?
> (keep it simple).

"State" comes from the same root as do "stationary" and "stay". When
an atom or electron or something is in a "state," it's not cycling
through a bunch of internal conditions, it's just "sitting there," if
you will. (Not necessarily in its lowest energy configuration.) If
you look at it, it will look a certain way, and if you look again, it
will look the same as it just did unless something makes it leave that
state. Multiple atoms in the same state are *indistinguishable* from
one another.

In contrast, you describe an entire trick, say, 5551, as a "state."
But in 5551, one cycles through a series of internal conditions, in
order, and if you take a snapshot of someone doing 5551 and then take
another at another time, the two pictures won't necessarily look the
same - you can distinguish them. "Why look, in this picture he's
throwing up in the air, but in that one he's throwing a direct hand-to-
hand toss." In fact, there are four states:"This guy just threw a 1,
that one threw the 1 one throw ago, she threw the 1 two throws ago,
and that guy threw the 1 three throws ago."[1] How can four different
people doing four different things, be in the "same state?" They're
doing different things! This is essentially different from the sense
of a collection of atoms being in a certain state, which all look just
the same as each other. What is much more analogous to an atomic
state is the landing schedule, which changes after every throw (except
in the case of the basic pattern, the only 1-period trick, which
therefore always stays in the same state). After you throw 4444, the
balls land in exactly the same configuration as they would if you
threw 5353 or 5551. Admittedly, the pictures won't look identical,
but the pictures aren't the states, the landing schedules are the
states. They all look the same after 4444, 5353, and 5551.

> To me “state” either means “ground” or “excited” or
> a set of numbers like 5551 or 71.
>
> Paul

-boppo

[1] I'm not suggesting that how long ago one threw a 1 implies what
state one is in, but instead only to make the point that the actions
of the four jugglers at this one instant are distinguishable. The
states are the landing schedules, not the past throw history (although
they are related - of course what the landing schedule is right now
depends on what throws were made in the recent past).

[Scott Seltzer]

ChaseMartin wrote:
>
> My latest thing is finding siteswaps in which certain balls obey a given
> rule over and over again. You can then juggle different colored balls to
> make this obvious to any audience. A simple one would be 642 (the 64 can
> be white then black then white etc.)
>
> For example, one of my favorites is a 88441753. Make the 8's one color and
> the rest another, and they get locked in place with all the low/crossing
> throws white and all the 8's black.

I did a similar-ish thing a while back. See the
645555663555666255666615666660666615666255663555 vid down in the video
section of http://www.juggler.co.il/scott/. Sorry, it's a download and not
a stream.

The classic performer variation is 633 with the 6's bounced and the 3's
being juggled are eggs.

> I've also played with mills mess color coded siteswaps, including
> transitions to make the location of each color change. These transitions
> are very difficult to work out, but I have some specific ones if any of
> you want to give it a try.
>
> I was kind of considering making an entire routine off this idea and
> competing with it, but then I remembered that I'm a teacher and have no
> free time lol.

Me, too (though I'm not a teacher, but I also have little free time).

645 was always one of my favorite patterns so I'd juggle 3 balls of one
color and 2 of another. Then the challenge was to figure out when to throw
from a cascade into 645 to make the 6's being the two odd colored balls.
There are different variations depending on the orders of the balls. Fun
stuff.

Another weird coloring idea I had was to juggle 3 white rings and 4 black
rings in front of a black background so it would look like I was
essentially doing constant flashes of the 3 white rings (I know that the
blacks would occasionally block the whites, but that's ok). Then I'd color
change the four blacks to white and magically turn it into a 7 ring juggle!

-Scott

[Q Juggler]

Hi Adrian and Boppo

I think I know what your saying. Are you saying that a certain pattern
is “ground state” because it starts from ground state and ends up in
ground state? What I think happened is that I call them ground state
level patterns, but at point of time it was translated to just ground
state. I still say that a four ball (4) is the only real ground state
and all others are excited. In the pattern 933333 if you stooped in
the middle of the pattern there would be a thud, pause, then a big
thud. The middle of the pattern is excited, only the just before the 9
and after the last 3 you are in ground state.

Adrian I'm not sure of your notation so correct me if I'm wrong.

111 = 4444
1101 = 4445 as in 44456262626234444 or 62
1011 = 4455 as in 445571471471424444 or 714
0111 = 4555 as in 455580448044804414444 or 8044
All of these are level one excited states because there is just one
zero in your notation

11001 = 4446 as in 4446822822334444 or 822 is a level 2 excited state.
110001 = 4447 as in 447A222A222A2223334444 or A222 is level 3.

The level is the same as the number of zeros.

Paul

[Adrian G]

Q Juggler wrote:
>
> Hi Adrian and Boppo
>
> I think I know what your saying. Are you saying that a certain pattern
> is “ground state” because it starts from ground state and ends up in
> ground state? What I think happened is that I call them ground state
> level patterns, but at point of time it was translated to just ground
> state. I still say that a four ball (4) is the only real ground state
> and all others are excited. In the pattern 933333 if you stooped in
> the middle of the pattern there would be a thud, pause, then a big
> thud. The middle of the pattern is excited, only the just before the 9
> and after the last 3 you are in ground state.

Yes, 933333 is certainly not ground state at any point in the pattern,
just at the very start (or end). but in general, it is a 'ground state
pattern' because it can be entered into from ground state.

On the topic, do you consider there to be a 5 ball (or any odd number)
synch ground state pattern?

> Adrian I'm not sure of your notation so correct me if I'm wrong.
>
> 111 = 4444
> 1101 = 4445 as in 44456262626234444 or 62
> 1011 = 4455 as in 445571471471424444 or 714
> 0111 = 4555 as in 455580448044804414444 or 8044
> All of these are level one excited states because there is just one
> zero in your notation
>
> 11001 = 4446 as in 4446822822334444 or 822 is a level 2 excited state.
> 110001 = 4447 as in 447A222A222A2223334444 or A222 is level 3.
>
> The level is the same as the number of zeros.
>

Your states are 3 ball states but patterns are 4 ball patterns, I get what
you'r saying though

OK, I thought it was slightly different what you meant, but with what
you're say, that makes sense.

The problem is that I would assume that you could transition straight from
a trick that ends on one level of excitation to another that starts on
that same level.
e.g. 62 and 714 are both L1 which makes it sound as if you can go straight
from one to the other.

I see now that you've defined it slightly differently, however, I'd say
the state 1101 is less excited than 1011 which in turn is less excited
than 0111, does that make sense?

Ben Beever has a method for calculating 'excitation' from the state, my
JuggleSim program uses something similar but only looks at the starting
state rather than the average of all of them.

From his book
(http://www.jugglingdb.com/compendium/geek/notation/siteswap/bensguide.html?page=3):

L(S) = sum(i x Si) - sum(1, 2, ..., B),

- where 'sum' is 'the sum of', Si is the ith digit of the state-string
(ordered from the left), and B is the number of balls. (A similar formula
can be devised for synchronous SSs - details left as an exercise.)

As an example, let's take the ground state for 3 balls: 111. Here, S1 = 1,
S2 = 1, S3 = 1, and Si = 0 for all other i. So L(111) = (1 x 1) + (2 x 1)
+ (3 x 1) - (1+2+3) = 6 - 6 = 0.

Another example: L(0111) = (1 x 0) + (2 x 1) + (3 x 1) + (4 x 1) - (1+2+3)
= 9 - 6 = 3.

This method allows every state to have a different excitation level which
in my mind is less ambiguous. What do you think of this method?

Adrian

[Chris Bendall]

Hoorah! I was hoping you would post on this, Forrest and I (from Orlando)
have been wondering what some good siteswaps for this are and now there is
a mini-thread.

Thanks Chase!

-Chris

[Chris Bendall]

On the topic of ground and excited states...

Is there a method that someone has developed when it comes to determining
entrances and exits for excited siteswaps?

e.g. - 3 ball shower = 4 51 23
- 4 ball shower =56 71 234

Those are obvious/learned through time but if I wanted to throw a 7131 or
a 717111 (both from the 3 ball shower) and I didn't know where to start
what would the suggested route be?

Thanks for the help.

-Chris

[Adrian G]

Chris Bendall wrote:
>
> On the topic of ground and excited states...
>
> Is there a method that someone has developed when it comes to determining
> entrances and exits for excited siteswaps?
>
> e.g. - 3 ball shower = 4 51 23
> - 4 ball shower =56 71 234
>
> Those are obvious/learned through time but if I wanted to throw a 7131 or
> a 717111 (both from the 3 ball shower) and I didn't know where to start
> what would the suggested route be?
>
> Thanks for the help.
>
> -Chris
>

I worked out a algorithm for ground state to excited states and back for
vanilla siteswaps only, for JuggleSim ( www.jugglesim.com ), however I
wanted it to work with everything (synch, multiplex, mixed synch & asynch)
so I changed the one used by Pedro at
http://cursomalabarismo.no.sapo.pt/jdb/ulbox.html to use with my program
and completely forgot my simple algorithm. Anyway, here's the description
Pedro gives:

Program done by: Pedro Teodoro - Portugal MAY-2006
http://cursomalabarismo.no.sapo.pt
Generation of Transitions (main Algorithm):

:: (1) Get SiteSwap 1,2 (SS 1,2) => State Patern 1,2 (SP 1,2)
SS1 = [76]20 => SP1 = 21011
SS2 = 960 => SP2 = 1101101
:: (2) If SP2!=SP1 (else there is no need of transitions)
Put SP2 under SP1 and move SP2 to the right until each element of SP2 is
equal or higher than each element of SP1.
Timebeat = 0123456789
SP1aux = 2101100000
SP2aux = 0001101101
:: (3) a = elements of SP1aux higher than SP2aux => a = 0,1,2
:: (4) b = elements of SP2aux higher than SP1aux => b = 6,7,9
:: (5) TT = transitions throws given by:
TT = b-a
TT[0] = 6-0 = 6; (SP1aux = 2 when a=0, so we can use a=0 twice)
(SP2aux = 1 when b=6, so we can use b=0 once)
TT[1] = 7-0 = 7;
TT[2] = 9-1 = 7;
TT[3] = 0 = 0; (whenever SP1aux = 0, for each a)
TT = [67]80 (transitions throws from to go fo from SS1 to SS2)
:: (6) Get another permut of b and repeat (5) to obtain another solution.
Do it for each permut of b
:: (7) Repeat (1) changing now SS1 to SS2

That actually generates all transitions, if you just want one then stop
after step 5.

Adrian

[Jason Lu]

I'd say there are 2 ground state in sync siteswap. Namely:
* *
** **
** and **

I.e. just pile the balls up evenly on both sides until the last ball,
which you can put anywhere.

This is an intuitional because every state in sync siteswap has a mirror
image. Every pattern which is valid on the first state is also valid on
the mirror image, hence why it'd make sense to have 2 ground states in
sync.

[Boppo]

On Nov 6, 12:44 am, gossey2...@cs.com.nospam.com (Chris Bendall)
wrote:

I posted a bunch of Boppo's Whiteboards on this topic.

http://www.youtube.com/watch?v=UEXn6j_HGGM is the intro, and look for
videos I made after this one. I'd like to draw your attention
especially to the brute-force way, shown in http://www.youtube.com/watch?v=pt-YbWZ399w
but I show several different ways of thinking about the problem,
spread over several videos.

The entire series is also at JTV. They all have "Boppo's Whiteboard"
in the title. Almost all have "transition" in the title too.

-boppo

[Boppo]

On Nov 5, 11:02 pm, Q Juggler <paulkli...@att.net> wrote:
> Hi Adrian and Boppo
>
> I think I know what your saying. Are you saying that a certain pattern
> is “ground state” because it starts from ground state and ends up in
> ground state?

Yes.

-boppo

[Boppo]

On Nov 5, 11:43 pm, ag...@hotmail.com.nospam.com (Adrian G) wrote:

> I see now that you've defined it slightly differently, however, I'd say
> the state 1101 is less excited than 1011 which in turn is less excited
> than 0111, does that make sense?

From 111 it takes only one throw to enter the state 1101, while it
takes at least two throws to go from 111 to 1011 and it takes at least
three throws to go from 111 to 0111. On the other hand the ground
state can be attained in just one throw for each of 1101, 1011, and
0111, by throwing 2,1,0 respectively. Don't penalize the 2 for being
a higher throw than the 0, because in state 111 one needs to throw a
3(!) to get back into the ground state, which would be an even higher
throw than a 2. I have always thought of level of excitation as the
minimal entrance sequence, personally, with the longer sequences and/
or higher throws belonging to the more highly excited tricks.

Of course a siteswap (in which sites are actually swapped, i.e. not
the null siteswap = the basic pattern = cascade or fountain) goes
through more than one state, so what the level of excitation is during
the trick changes. Two patterns might have the same number of zeros
in the landing schedule, but still have no common states, so even
though they have the same zero-count level of excitation, you can't
get from one of them to the other at any point without using
transition throws. 714 and 741, for example. In that sense the term
"level of excitation" as counted by zeros is a bit misleading, because
you'd think that tricks with the same level are somehow equivalent and
could be seamlessly transitioned between, but that's not necessarily
true. But using the entrance sequence (= state) as a level of
excitation still works, because it is possible to jump from one trick
to another at the point at which the states are the same, with no
transition throws (but not necessarily at the end of one trick and the
start of the other). This particular fact is the subject of one of my
videos, http://www.youtube.com/watch?v=A3-3Fhe6RLk.

-boppo

[capricornwhite]

ooh thanks for the tips, you were almost there with that animation of the
staggered [43],2,3 pattern, except the 3s were passes as well, but that's
not much of an alteration.

Yeah, i think with siteswaps a lot of it is theoretical rather than things
actually being juggled and it's interesting to see... maybe even attempt.
Though i'm sure those passing patterns can be juggled in real life,
although i've never seen it done.

As for how i juggle these patterns, i actually do a lot of my juggling in
front to back formation in a shared pattern with balls than doing it with
clubs so singles/doubles don't really mean too much, just making sure the
heights are right for the patterns.

But yeah social siteswaps, dropswaps, bouncing/juggling combinations.
There's a whole world of it, i don't think it's entirely being discovered
as yet. Recently there's been negative siteswaps too, though personally i
think it's just normal siteswaps transposed and repackaged to make it look
like something new.

But i think we need to keep juggling, keep siteswapping, innovating and
inspiring others to do the same. If you look at how juggling has
progressed in the last 25 years, thanks to siteswaps and mess patterns and
then.

Thanks again for the tips though, hope you've found something new. :)

[Q Juggler]

Good morning yall

The way I look at it is that there are three types of transitions,
excite, maintain, and decay. In the pattern 4444574174134444 the 5 is
the excitation, 741 is a level one excited state and the 3 is the
decay. Now in the pattern 44445571471424444 the first 5 is the
excitation, the second 5 maintains the current excitation (and in the
process it shuffles things around). It takes the same amount of energy
to juggle a 741 as a 714. In the pattern 44445741741571471424444 the 5
in the middle makes the transition between them. You can only maintain
a excited state for so long as in 444455550444. The high level excited
states like 741 can last a long time because the 7 keeps on pushing up
the energy level and the 41 is the decay but it doesn't fully decay to
ground state until the 3 happens. If you look at my notation, the
height of valleys indicates the energy levels.

Paul

[ChaseMartin]

On the plane home from IJA, I came up with the name "lockswaps" for when
each particular color gets locked in one state.

A brief list:

4 balls, 3 white 1 black
534
723
6451

4 balls, 2 black 2 white
642
750 (they switch every throw)
53 (same idea, if I remember right)
7441 (4's get locked, very pretty. easily switched)
half mess (white one side, black on the other)
633
9151

5 balls, 4 white 1 black
663
753
64645
84445

5 balls, 3 white 2 black
(6x,4)*
88441753
84486415


I could go on for a while. The color coded tricks aren't super interesting
by themselves, but if one could string together 3 or 4 patterns from each
category, a sweet routine could be made.

The thing that makes this a nightmare is that a single drop means the
sequence must start all over again, because the sequence really IS the
trick. Fun to play with though, and it definitely caught people's eye at
IJA this year when I did some of these. Might even consider offering a
nerdy workshop on it if I can come up with more patterns.

[Boppo]

On Nov 6, 11:53 am, cha...@email.unc.edu.nospam.com (ChaseMartin)
wrote:

An idea for an act I had a long time ago was to merge the patterns of
two jugglers, each using balls of one color only, into one (usually
synch) pattern. So for example one person could be doing 1 1 1 1
1 ... and the other doing 3, then they bump into each other (and a
steal happens) and one guy is left with nothing and the other the
sprung cascade. Two three cascades could merge into the 6 wimpy
pattern, etc. My dream-like big finish would have been to have a guy
doing 3 and another 5, merge into the 8-ball pattern (10x,6x)* (sorry,
that's (ax, 6x),* ugh)[1], which seemed impossible at the time but now
seems within the reach of several people. A little more easily, two
people showering three in opposite directions could merge into the 6
shower. (Or, two people showering two merge into the four shower.)
They need to be in the opposite direction so when the two jugglers
face each other, the balls are all going the same way so the steal
works.

Later, the empty-handed person could steal either his own balls back,
or the other person's, and they go their separate ways, each once
again having balls all of the same color.

-boppo

[1] Is one supposed to put the comma inside the conjugate or outside
it? (ax,6x),* or (ax,6x)*, when it is to be followed with another
clause? My grammar book doesn't have conjugate siteswap examples in
it.

[Jason Lu]

I have made some progress to develop a nice way of analysing the effect of
throwing siteswap on the order of balls. The application here is that if 1
ball stayed in the same position, then that ball would be "locked". If you
think of any siteswap as a permutation on the order of the balls, then you
can find out lots of thing about the said siteswap using some basic
theorems and propositions about permutations in Sn.

Has anyone else used this? The method is not without is flaws, e.g. the
function that sends a siteswap to its related permutation is not
injective. For example for 3 balls, there are only 3!=6 permutations
possible, but of course there are more than 6 siteswaps. This indicates
that this function loses information and that multiple siteswaps which
permutes the balls in different ways gets sent to the same overall
permutation.

J

[Q Juggler]

If a particular number is the equal to or is multiple of the period,
it is locked. 633 is a good example.

Paul

[Adrian G]

Just to expand on that, there are some siteswaps that have balls 'locked'
in multiple throws, in other words, if by taking the throws mod the period
and keep counting over the siteswap, you get back to the same throw, a
ball will be 'locked' with those throws.

For example, 642. The six will stay in its own place. by counting from the
two, we get to the four and then taking four mod three (1) and counting
one place we get back to the two. in other words, there will be two balls
locked in the sixes and two in the four-two orbit.

[Emman]

Where's Norbi?

[Miika]

TABjuggler wrote:
> Miika wrote:
> > Hello everybody :-)
> >
> > I like juggling. I also like thinking about juggling. In particular I
> > like thinking about siteswap theory. Stuff like the notation, states,
> > transitions, transformations, relationships between siteswaps, and some
> > other mathematical aspects are really interesting to me. But after playing
> > around with these concepts for close to ten years now, it's hard to come
> > up with new stuff that still excites me and makes me spend hours trying to
> > wrap my head around how and why it works. Yet I know there is lots more to
> > figure out, if we just find the right approach.
> >
> > My question today is, what else is there? Do you know some weird aspect
> > of siteswaps that I haven't encountered before? Have you had any ideas
> > like this that you thought were cool but haven't really shared with
> > anyone? Do you know someone that has? What mathematical results haven't
> > been published yet?
> >
> > Even if it's some small thing, I'd love to hear it. Or if you have some
> > confusion about some aspect of siteswaps, perhaps sharing it here would
> > help figure it out. Or if you can think of an unanswered question about
> > something related to siteswaps, we could try that too.
> >
> >
> > Swap away,
> >

> > -Miika
> >
> > --
> >

> > Siteswaps of the day: pear , apple , mango
> >
>

> I just made this post in another thread. Some of the info might be
> relevant here:
>
> "Important Observations:
> *Even numbers never cross but throw to the same hand they threw from
>
> *Odd numbers always cross
>
> *If you add up the total of a siteswap and takes its average, if the
> average is a whole number, than that tells you the number of balls
> required to perform the siteswap (ex: 531 = (5+3+1)/3 = 9/3 = 3 balls) and
> (ex: 7441 = (7+4+4+1)/4 = 16/4 = 4 balls)
>
> *The order of a siteswap's numbers do not matter, but are usually written
> with the largest number first (ex: 423 = 234 = 342)
>
> *Just because the average is a whole number does not mean that the
> siteswap is valid. However, you can always take a permutation of the
> numbers and get a valid siteswap. (ex. 432 = (4+3+2)/3 = 9/3 = 3 Balls,
> but the siteswap is not actually possible. So you have to look at all the
> combinations of those numbers (423, 432, 234, 243, 342, 324). Due to the
> above observation (423 = 234 = 342 and 432 = 324 = 243). That means that
> you only really have two possible permutations of the siteswap numbers and
> one doesn't work. That means that 423 = 234 = 342 are all valid
> siteswaps.
>
> *Another cool observation of the above is that all two period siteswaps
> that have whole number averages are valid siteswaps. (ex: 51 can only
> permutate to 15, which is just 51 in a different order -> 51 = 15)
>
> *Any siteswap that is written as descending sequential numbers is invalid.
> (ex: 432, 765431, and 987 are all invalid). There is another fact about
> that that is best explained with an example (If you take any of the above
> examples of invalid siteswaps and change out any of the numbers, the
> pattern is still invalid if at least 2 of the numbers would be sequential
> and descending..... ex: (765431, 7x5xxx, 7xx4xx, and 7xxxxx1 are all
> invalid no matter what numbers x is equal to in those examples)
>
> Hopefully that isn't too much info, but its most of the relevant parts of
> siteswap that will actually help you.
>
> If all of this wasn't too bad I'll give you the most useful formula out of
> Mathematics of Juggling that I have found so far. You just had to take it
> at face value that 432 and all descending siteswaps are invalid and that
> all two period swaps with correct averages are valid. This formula lets
> you do the math for valid vs invalid in your head without too much
> difficulty. "
>

Hi TAB!

Thanks for your post; it was interesting to see what you've discovered in
your first steps into deeper siteswap theory :-) Most introductions to
siteswap are explanations of how juggler's can use it to describe some
patterns. They often include some of the things you said, though you do
put your own twist on a couple. However, your post has reminded me that
there really isn't a decent introduction to the mathematics and theory of
siteswaps for jugglers.

As for your way of checking if a siteswap is valid and excited tricks, I
suggest you first mod out the high throws (=subtract out the period enough
times to just have throws under the period). Then you can also repeat the
pattern and check for collisions by comparing to the descending numbers.
This way you don't even need to check the average of the pattern!

Keep working that seven,

-Miika

--

Siteswaps of the day: moth , fowl , kitty

[Miika]

Emman wrote:
>
> Where's Norbi?
>

You mean that little guy who posts three ball stuff from his bedroom? I
bet he was waiting for the thread to reach 100 posts :-)

On that note, thanks to everyone for all the interesting ideas and
discussion you've shared here! It's been much more than I anticipated :-)
I've slowly been making my way down the thread with some replies, but at
this rate it'll be Christmas before we're done.

-Miika

--

Siteswaps of the day: wise surf hair