Hello, I'm new to the juggling edge.

At the moment I'm working on 5 clubs, 7 balls, (in theory 6 rings...,) and being a generally better juggler. I'm super interested in theory, I've put a lot of effort into studying transformations between siteswaps and stuff in that realm.

Hi & welcome, Rusty.

Most probably you then know about Ben Beever's Axioms of juggling for transforming siteswaps.

RustyJuggling - - Genitore #

I wish!

Information about juggling can be harder to find than it should be.

The axioms really interesting. My first thought is with the axioms about switching any two throws. I think you could put patterns into a kind of path/orbit notation to make that transformation really easy to visualize. Like 441, notated as (destination in the pattern)/(# of orbits) would be 2/1 3/1 1/0, and then if you switched the second and third throws it would just be 2/1 1/0 3/1 (IE; 423) with no addition or subtraction needed. So I guess to put that into terms of an axiom you could use a different prefix, like if we call that notation D/O, axiom 2 might look like

A2: DO(abc)- DO(acb)

Unrelated to that, you could also construct a multiplex-making axiom like:

A(m): SS(A), SS(B)- SS([AB])

where A and B have the same period.

I think there might also be an axiom for converting negative throws into multiplexes? Testing 55-1 and 44-2 to 5[51][22] and [42]4[11] with D/O shows a pretty nice pattern but I'm not sure how it would be converted into axiom notation, because throws with different destinations would end up multiplexing with different throws.

That might be a weakness with the axiom notation, I think, is that it struggles to show every application of a given transformation, even when we see them as the same thing.

I'm going to play with this some more, thanks so much for showing it to me!

RustyJuggling - - Genitore #

After a little playing, I think I have a modification that helps axiom notation's descriptive power a bit, for my purposes anyway. If you use one letter and its position in the pattern, it really opens up the notation. For example SS(abc) becomes SS(y1 y2 y3). This might mess a bit with the capital letter part of the notation? Not sure at the moment. I've been subscripting the numbers in my notes, which is cleaner than on computer, where I'll put the numbers in //.

Using this, the axiom that any two throws can be swapped is thus:

SS(...y/a/...y/b/...)- SS(...y/b/+(b-a)...y/a/-(b-a)...)

with 'a' being a digit in the pattern before digit 'b'. So basically this allows for the pattern to have any period, and indicates any two throws, with any distance between each other, being switched. In destination/orbit format the axiom is:

DO(...y/a/...y/b/...)- DO(...y/b/...y/a/...)

If you use some janky notation I think I have the axiom that describes converting a negative throw into a multiplex throw (assuming that the negative throw is formatted as the last throw). An apostrophe after a throw indicates its "absolute value," that is, the throw with the period of the pattern subtracted as many times as possible without entering negatives. A throw being negative is indicated in the axiom, because it relies on the presence of a negative throw. Using this, the axiom is:

DO(...y/b-(y/b/')/...-y/b/)- DO(...[y/b-(y/b/')/P]...[y/b/ y/b/])

with P representing the period of the pattern, orbiting 0 times. In the future the number of orbits might be included within the axiom? I don't know how often such a thing would be needed so it's hard to say.

Also, for convenience, "y/b-(y/b/')/" just represents the destination of the negative throw. I realize that it looks a bit revolting. There might be a better symbol to use than // for writing these on computer? I don't know.

Anyways, those are my ideas on a modified axiom notation.

I think you could put patterns into a kind of path/orbit notation to make that transformation really easy to visualize. Like 441, notated as (destination in the pattern)/(# of orbits) would be (IE; 423) with no addition or subtraction needed. So I guess to put that into terms of an axiom you could use a different prefix, like if we call that notation D/O, axiom 2 might look like

A2: DO(abc)- DO(acb)

So nifty!

I have a whole mess of remarks about this:

3•) - - {{ I think, by 'orbit', here, you mean 'period', ( or 'length of the siteswap' )? ... as '*orbit*' usually refers to the path that the single balls take through the pattern }}

#**1**) - - How does swapping 2/1 3/1 1/0 to 2/1 1/0 3/1, 'visualise' any throws to be done or the sequence of throws to relative heights?

n#) - - in order to use this simplification, you have to convert 441 to 2/1, 3/1, 1/0, and reconvert back when swapped, soas to know your digits to be juggled - all in all that seems more complicated, a detour. (?)

•√) - - the little adding and subtracting that you need when juggling and playing with altering siteswaps is usually trivial enough to be done in mind. So ... as elegant as it is - where's the benefit of it all, of D/0?Unrelated to that, you could also construct a multiplex-making axiom like:

A(m): SS(A), SS(B)- SS([AB])

where A and B have the same period.

This, it seems, is a pure notation issue, isn't it? Let's see ... in siteswap notation

3 and

1

become [31],

333 and

423

become [34][32][33], ...

hm, yeah, why not.I think there might also be an axiom for converting negative throws into multiplexes? Testing 55-1 and 44-2 to 5[51][22] and [42]4[11] with D/O shows a pretty nice pattern but [...]

I had been thinking about this too - you can relate negative digits to the next period of the siteswap repeated, for example [*please ignore the spaces as a go around autogreen for stickman*!]:

5 **5** -1, where that throw can't land anymore (backwards in time)

should and can simply land

5 5 -1 5 **5** -1 ...

thus 2 beats ahead in time

5 5 +2 !

Switching**0 1** 2 3 4

to**2 -1** 2 3 4 ... would actually then land on the first digit, 4 beats later. Making it...

2 4 2 3 4

q.e.d. ( well no lol - it's no proof, just one example where I hope it works ((didn't check yet)) )

Ladder diagrams are also most helpful to see, visualize swaps, and many other solutions for siteswaps quested for.[...] a modification that helps axiom notation's descriptive power a bit, for my purposes anyway.

As not a mathematician, I see your generalized axiom notion - to which behalf, I see only nebula. Would it for example make writing software easier or more generally applicable to also other notations than siteswap, maybe dance steps or sth or knitting?If you use one letter and its position in the pattern, it really opens up the notation. For example SS(abc) becomes SS(y1 y2 y3).

Aoutch! ... now I'm out ( I might look through this, lotsa spare time given, but my needs for calculating with siteswaps are mostly if not fully covered by what there already is notations ).

But I have a slight impression that what you're finding out there might overlap with Beever's full original publication, http://www.jugglingedge.com/pdf/BenBeeversGuidetoJugglingPatterns.pdf [ section 8. ]SS(...y/a/...y/b/...)- SS(...y/b/+(b-a)...y/a/-(b-a)...)

[...] allows for the pattern to have any period, and indicates any two throws, with any distance between each other, being switched.

Yes, I think this is implied in the axioms.In destination/orbit format the axiom is:

DO(...y/a/...y/b/...)- DO(...y/b/...y/a/...)

A simplification then, one level higher or more abstract or more basic. (?) Just, I do still wonder how to make use of it..[...] janky notation [...] apostrophe after a throw indicates its "absolute value," that is, the throw with the period of the pattern subtracted as many times as possible without entering negatives. A throw being negative is indicated in the axiom, because it relies on the presence of a negative throw. [...]

[...] "y/b-(y/b/')/" just represents the destination of the negative throw. [...] revolting. [...]

°*Ouffh*° ... you knocked me out X-D

one ball more then - the price you pay for messing with negative digits, now

RustyJuggling - - Genitore #

So nifty!

I have a whole mess of remarks about this:

3•) - - {{ I think, by 'orbit', here, you mean 'period', ( or 'length of the siteswap' )? ... as 'orbit' usually refers to the path that the single balls take through the pattern }}

#1) - - How does swapping 2/1 3/1 1/0 to 2/1 1/0 3/1, 'visualise' any throws to be done or the sequence of throws to relative heights?

n#) - - in order to use this simplification, you have to convert 441 to 2/1, 3/1, 1/0, and reconvert back when swapped, soas to know your digits to be juggled - all in all that seems more complicated, a detour. (?)

•√) - - the little adding and subtracting that you need when juggling and playing with altering siteswaps is usually trivial enough to be done in mind. So ... as elegant as it is - where's the benefit of it all, of D/0?

1. Yeah I've been pretty isolated thinking about juggling so my terminology can be a little off sometimes. That's what I mean, is the number of extra periods it has had added to it, from the lowest positive number.

2. I suppose the I would use D/O (Or D/something else I guess) as a different way of seeing the same siteswaps. Like as a way to look at two patterns and see certain relationships more clearly. (Sidenote, with two seconds of looking through the HTML I can subscript things now.)

So like in the example I might not be able to look at 441 and 423 and say 'yeah the last two are switched but in D/(?) it's fairly clear: 2_{1} 3_{1} 1_{0} to 2_{1} 1_{0} 3_{1}.

So I guess it's like a way of looking at different properties of the pattern. Not useful for juggling a pattern but more so for studying relationships between destination and added periods. Or maybe for finding/writing axioms, which I think tend to be cleaner using it, but honestly I'm not sure yet, we'll see.5 5 -1, where that throw can't land anymore (backwards in time)

should and can simply land

5 5 -1 5 5 -1 ...

thus 2 beats ahead in time

5 5 +2 !

Switching

0 1 2 3 4

to

2 -1 2 3 4 ... would actually then land on the first digit, 4 beats later. Making it...

2 4 2 3 4

That's really interesting, I hadn't considered that kind of transformation. It's like adding the period to a normal throw- which would be where that extra ball comes from. Ladder diagrams are also most helpful to see, visualize swaps, and many other solutions for siteswaps quested for.

Super helpful, ladder diagrams have helped me a bunch with conceptualizing negative siteswaps. Useful for sync transformations too.Yes, I think this is implied in the axioms.

Yeah, looking back to the axiom notation page it's actually way more clearly defined than I thought. I missed the part where A could be empty and I thought "well what if I want to do things with the first number?" So ellipsis in the modified axioms are basically the same as capital letters in the original, meaning that the D/? switching axiom could just be:

D/?(AbCdE)- D/?(AdCbE)

Which is exactly the same as how it was originally written, just with addition/subtraction omitted.

So partially the modified notation is just me being silly and not reading well enough, but I do think it's better-suited for describing the specifics of where in the pattern a transformation occurs. Like the negative-converting axiom relies on where specifically a throw occurs in the pattern, which is why I couldn't fit it into the original system.

Also I think there might have been something off with my earlier negative-multiplex axiom, so here's a revised version, in D/? because it's way cleaner than writing it in siteswap. Doesn't require that janky apostrophe. Note, I'm going to use y/z instead of just y to mess with parts of the throw separately.

D/?(...(y/z)_{a}...-(y/z)_{b})- D/?(...[(y/z)_{a} P/(z_{b})]...[(y/z)_{b}(y/z)_{b}])

With P being the period of the pattern.

I also added parenthesis to distinguish when a subscript refers to the whole thing and when it only refers to the number of periods added.

Which also needs a name, so I can stop using D/?. Magnitude? Loops? I don't know. Something shorter than "the number of times that the throw has had the period of the pattern added to it".

So like in the example I might not be able to look at 441 and 423 and say 'yeah the last two are switched but in D/(?) it's fairly clear: 2

_{1} 3_{1} 1_{0} to 2_{1} 1_{0} 3_{1}.

Oh! That's very cool.[...] it's [

*D/O notation*] better-suited for describing the specifics of where in the pattern a transformation occurs.

Yes, now I see it too, while earlier didn't, that a relation between 441 and 423 is not exactly apparent, but D/O nudges your nose right on it.

All in all I find it great, that your D/O notation and this way to look at siteswaps is possible, that it exists - an enrichment.

Thanks for presenting & sharing it here on The Edge!

RustyJuggling - - Genitore #

All in all I find it great, that your D/O notation and this way to look at siteswaps is possible, that it exists - an enrichment.

Thanks for presenting & sharing it here on The Edge!

Thanks, I really appreciate that! And thanks so much for introducing axiom notation to me, it's really interesting and the knowledge is invaluable. 'Loop' sounds good to me, D/L then

Yeah, 'loop' is what I would use.

And as a last note mainly for myself, the revised siteswap version of that pesky negative axiom, because the other one isn't as good and I don't want it to be the final version:

SS(...y_{a}...-y_{b})- SS(...[y_{a} y_{b}]...[(P-y_{b}')(P-y_{b}')])

Where a=b-y_{b}' and P=period

Which also needs a name, so I can stop using D/?. Magnitude? Loops? I don't know. Something shorter than "the number of times that the throw has had the period of the pattern added to it".

Well, it's your work, your research, so you can call it anything you like in the first place - the more that you must very much have gotten used to thinking it as 'D/O' ...

( ...but I really think 'Orbits' is misleading, but I wouldn't have a problem calling it that ).

What's wrong with calling it 'D/P' for 'Period'?

There's anyway a little mess about *period* - length of the ss

. . . ( misconceivable in itself again also seen that e.g. 3 = 333 ),*round* - mostly used synonym to 'period' I think, but depending on the context, ( e.g. 'round(s) of the siteswap' ),*cycle* - also ( and I find wrongly ) used for 'period \ round', as more precisely used for 'until all balls ( especially of different colors, or else also of mixed props ) are back where they started off', which can be many roun... errh periods.

'Loop' sounds good to me, D/L then, while I don't at all feel entitled to give statements (just "thoughts") on such realm's namings, as many others have worked on this a lot more before.

I love the level of juggle-nerdism on display here. I understand very close to none of it but I bet Ben would have loved it (and understood/had comments) too. Bravo.

When throwing to different heights with props passing one another by,

you can exchange two of a series of throws to land at the moment where the other did.

Just, for both then also being thrown at the moment, the other got thrown ( one then earlier, one later ), their flight-time needs prolonged \ reduced by that difference ( often simply = 1 )

'Swapping digits' of a siteswap, that is.

This is good for playing with altering patterns or intendedly changing them for a purpose e.g. as exercise, and I presume, also for coordinating numbers of passers into a given pattern.

;-D

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