Juggling Equations

RustyJuggling - 2025-02-20 18:54:49 UTC #

Juggling Equations

I've been looking at juggling through a more mathematical lens than I usually do, and I've found some interesting relationships. The variables I'm going to use are as follows:
S- siteswap value
P- Period
X- start location in the pattern (ex. x in any period 3 pattern is 123)
D- End location in the pattern (ex. D in 531 is 321)
L- the number of times something 'loops' through the pattern (ex. the 4s in 441 loop once, the 1 doesn't loop). The mathish way to say this would be S mod P, but it makes more sense here to just say L. (Note: L can also be based on a value that isn't S)
W- the number of times a throw crosses the right 'edge' of a pattern, including after it 'loops'. This number can be similar to L but is NOT the same.
O- the number of objects in a pattern.
I- (If the font doesn't distinguish, this is an i, loop is an l) The total 'distance' a throw travels in the pattern. (ex. in 441, I is 111.)

That's a pretty good heap of variables. Using these and some logic, we can define some relationships. I'm going to use # as a summation sign, meaning the sum of all of that value in a given pattern. For example, #S of 531 is 9. I hope that doesn't mess with the hashtag system... On paper, I use a capital sigma like in math.

First of all, the one everyone knows:

O= (#S)/P

Which gives you the number of objects in a pattern.
We can also say that

I= S-LP

because -LP essentially takes the loops out of a siteswap value, leaving only the smallest possible value, which is what I is.
A helpful way to check validity is to see that there are as many of each D value as there are of each X value. Translation: each throw has exactly one start and one end. Because of this rule, we know that

#D = #X

But the most important relationship I've found is that

WP+D = S+X

which I got to via experimentation and logic. I don't really remember all of it, but trust me. (I think if you try to find a way to get from S to D mathematically, this is probably what you'll come to.)
Anyway, now that we have these relationships, we can substitute some of them together. For instance:

WP+D=S+X --> WP-S = X-D
--> #W P - #S = #x - #D
--> #W P - #S = 0 (not object number, value zero)
--> #W P = #S

P behaves a little bit strangely, but best I can tell, summing it does nothing because it's one value throughout the whole pattern. Questionable logic, I'm aware.
But we can then take that equation and see that

#W P = #S --> #W = (#S)/P
and therefore:
#W = O (Object number, not value zero)

and that's the basics. In summary,
O=(#S)/P
#D=#X
WP+D=S+X
#W=O

The following gets crazy and probably hard to follow:
And now, for a stranger and more theoretical equation. It can be used to convert siteswap values into state values, and it is pretty janky.
Here is some notation I will use and other details:
- Each value in a pattern can be treated as a set of values. For 441, S= [441], X=[123], W=[111], L=[110], and so on.
- A subscripted number can be used to indicate what item of a set is to be used. for example S₃ of 441 is 1. When a variable is subscripted, each value of that variable is used to create a set with one more 'dimension'. Single value--> list --> matrix --> so on.
- There's a mystical magical set that I usually call psi, but on the computer I will use %. %=[123456...]
- A range of values can be indicated in a subscript like so, S_1~4, meaning that values 1 through 4 of the set are to be used.
- A different base value for L can be indicated using //, example L/X+3/. (For any period 3 siteswap that would be [120] because X+3=[456])
- S_t is a state value. State is notated by giving the position of each object. For example xxox would be 124.

With that out of the way:

S+X-P(L/S+X/ - %_1~L/S+X/)= S_t

Which took entirely too much time to find. But we can also rewrite that as

S+X-PL/S+X/+P%_1~L=S_t
and
I+X-P%_1-L/I+x/=S_t

I use 'music rules' for this, meaning you only define L the first time you use it, unless it changes. Also good to note that for some reason, the P% in one is positive and in the other is negative. I think that has to do with L being based in different numbers, but I honestly don't know. I'm a bit too tired to find out at the moment.
Anyways, if you read through that, really really well done. I struggle to understand some of this sometimes, and it's far far cleaner on paper. These are the most refined siteswap to state conversions I have found so far (there are others but I don't like them as much). If you find an error in the math part of it, that would not surprise me, but they work as far as I can tell. Further testing required I suppose.
Anyways. There's that.

("cleaned" versions of the equations:)
S + X - P (L_/S+X/ - %_1~L) = S_t

S + X - P L_/S+X/ + P %_1~L =S_t

I + X - P %_1-L/I+x/=S_t