Jugglometry
Researching siteswap transformations, I needed a good way to quantify what a given transform actually did. This resulted in the creation of what I call jugglometry: the study of different aspects of siteswap patterns. It centers around equations relating different parts of a pattern. I wish I could say that there's a lot of juggling involved, but what follows is mostly math, (not hard math though) related to how juggling works.
The most intuitive relationship is that S + X = B, which means that the siteswap value, or duration of a throw, plus its position in the pattern (1st throw, 2nd, 3, or so on,) gives you the ending position of the throw, or when that object will be thrown next, B. We can see that the throws in 441 end at positions 5, 6, and 4.
Maybe the most central equation of jugglometry is that B= T + X + WP. T is the displacement of a throw, or how much it moves within the pattern. For example the 5 in 531 is next thrown as a 1, meaning it has moved forward 2 positions. So T for all of 531 would be 20-2. P is the period, and W is a tricky variable to define. It measure it by the number of times that a throw crosses the back of the pattern before 'landing'. W for 612 is 201, and W for 441 is 111.
Knowing B=T+X+WP and S+X=B, we can say that S = T + WP, which is helpful.
The last major variable is D, the ending position of a throw, relative to the pattern. For example D of 423 is 213. D is equal to T + X, which makes sense intuitively. I imagine it as drawing an arrow from a point, the point is X and the arrow is T. Where it points to is D.
The last bit of helpful knowledge is an operation I'll use || for. That's not how I write it on paper but it'll have to do for now. Putting something inside of || means to subtract the period from each digit as many times as possible without going below either 1 or 0. ||' will mean stop at 1, ||. will mean stop at 0. For example |423|' is 123 and |423|. is 120.
There's also the counterpart operation, (I'll use \\ for it) which turns each digit into how many times you can subtract the period from it. So \423\' is 100 and \423\. is 101.
Interesting relationships
This part is more interesting. We can convert the average rule of siteswaps into jugglometric terms, using J for number of objects and ^ to mean "sum of".
(S^)/P = J
(I would use O for objects, but that would get confusing with zeroes.)
Also, W^=J.
Another one is that T^=0.
Therefore, we can derive the average rule by summing S= T+WP:
S^=T^+W^P
S^=JP
S^/J = P
In my experience, the king of jugglometric relations is that D^=X^. In language, "For every beginning there is an end". This is basically what defines if a siteswap is valid. We can prove that it works by using D=T+X:
D=T+X
D^=T^+X^
(T^=0)
D^=X^
The other really big ones are that D= |B|' and W= \B\'. This comes into play with transforms.
Transformations
So, onto the main focus. I imagine transforms as ways to alter jugglometric components without ever risking invalidity. The primary transformations, as I see it, are Lifting (441 to 552), Cycling (441 to 414), and Swapping (441 to 522). I don't count anything that changes the period, especially adding a basic throw to the pattern, because compositing siteswaps seems like a different kind of transformation, and messes with the math.
I have found that each transform is basically dependent upon one key change on one variable in the pattern. If you know the equation for that change, you can determine the equation for every other variable's new value. For example, the basis of Lifting is that S+1=S', where S' is the transformed pattern.
X never changes, so X=X'.
Because S'+ X'=B' and S+X=B, we know that B+1=B'.
Because D= |B|',|B+1|'=D'
Because W=\B\', \B+1\'=W'
Because D'-X'=T', |B+1|'-X=T'
The assumption for Swapping is that are that B'=BF, where BF means the next value of B, or B|X+1|' in math terms. For Cycling, it's that S'=SF.
It should be noted that Swapping is odd, because you can apply it to smaller parts of one pattern. You could swap the 4's in 441 instead of swapping every single number. But the math works out the same either way.
As far as I can tell, these tranformations work because they don't violate D^=X^. |B+1|'^=X^, |SF+X|'^=X^, and|BF|'^=X^. So in theory, there could be many more viable transforms, as long as D^ is always equal to X^, but I don't have a good way to know what fits this rule and what doesn't. Anyways, I think it's a great option for looking at transforms and understanding how and why they work on a much deeper level.
I'd encourage you to explore this for yourself; maybe try and find a new transform or use them to generate new siteswaps, or explore anything you might wonder about. I promise it's more interesting than I make it sound!
Note about state
One more thing, you can use jugglometry to find the state of a pattern from just the Siteswap value. The equation is B-yp= St, where St is state and y basically means 'all positive numbers starting at 1'. Keep in mind that because of that, it gives you an infinite version of the state, so you have to just take the useful part. In practice, you would say, for something like 741:
S+X is B, so 741 + 123 is 864
864 - 333 is 531
531 - 333 (ignoring negatives and repeats) is 2.
So there has to be an object at 1, 2, 3, and 5, making the state XXXOX.
This isn't really the fastest way to calculate state, but it's at least interesting to put it in mathematical terms.
Subscribe to this forum via RSS
1 article per branch
1 article per post