Started juggling 2021-08
Juggler. Decent but definitely a work in progress.
I like theory.
Theory developments that I don't want to make a special point of posting:
Newest: I found equations. Here's variables:
S= the value of a given siteswap throw
X= the position of a given throw within a siteswap (example x of the 3 in 531 is 2. The full X for a pattern is always [123...P])
P= the period of a pattern
L= the number of times a pattern 'loops', or how many times you can subtract the period from a given throw.
D= the destination in the pattern of a given throw. (Example the D of 5 in 531 is 3)
w (technically I use lowercase omega)= the number of times a given throw 'crosses' the 'end' of a pattern before being thrown again. (example the 5 in 531 crosses the end of the pattern 1 time)
I= the total distance a throw makes through the pattern, or the throw value minus the period as many times as possible (no negatives) (example the Is of 531 are 201)
# (technically I use capital sigma, like summation) = all of the values of a given variable for a given pattern added together (example #S for 441 is 9.)
O= number of objects in a given pattern
Now, given these variables there are two primary relationships, one that I found and the other being the average rule of siteswap:
I+X+wP=D
(#S)/P=O
It is also good to note that S=I+LP.
And from these I derived some more relationships:
#D=#X
#w=O
The first of these is a nifty way to check validity, and the second is an interesting but less efficient way to find the number of objects in a pattern (w is generally pretty tedious to find for a pattern)
Older and quite possibly irrelevant stuff:
1.
One axiom that allows 333 into 441 is=
A: ss(Y1 yp)- ss((Y1+1) (yp+1-p)
where p is the period.
However, this is actually a composite of these axioms:
A:ss(y1 Ym yp)- ss(y1 Ym (yp-p))
A:ss(y1 Ym yp)- ss((y1 Ym yp)+1)
Where Ym is all digits between the first and the last.
This is because y1 with Ym together is equal to Y1. The +1 goes to both terms, the -p goes to just the last one.
2.
It is possible to encode the most essential variables into a single digit, meaning that it is possible to notate with no regard for order.
Siteswap takes each y of a sequence and makes it (y+Lp). Turns out you can take a similar step with x. The first conversion works (ie. you can retrieve both of the original components) because y is always less than p. Never greater than or equal, because then you could subtract the period. Well, x is always less than or equal to the period. Because x is allowed one number higher than y, we can turn (y+Lp) into (y+Lp)+x(p+1). So then, 441 can be called 8gc. Or 8cg. Or cg8. Or literally any other combination of those three characters.
Which isn't helpful. But it is interesting, so there you have it.
(side note: for numbers larger than 36 [which can occur fairly often using this method], I use capitals, using only lowercase letters before that point. So 88441 = CHjo6. Or jH6Co. You get the picture.)
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