#siteswap

(here I've set "colors = mixed" so Juggling Lab makes all the balls different colors, which makes it easier to tell what's going on.)

notice that in this pattern, the juggler throws the balls in the sequence red, green, blue. crucially, the amount of time every ball spends in the air is exactly the amount of time it takes to catch and throw two other balls.

that is what the throw "3" means in siteswap: throw a ball high enough such that you will have thrown two other balls by the time you catch this ball again.

this can be generalized to any number, although any real juggler has a hard limit to how high they can throw a ball. here's the pattern "4":

and the pattern "5":

every digit is defined the same exact way: throw the ball high enough so that you have time to catch and throw n-1 other balls by the time you catch the ball you're currently throwing.

the fun part comes from stringing together these throws into more complicated patterns! for example, here is the pattern "53":

note that this is not a pattern juggling fifty-three balls. the pattern "53" involves alternating between doing a "5" throw with one hand and doing a "3" throw with the other hand.

so, while the "4" pattern had the juggler throw the balls in the repeating cycle of red, green, blue, yellow (orange standing in for yellow here for legibility), "53" has the more complicated cycle red, green, blue, yellow, green, red, yellow, blue.

if this sounds confusing, it may be easier to work backwards from the sequence of which specific balls are thrown and work out the sequence of throws from there.

the first throw in this cycle is the red ball, and the next time the red ball is thrown is five throws later (after four other throws), so this must be a "5" throw. next is the green ball, and there are two other throws between this one and the next time the green ball is thrown in the cycle, so this is a "3" throw. doing that for the whole cycle, you can find that this is indeed a repeating "535353..." pattern.

what this means is you can start with some sequence of what order you want to throw the balls in, count how many throws there are between each time you throw a specific ball, and use that to get a valid siteswap pattern!

for example, let's say you want to throw three balls in the sequence ABCCBA. by counting gaps between throws, we can convert this into the siteswap pattern "531531" (or just "531"), which looks like this:

notice that this pattern makes use of a "1" throw, which implies the existence of a simple juggling pattern with one ball:

although this isn't quite as unimpressive looking as the "2" pattern:

this "2" throw is considered by jugglers to be so pathetic looking that by convention they don't even bother doing it when a pattern calls for it and instead just hold the ball for a beat without throwing it at all:

indeed, in Juggling Lab you have to explicitly tell it to use "2T" instead of "2" if you want the dinky little throw (actual terminology used by jugglers to describe it!) implied by the mathematics of siteswap.

anyway, those are the basics of siteswap! there's more advanced stuff that you can read about on the Juggling Lab website, but these fundamentals should be enough for you to be able to get started making funny juggling gifs. have fun!