How to Quickly Validate a Siteswap in Your Head
The following is an explanation of a simple method for validating
siteswaps that does not involve any calculations and is ideal for doing
in your head. It also gives you some helpful insight in that it tells
you which throws are causing problems.
A sufficient criterion for a validation of any siteswap is that for
every beat, the number of balls thrown equals the number of balls in
your hand at that beat. For vanilla siteswaps, since at most one ball
is thrown at a time, this means that no two throws can land at the same
time and if a throw lands, it must be thrown again at the next beat
(i.e. the next beat must contain a nonzero siteswap number).
The actual time that a given throw lands is going to depend on the
jugglers dwell ratio, but for ease of understanding, we can assume that
a throw of height x will land x beats after the throw. The amount of
time between a throw of height x and the next throw of that same object
is equal to x+1 and is independent of the dwell ratio.
For this validation method, each throw has to be counted backwards.
Let's take a look at a countdown for a particular throw and see what it
tells us. Here we have the siteswap sequence 534. For the first throw
I have counted down from 5 to 0 since 5 is the height of the throw. At
beat 5, countdown1 reaches 1 and object is caught. At the same time
that this catch is occurring, the other hand is throwing an object at
height 3. On the next beat (beat 6), countdown1 reaches 0 and the
object is once again tossed into the air, this time with a height of 4.
Countdown2 & 3 are for the second and third throws respectively.
Spotting an Invalid Siteswap
Siteswap 534 is a valid sequence. What happens if we look at an invalid
siteswap? Let's take a look at 6054. You can see that countdown3 and
countdown4 show that both throws are landing at the same time and are
supposed to be thrown again at the same time on beat 8. Of course, the
siteswap shows that only one object is to be thrown at that beat, thus
the '5' from beat 3 is colliding with the '4' from beat 4.
We can recognize this collision problem without even looking at
countdown4. If we just look at countdown3 and the siteswap, we can see
that the '4' from the countdown matches up with the '4' in the siteswap
on beat 4. Any time that a siteswap number matches up with a countdown
number, you can be sure that the siteswap is invalid, and you can tell
which two throws are colliding.
What happens if we collide with a zero?
Now lets look at another example: 3560. We can see here that throw '3'
from beat 1 is colliding with throw '0' from beat 4. Of course,
technically this is not a collision. Basically, the countdown says that
the throw '3' from beat 1 is supposed to be thrown again at beat 4, but
the siteswap says that there is no ball at beat 4. This is obviously a
contradiction and the siteswap is invalid.
Beats: 1234(countdown2,3&4 omitted)
Are countdowns sufficient to insure validity?
If countdown is performed for every throw in the siteswap sequence and
no collisions were found, then the siteswap is most certainly valid. A
countdown will only check for collisions between the throw being counted
down, and all throws smaller than it. Since the larger throws are
already checked against the smaller throws, the smaller ones don't need
to be checked against the larger ones. So basically this method of
checking all throws against all other throws without being redundant.
Refining the Process
So now you should have an idea of how to use this method to validate a
siteswap sequence. There are some shortcuts that can be taken. Most
notably, you do not need to countdown all the way to zero if your
siteswap does not contain any zeros. You only need to count down until
you reach the number that is equal to the smallest number in the
siteswap sequence. After time, you should be able to notice some of
these collisions automatically even before you begin any countdowns.
Let's look at 3651 for example. Before you even begin your countdown on
'3', you should notice right away that 6 and 5 collide with each other.
You should never have two numbers next to each other in decreasing
order that differ by one.
||Take note of the smallest throw in the siteswap sequence.
||For each throw, countdown from the throw height to the number
from step 1 and check to see if any of the countdown numbers
match up with the siteswap numbers on the same beat. If not,
the siteswap is valid. Otherwise it is invalid.
That's it! It's that simple :-)
© Nathan Peterson 2002