Causal Diagram

Causal Diagram: The arrows in this diagram point to the hand/throw that is caused by the object. Like "This throw has to be done so I have a free hand to catch the object".

Description

The causal diagram has a direct relation to the siteswap diagram, in which the arrows just represent the siteswap-number: As the arrows always point to the throw that needs to happen for the hand to become free to allow the catch, the number represented by the arrow is the siteswap number minus the number of beats until the same hand throws again.

This makes the graph often look a bit counter-intuitive (specially if one expects the arrows to represent the siteswap numbers):

• arrows for zips point to the left, i.e. are pointing back in time (because the throw to free the hand actually happens before the zip)
• "zero", i.e. empty hands go even further back in time
• flips have an arrow that points back to itself - which just shows the fact that nothing forces you to throw a flip and you can replace it by a hold

Disadvantages of causal diagrams

• counterintuitive
• hard to see where an object ends up (i.e. some things are only obvious in a siteswap diagram)

Advantages of causal diagram

• shorter arrows make the resulting diagrams much less cluttered
• less overlapping arrows make it easier to validate 3 or more person patterns
• directly show which throw triggers (and maybe causes) a bad throw happening in the pattern. Often the obvious bad throw that ends the pattern is caused one throw or several backwards in the causal chain of throws.

Examples

2-Hand 3 Ball Cascade

The siteswap diagram, in which the arrows represent the siteswap number looks like this:

Siteswap diagram. Each arrow represents the siteswap number

In the causal diagram, the arrows point to the last beat that happened on the same hand. As this example has two hands, the arrows are always shifted by two beats

Causal diagram. Each arrow represents the siteswap number minus the number of hands - i.e. the same hand as the siteswap, but one throw earlier

As you see, the diagram becomes much less crowded. The more complex the diagram becomes, the more helpful this is.

531 - Example with a Zip

One counter-intuitive thing is, that causal arrows can point backwards in time if zips are involved.

Note that the 3 is magic in this pattern, which you can easily see in the siteswap diagram, but not anymore in the causal diagram.

Siteswap diagram of 531. The 3 is magic

And as causal diagram:

Causal diagram of 531. The the arrows for zips point backwards in time, because you have to free the hand before you make the zip

While arrows pointing back in time *seems* counter-intuitive, it actually makes sense: As a zip arrives one beat after it was thrown, the hand has to be freed on the beat before the zip. So the zip "causes" the throw one beat earlier. You have to make that throw, else you cannot do you zip!

522 - Example with Flips

Another thing that can be seen here is, that 3 and 1 cause each other. As a consequence, if you don't do one of the throws, you can (and must) also not do the other. In other words: you can just hold (or flip):

Causal diagram of 522 - this is what you get when you "skip" doing the 31 of above

Again, what you see here in the diagram is that flips cause themselves - or in other words, you can do a flip or you can just leave that be and hold the club until it's this hand's turn again.