Causal diagram

A causal diagram is a 2-dimensional visual representation of a juggling pattern, most commonly used for passing patterns. This type of diagram was invented by Martin Frost.
Unlike ladder diagrams, causal diagrams are usually drawn with time running from left to right, instead of top to bottom. Rather than tracking the paths of all the props, a causal diagram just shows which throws "cause" other throws in the pattern. This results in a much simpler diagram, which can give you a better idea of what a pattern looks like than a ladder diagram would.
If throw A causes throw B, that means that if you do throw A, you have to do throw B to keep the pattern going properly. Throw A is the reason for doing throw B.
The causal diagram for a 2-handed pattern looks similar to a ladder diagram for the same pattern, drawn with two beats of hold time between throws - or, a ladder diagram for the pattern made by subtracting 2 from each number in the siteswap.
3 ball cascade
When you juggle 3 balls, you normally want to avoid catching a ball with a hand that is already holding a ball, so whenever one hand makes a throw, that causes the other hand to have to make a throw on the next beat. The arrows in the diagram point from one throw to the throw that it causes.	
423
When you throw a 4 in a siteswap, that causes the same hand to have to make another throw two beats later, so the arrow from a 4 stays on one side of the diagram. The beats where you do 2s have arrows that point back to the same beat, which means that the twos aren't causing any other throws, or being caused by any other throws - you're just holding a ball because you have a ball to hold.	
522
When you throw a 5, that causes a throw on the other side of the pattern, but not until three beats later. In the causal diagram for a 2-handed pattern, the arrow from a throw of siteswap value N always points to the throw (N - 2) beats to the right. A 5 causes a throw 3 beats later, a 4 causes a throw 2 beats later, a 3 causes a throw 1 beat later, a 2 causes a throw 0 beats later (it causes itself)...
2 ball shower
...and a 1 causes a throw -1 beats later: one beat before the 1 was thrown. The arrow from a 1 points one beat to the left. In the 2 ball shower, throwing a 3 causes you to throw a 1 on the next beat, so your hand will be empty and ready to catch the 3, but the only reason for throwing the 3 was to empty your hand so it can catch the 1, which you haven't even thrown yet when you're doing the 3. The 3 and the 1 cause each other.	
441
Some of the throws in this pattern are caused by 1s from the future - the only reason to throw the second 4 is that you will need an empty hand to pass the 1 into on the next beat. There is a path of arrows that continuously travels to the right. In patterns that don't have a continuous path of causal arrows moving to the right, such as the 2 ball shower, it's possible to pause in the middle of the pattern (but not in the middle of an arrow) for as long as you want, without having to catch too many balls in one hand, and then continue the pattern normally, without any throws or catches happening in the wrong order.	
531
This pattern is the same as 522, except instead of having two beats in a row that each cause themselves, there are two beats in a row that cause each other. The only reason to throw the 3 is so that you can do the 1, and vice-versa.
4440
A 0 causes a throw to happen two beats before the 0. In this pattern whenever the right hand throws a 4, that causes a 0 two beats later: because that hand threw a 4, it will be empty two beats later. When it does a 0, that causes a 4 two beats earlier: for that hand to be empty on the 0 beat, it has to have thrown the ball two beats earlier. The 4 and the 0 cause each other.	
330
This pattern has cycles of three throws that all cause each other. Throwing the first 3 causes you to make another throw on the next beat, so you'll have a free hand to catch the first 3. Throwing the second ball as a 3, rather than passing it straight over as a 1, causes the hand that threw first to be empty on the next beat. For the third beat to be a 0, you also have to have done the first throw, so the first throw is caused by the 0.	
504
This is another variation of 522. Instead of the beat before a 5 and the beat after the 5 each causing themselves, those two beats cause each other. Doing the 0 is the only reason for throwing the 4, and vice-versa.
6 ball 2-count
In a causal diagram for a passing pattern, each person's throws are all drawn in one line, so there are two lines of throws in the diagram for a 2-person pattern. Each throw is labeled "R" or "L", so you can still tell which hands are making the throws. The arrow from a pass points to the other juggler's line, and the arrow from a self throw points to another throw in the line it came from.
7 ball 2-count
In this pattern, one person's right hand throws are made at the same time as the other person's left hand throws.
8 ball 53 2-count
If you draw a vertical line anywhere on a causal diagram, the number of arrows pointing to the right that it crosses minus the number of arrows pointing to the left that it crosses plus the number of hands used for the pattern equals the number of objects used in the pattern. A vertical line placed anywhere on this diagram would pass through four lines, all pointing to the right, and this is a 4-handed pattern, so this pattern uses 8 balls.
A left-handed diagonal 4 in a right handed 6 ball 2-count
The passed 4 is thrown a beat earlier than you would normally pass in this pattern, but since it's higher than the rest of the throws, it doesn't cause the other person to throw any earlier than they would if you had just kept doing the 2-count normally.
9 ball triangle 2-count
This is a 3-person pattern, so there are three lines. Each person is passing to a different person.
9 ball feed
The middle-line person is feeding the two outer-line people.
==External links==
* Causal Diagram editor on the Juggling Edge
* Passist passing siteswap animator, with causal diagrams
Category:Notation