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Ladder diagram

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A ladder diagram is a 2-dimensional visual representation of a juggling pattern. The vertical dimension of the diagram represents time, and it is conventionally drawn so that going down the ladder corresponds to going forward in time, with the "rungs" of the ladder separating evenly spaced beats in the pattern. (The ladder diagrams that Juggling Lab creates don't show the rungs.) Turning a ladder diagram of a pattern upside-down turns it into a diagram of the time-reversed version of the pattern.

The lines going down the ladder represent the paths that the props take through time and space (ignoring the vertical dimension of physical patterns). The number of paths that intersect a horizontal line anywhere in the diagram is the number of props in the pattern. Throws and catches made with the right or left hand are shown on the corresponding side of the diagram.

3 ladder

This is a ladder diagram of the 3 ball cascade. The throws are made alternately from the right and left sides. Every throw crosses to the other side of the pattern, and is thrown again three beats later. No dwell time is shown in this diagram.

3 ladder 1dwell

This is another diagram of the same pattern, showing one beat of dwell time per throw, so that each throw has two beats of air time and then is held for one beat before it's thrown again. Each prop is still thrown every three beats. Each hand is alternately empty for one beat, and full for one beat.

When dwell time is shown in a ladder diagram, the throws and catches are drawn separately. The empty circles represent throws, and the filled circles represent catches.

3 ladder 2dwell

A ladder diagram with two beats of dwell time per throw (so that the hands would never be empty) looks similar to a causal diagram of the pattern.

0123456 ladder

This is a diagram of the siteswap 0123456, showing what each type of throw in a siteswap looks like in a ladder diagram.

The siteswap value of each throw is the number of rungs the line goes down the ladder before it's thrown again. The odd number throws are straight lines that cross to the other side of the diagram, and the even number throws stay on the same side, and have to be drawn as curved lines to make the diagram readable (except the 0s and 2s).

A 2 is drawn as a short vertical line representing a prop that is held for two beats before it's thrown again. (If dwell time was shown in the diagram, a 2 would be an object held for two beats longer than the usual dwell time.) A 0 is an empty beat, where no props are thrown or caught.

4 async-sync-async ladder

This diagram shows transitions between an asynch fountain and a synch fountain with 4 balls. The first throw that is different from the asynch fountain (the 5th beat from the top) is a non-crossing 5 (written as 5x), followed by a 4, and then a one beat hold (a 1x in siteswap) before the synch pattern starts.

...4444 5x41x (4,4)(4,4)...

The synch fountain alternates between a beat where two 4s are thrown at once, and an empty beat (a beat with no throws). The first beat of the transition from the synch fountain (the 8th beat from the bottom) contains a non-crossing 5 (a 5x) from the right, and in the next "synch pair" only one ball is actually thrown, and the other is held for one beat (a 1x). The asynch fountain starts immediately after this - an exclamation mark in the siteswap indicates that that pair of "throws" is not followed by the usual empty beat.

...(4,4)(4,4) (4,5x)(4,1x)! 4444...

55-1 ladder

This is a diagram of the physically impossible 3 ball siteswap 5 5 -1. A -1 is caught one beat before it is thrown (in a diagram that ignores dwell time), so those throws go one rung up the ladder, causing a ball to exist in more than one place at the same time. The paths of the balls traveling backward in time (called "antiballs") are shown in red. The average of the numbers in the siteswap, (5 + 5 + -1) / 3 = 3, equals the number of (forward traveling) balls that exist minus the number of antiballs that exist at any point in time in the pattern.

If this pattern could be juggled, the parts with the negative throws would look like this: Two balls (actually the same one in ball and antiball form) appear in one hand out of nowhere (this is when the ball changes direction to go forwards in time), and are thrown as a sliced multiplex. Then one of these balls (the antiball) lands in the other hand on the next beat, landing at the same time as another ball (which is also really the same ball) as a squeeze catch, and those two balls disappear (this is when the ball changes direction to go backwards in time).

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