Siteswap state diagrams

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Siteswap state diagrams

Hans Lundmark ([email protected])

October 4, 2004

This document contains diagrams that can be used for quickly finding valid siteswap juggling patternsand transitions between different such patterns. These diagrams contain all possible siteswaps for

juggling with 3, 4, or 5 balls using throws of height at most 7.

www.juggling.org/help/siteswap/ is recommended reading if you are not familiar with siteswapnotation.

How to use

Decide how many balls you want to juggle and the highest throws you want to use, and find thecorresponding diagram.

The arrows are important. Each arrow is labeled by a digit which represents a throw, or by a stringof digits which represents a sequence of throws.

The numbers in the boxes are less important. They represent so-called juggling states, and thenumbering is explained in the technical notes below. The state with the lowest number in each

diagram is called the ground state and the others are called excited states.

Note: A Bx EEy EE

zmm is drawn as A Bx,y z in order to save space.

Any closed loop corresponds to a valid siteswap pattern, and vice versa. In other words:pick an initial state, then start walking around the diagram from there, following an arrow of yourchoice from one state to the next. For each step, remember the digit(s) on that arrow. When youreturn to the state where you started, the sequence of digits so obtained constitutes a valid siteswap.(And every valid siteswap with throws up to the chosen height can be found in this way.)

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Examples

In these examples, I will use the diagram for 3 balls, up to height 5. This is already enough formost well-known 3 ball tricks. The illustration below shows where the siteswaps 3, 441, and 531 canbe found in the diagram.

7

11 13

3

Siteswap 3 (cascade)

7

11 13

4

4 GG

1VV

VVVVV

Siteswap 441

7

11 13

53

""1

Siteswap 531

These three siteswaps all begin and end in the same state (namely the ground state 7). This meansthat they can be combined freely: 3 3 441 3 531 441, for example, is a valid siteswap.

The shower, on the other hand, starts and ends in an excited state, so one cant go immediately froma ground state pattern (like the cascade) to the shower; throwing 3 51 will cause two balls to landsimultaneously in one hand. Some extra throws must be inserted as a transition between the twopatterns, for example 3 4 51 or 3 52 51. As can be seen in the state diagram, there are four suchtransitions from the ground state 7 to the shower state 11 using throws no higher than 5:

7

11 13

51

Siteswap 51 (shower)

7

11 13

4,52

Transitions: 4 or 52.. .

7

11 13

53,551

VV

((VVV

VV

50oo

. . . or 5350 or 55150

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Inserting one cycle of the shower at the point where the siteswap 441 passes through state 11 producesthe siteswap 4 51 4 1:

7

11 13

4

51

4 GG

1VV

VVVVV

Siteswap 45141

The nice-looking number 12345 also happens to be a siteswap. When juggling this, it is better to think

of it as 3 4 51 2 since that is a ground state pattern:

7

11 13

3

4

51

2

ii

Siteswap 34512

Here are two ways of juggling 3 balls out of a 5 ball shower:

7

11 13

55500

Three ball flash

7

11 13550 DD

50ll

The snake

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Technical notes

These diagrams cover only standard siteswap juggling (asynchronous, no multiplex).

Juggling states are usually thought of as binary numbers with as many ones as the number of ballsbeing juggled; a one at the kth position from the right means that a previously thrown ball will belanding k time units from now, a zero means no ball. In the diagrams, the states are labeled by thedecimal and binary representations of that number.

The usual way of drawing state diagrams is to have each arrow denote only a single throw. However,the diagrams here are reduced in the sense that states having only one arrow coming in or only onearrow going out have been removed and replaced by extra (multi-throw) arrows. This leaves only theinteresting states where there is a true crossroads of choices, and reduces the total number of statesin the diagram from

dn

to

d2n1

, where n is the number of balls and d is the maximum height:

1 2 3 4 5 6 7 8 balls

height 2 13 1 14 1 2 15 1 3 3 16 1 4 6 4 17 1 5 10 10 5 18 1 6 15 20 15 6 19 1 7 21 35 35 21 7 1

Number of states required in a reduced state diagram.

I discovered this simplification on my own, but later I found a paper on logic and juggling by Steven deRooij1 from 2001, where the possibility of making this kind of reduction was pointed out in a footnote.I also suspect that graph theorists have thought of similar things in non-juggling contexts long beforethat. So I dont claim to be the first one with this idea, although I might be the first one to reallyexploit it for juggling purposes.

This document was produced using the LATEX typesetting program with the XY-pic package. I wrote aPython program which generated XY-pic code for all the arrows and labels in the diagrams. The layoutwas then manually fine-tuned to avoid collisions. The first version saw the light of day on January 14,2003.

1See homepages.cwi.nl/~rooij/.

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3 balls, up to height 4

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Siteswap state diagrams