Siteswap State Diagrams

Siteswap state diagrams

October 4, 2004

This document contains diagrams that can be used for quickly ﬁnding valid “siteswap” juggling patterns and 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 siteswap notation.

How to use

Decide how many balls you want to juggle and the highest throws you want to use, and ﬁnd the corresponding diagram. The arrows are important. Each arrow is labeled by a digit which represents a throw, or by a string of digits which represents a sequence of throws. The numbers in the boxes are less important. They represent so-called juggling states, and the numbering 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.

x - y - x,y z Note: A m z B is drawn as A B 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 your choice from one state to the next. For each step, remember the digit(s) on that arrow. When you return 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.)

1 Examples

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

7 7 7 \8 V ÖÖ 8 53 4 88 Ö 8 Ö 1 Ö 8 1 ÒÖÖ 8 11 13 11 4 / 13 11 13

Siteswap 3 (cascade) Siteswap 441 Siteswap 531

These three siteswaps all begin and end in the same state (namely the ground state 7). This means that 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 can’t go immediately from a ground state pattern (like the cascade) to the shower; throwing 3 51 will cause two balls to land simultaneously in one hand. Some extra throws must be inserted as a transition between the two patterns, for example 3 4 51 or 3 52 51. As can be seen in the state diagram, there are four such transitions from the ground state 7 to the shower state 11 using throws no higher than 5:

7 7 7 Ö 8 4,52 53,551 51 Ö 88 ÖÖ 88 ÒÖÖ 8 11 13 11 13 11 o 50 13

Siteswap 51 (shower) Transitions: 4 or 52...... or 5350 or 55150

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

7 \8 ÖÖ 8 4 88 51 Ö 8 Ö 1 Ö 8 ÒÖÖ 8 11 4 / 13

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 E 4 51 2 Õ 11 13

Siteswap 34512

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

55500

7 7

550 11 , 13 11 13 l 50

Three ball ﬂash The snake

3 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 balls being juggled; a one at the kth position from the right means that a previously thrown ball will be landing k time units from now, a zero means no ball. In the diagrams, the states are labeled by the decimal 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 one arrow going out have been removed and replaced by extra (multi-throw) arrows. This leaves only the interesting states where there is a true “crossroads” of choices, and reduces the total number of states d d−2 in the diagram from n to n−1 , where n is the number of balls and d is the maximum height:

1 2 3 4 5 6 7 8 balls height 2 1 3 1 1 4 1 2 1 5 1 3 3 1 6 1 4 6 4 1 7 1 5 10 10 5 1 8 1 6 15 20 15 6 1 9 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 de Rooij1 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 before that. So I don’t claim to be the first one with this idea, although I might be the first one to really exploit it for juggling purposes. This document was produced using the LATEX typesetting program with the XY-pic package. I wrote a Python program which generated XY-pic code for all the arrows and labels in the diagrams. The layout was then manually fine-tuned to avoid collisions. The first version saw the light of day on January 14, 2003.

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

4 3 balls, up to height 4

3,42,441,4440

7 111

3 balls, up to height 5

3,5520,55500

7 111 99 ÕÕÕ 999 ÕÕÕÕ 99 4,52Õ 53,551 Õ 999 ÕÕÕ 999 ÕÕÕ 999 ÕÕÕ 999 ÕÕÕ 999 ÕÕÕ 999 ÕÕÕ 999 ÕÕÕ 999 ÕÕÕ 999 ÕÕÕ 999 ÕÕÕ 999 ÕÕÕ 999 ÕÕÕ 99 51 2,530Õ 1,40 999 ÕÕÕ 9999 ÕÕÕ 9 11 13 4,550 50 1011 1101

5 3 balls, up to height 6

3,66300,666000

7 111 '' ?? '' ???? '' ??? '' ???? '' ? 4 '' 6620 63 64,661 ???? '' ?? 5,62 '' ???? '' ??? '' ??? '' ??? ''' ??? ' ' ??? ''' ??? 30 '' ??? 60 '' ??? '' ??? '' ??? '' ??? '' ??? 21 '' ??? 10101 '' ?? WWWW '' ??? qq S' 50 W '' ??? 1 qq ' WWWW '' ??? q ' WWWW '' ??? qqq ' WWWW''' ? 2 qqq ' W' WWW 1,40 qqq ' ''' WWWW 630 qqq ' ' ' WWWW ???? 5 qq ' ' WWW ?? ' ''' WWWW ??? qqq ' ' ' WWWW ??? qqqq ' ''' WWWW 11 q ' '' WWW 13 4,640 ' '' 50 + 1011 WWW ' '' 1101 WWWWW ' '' q8 ???? W 660 WW ' '' qq ??? WWWWWW ' '' qq ???? WWWWWW ' '' qq ??? WWWWWW ' qq WWWWWW '' 20,500 qq 61 WWWWW'W '' qq 60 ?? W'WWWWW '' qq ??? ' WWWWWW '' qq ??? ' WWWWWW '' 1 q ??? ' WW 600 q ??? ' WWWW qq ??? ' W 25 ??? ' 11001 ??? ' ??? ' F ???? '' ???? ' ??? ' ??? '' ???? ' ???? ' ?? ' ???? ' ?? '' ???? ??? 61 6400 ? ' 5 2,620 ' 3,6600 ' ???? ' ???? ' ?? '' ?? 19 10011

6 3 balls, up to height 7

3,774000,7770000

41 7 35 400 74 6,72 75000 101001 l 111 100011 L W/ < OOO L %%// ?? 71 oo / << OOO %% // ???? ooovv Ù / << O %% /// ???? ooo vv ÙÙ // 600 %%%% /// ??? ooo vv Ù / 1,700 OO %% // ? oooo3,720 ÙÙ 20 / < OO 4 %% // 77300 ooo Ù 77000 / < OO %% // ?? 7300 o v 4 / < OO 73 // ?? vv / << OOO 7720 // ???? oo vv Ù / < OO %% 75,771 ?? oo v Ù / < OO %% ???ooo vv Ù / << OO 5 %% // o?o??o vv ÙÙ 2 6 // < OOO %% // oooo ??? vv Ù / << OO %% //o/oo ??? vv Ù / < OOO %% oo/o// ?v?? ÙÙ / << OO %%ooo // vv??? Ù / < OO oo%%o // vv ??? Ù // < OoOoOo %%% // vv ??? ÙÙ / << ooo OO %%% /v//v ??? Ù / < ooo O%O% vv/// ??? Ù / << ooo %%OOOvv // ???Ù / < ooo %% v OO // Ù?Ù?? / << ooo v%%v OO // Ù ??? // < ooo vv%% OO/O/ Ù ??? /

720

7 4 balls, up to height 5

4,53,552,5551,55550

15 1111

4 balls, up to height 6

4,66620,666600

15 1111 ? ???? ??? 5,63 663,6661 ???? ???? 64,662 ???? ???? ???? ???? ???? ???? ???? ???? ?? 62 3,6630 1,50 ?? ???? ?? 23 29 64,6660 60 10111 11101 ??? ? ???? ? 5,661 ???? ???? ???? ???? ???? ???? ???? ??? ??? 2,640 ???? ???? 61 5 ?? ???? ?? 27 11011 _

660

8 4 balls, up to height 7

4,777300,7777000 71

43 15 57 o 74 774,7771 20,600 101011 k 1111 111001 OO nnn PP H 2 773 nn G O // ??? 6,73 ooo E V 22 4,750 ///?? P 700 o nn 5 / / 77720 PPP 6,770 OOO nnnn / ? PPPP ooo 2 OO nnn 75,772 ??? PPP ooo 2,730 2 OO nnnn ?? PPPP ooo 1 2 OO nnn // ???? PPP ooo 2 OOnnn // ?? PPPoo 75 22 nnnnOO // ??? ooPo 2 30 n OOO /// ??? ooo 77400 nn OO / / ?? ooo 72 PP 53 n OO / / ?o?oo 39 O / oo??? 771 110101 o OO // oo ?? 100111 S OO //ooo ??? k J . S 60 OO oo//o ?? kkkk ** . SS OO ooo // ??? 3 kkk | ** 70 . SSS OOO ooo /// ???kk || ** 6 1 SS O oo / kkk??k | ** SS OoOoo // kkk ??? 73 . SSS ooo OO // kkkk ?? 4,7720 77700 . SSS ooo OO kk//kk ???| ** . SSSooo OOOkkkk /// |??? ** . oooSSS kkkkOO /// || ??? ** . ooo SSS kkkk OO // | ??? ** .. ooo kkSkSSk OO// || ??? *** . oooo kkkk SSS O/O/ O | ??? *** . ooo kkkk SSS // OO|| ??? ** . ooo kkkk SSS ///| OO ??? ** . ooo kkkk SSS |/|/ OOO ??? ** .ooo k kkk S|S/// OO ??? *** 6 75 o.o kkk | SS/ S OO ?? * 3 oo . kkk || / / SS O ? ooo . kkk | / SS OO 1,50 70 7770 o kkk. k | // SS OO * kkkk . | // SSS OO ?? *** o 72 kk . | // SSS OO ??? ** ooook . || // SSS OO ??? ** oookkkk . || /// SSSOO 23 kokkk . | // SS' 29 k 7740 . | / / 60 ) 10111 . | / / 11101 OO . || /// ????OOOO . || // ? O ??? O .. | // ???? 771 . || // ? OOO . | /// 5,7730 OOO . || / / 74 ? OOO . | /// ??? OOO . || // ??? OOO . | // ??? OOOO .. || // ??? OOO . || /// ??? OOO. | /// ?? O|.O|OO // ???? |. OOO // ??? || . OOOO // ??? | .. OOO /// ??? || . OOO / / ??? | . OOOO /// ??|?| . OOO // ||??? . OOOO // | ??? .. OOO /// || ??? . OOO / / | ??? . OOO /// || ??? . OOO // || ??? . OOOO // | ??? . OOO // || ??? .. OOO /// | ??? . OOOO /// || ??? . OOO 1 | ??? . 2 71 OOO 7500 3 70 40 || . OOOO // 6 | 740 . 5 OO // || ??? . 720 /// | ????.. OO /// ~|| ? OOO Ù 45 27 51 60 6 770 2,7700 101101 i 11011 110011 _ 71

750

9 5 balls, up to height 6

5,64,663,6662,66661,666660

31 11111

5 balls, up to height 7

5,777720,7777700

31 11111 ** JJJ ttttt ** JJJ 6,74 t ** 7773,77771 *** J ttt *** JJJJ tttt ** JJJJ tttt ** JJJJ tttt ** JJJJ tttt JJJJ tttt 75,773 774,7772 JJJJ tttt ** JJJJ tttt ** JJJJ tttt *** JJJJ 73 ttt * * J 4,77730 *** 1,60 tt *** JJJJ tttt *** JJ 47 ** 61 774,77770 ** 70 101111 *** 111101 '' EE *** < K '' EEEE ** yy ''' EEEE ** y ''' EEEE ** yy '' EEEE ** yy ** yy 6,772 75,7771 *** y '' EE *** yy '' EEEE ** yy '' EEE ** yy '' EEEE ** y ''' EEE ** yy ''' EEEE ** yy '' EEE ***yy '' EEEE y* * '' EEE yy*** '' EEEE yy ** ''' EEEE yy ** ''' EEE yy ** '' EEEE y *** '' EEE yy *** '' EEEE yy ** '' EEE yy ** '' EEEEy ** '' yyEEE ** ''' yy EEE ** ''' y EEEE ** '' yy EEE *** '' yy EEEE * '' 3,7740 y EE 2,750 '' yy EEEE '' yy EEEE ** y *** 72 75 71 *** 6 ''' y EEEE ** ''' yy EEEE ** '' yy EEEE ** ''' yy EEEE *** ' yy EE * 55 6,7770 770 59 110111 111011 _

771