479:
547:
2645:
29:
969:
2385:
3849:
421:. The number of beats a ball is in the air usually corresponds to how high it was thrown, so many people refer to the numbers as heights, but this is not technically correct; all that matters is the number of beats in the air, not how high it is thrown. For example, bouncing a ball takes longer than a throw in the air to the same height, and so can be a higher siteswap value while being a lower throw.
523:
2724:. The longest prime patterns with height 22 (with 3 ball maximum), for 9 balls (with 13 maximum height), and for heights and ball counts in between, were enumerated by Jack Boyce in February 1999 using a program called jdeep. The full list of longest prime siteswaps generated by jdeep (with '0' throws represented by a '-' and maximum height throws represented by a '+') can be found
955:, as shown on the right. To generate a siteswap, pick a starting state row. Index into the row via the corresponding throw column. The state entry at the intersection is the transitioned to state when that throw is made. From the new state, one can index into the table again. This process can be repeated so that when the original state is reached, a valid siteswap will be created.
2661:
is prime, as 1, 4, 41, and 44 are not valid three prop patterns (as 1/1≠3, 4/1≠3, (4+1)/2≠3, and (4+4)/2≠3). Sometimes this process does not work; for example, 153 (better known by its rotation 531) looks like it can be split into 15 and 3, but checking that the cycle has no repeating nodes in the graph traversal indicates that it is prime by the more rigorous definition.
2380:
2640:{\displaystyle c_{i}={\begin{cases}a_{i+1}+1,&{\text{if }}i{\text{ is even and }}a_{i}{\text{ crosses hands}}\\a_{i-1}-1,&{\text{if }}i{\text{ is odd and }}a_{i}{\text{ crosses hands}}\\a_{i+1},&{\text{if }}i{\text{ is even and }}a_{i}{\text{ does not cross}}\\a_{i-1},&{\text{if }}i{\text{ is odd and }}a_{i}{\text{ does not cross}}\end{cases}}}
502:, meaning one hand continually throws a low throw or 'zip' to the opposite hand, while the other continually makes a higher throw to the first. Sequences of bracketed pairs are written without delimiting markers. Patterns that repeat in mirror image on the opposite side can be abbreviated with a *. For example, Instead of
2214:
604:. (note that in the latter case, 4.5 will be straight passes from one juggler, crossing passes (i.e. left to left or right to right hand) from the other juggler. Social siteswaps can also be created for more than 2 jugglers (e.g. 4p 3 3 or 3.7 3 for 3 jugglers, where 3.7 is meant to mean 3.66666.... or 3
2660:
A non-rigorous but simpler method of determining if a siteswap is prime is to try to split it into any valid shorter pattern which uses the same number of props. For example, 44404413 can be split into 4440, 441, and 3; therefore, 44404413 is a composite. Another example, 441, which uses three props,
714:
on the right. This leaves us with x--x-. Since we caught a ball (the x we removed from the left) we can't "throw" a 0 next. We also can't throw a 1 or a 4, because there are already balls scheduled to land there. So assuming that the highest we can accurately throw a ball is to a height of 5, then we
99:
in 1985 (whence comes an alternative name). Hamkins wrote computer code in 1985 to systematically generate siteswap patterns—the printouts were taken immediately to the
Athenaeum lawn at Caltech to be tried out by himself, Tiemann, and Magnusson. The numbers derive from the number of balls used
976:
Not all siteswap sequences are valid. All vanilla, synchronous, and multiplex siteswap sequences are valid if their state transitions create a cycle in their state diagram graph. Sequences that do not create a cycle are invalid. For example, the pattern 531 can be mapped to a state diagram as shown
722:
Another method of representing siteswap states is represent a ball with a 1 instead of an x, and represent a spot where there's no ball scheduled to land with a 0 instead of a -. Then the state can be represented with a binary number, such as binary 10011. This format makes it possible to represent
718:
The diagram shown illustrates all possible states for someone juggling three items and a maximum height of 5. From each state one can follow the arrows and the corresponding numbers produce the siteswap. Any path which produces a cycle generates a valid siteswap, and all siteswaps can be generated
139:
This pattern can be described by stating how many throws later each ball is caught. For instance, on the first throw in the diagram, the purple ball is thrown in the air (up out of the screen, towards the bottom left) by the right hand, next the blue ball, the green ball, the green ball again, and
692:
Just after throwing a ball (or club or other juggling object), all balls are in the air and are under the influence of gravity. Assuming the balls are caught at a consistent level, then the timing of when the balls land is already determined. We can mark each point in time when a ball is going to
635:
Multi-hand notation was developed by Ed
Carstens in 1992 for use with his juggling program JugglePro. Siteswap notation in its simplest form ("Vanilla siteswap") assumes that only one ball is thrown at a time. It follows that any valid siteswap for two hands will also be valid for any number of
63:
that specify the number of beats in the future when the object is thrown again: "The idea behind siteswap is to keep track of the order that balls are thrown and caught, and only that." It is an invaluable tool in determining which combinations of throws yield valid juggling patterns for a given
430:
of the pattern. The period is the number of digits in the shortest non-repeating representation of a pattern. For example, the pattern diagrammed on the right is 53145305520 which has 11 digits and therefore has a period of 11. If the period is an odd number, like this one, then each time the
127:
Its simplest form, sometimes called vanilla siteswap, describes only patterns whose throws alternate hands and in which one ball is thrown from each hand at a time. If one were juggling while walking forward, something like the adjacent diagram would be seen from above, sometimes called a
1691:
New valid vanilla sequences can be generated by swapping adjacent elements from another valid vanilla siteswap sequence, adding 1 to the number being swapped to the right and subtracting 1 from the number being swapped to the left. The swap property will convert the valid sequence
120:
2647:
The slide property gets its name by sliding the throw times of one of the hands by one time unit so that the throws align asynchronously. For example, the siteswap (8x,4x)(4,4) would create two asynchronous (vanilla) siteswaps using the slide property: 9344 and 5744.
1953:
For example, the swap property performed on the inner two throws of the sequence 4413 would move the 4 to the right subtracting 1 from it to become 3 and move the 1 to the left adding 1 to it to become 2. This produces the new valid siteswap pattern 4233.
2375:{\displaystyle b_{i}={\begin{cases}a_{i}+1,&{\text{if }}i{\text{ is even and }}a_{i}{\text{ crosses hands}}\\a_{i}-1,&{\text{if }}i{\text{ is odd and }}a_{i}{\text{ crosses hands}}\\a_{i},&{\text{otherwise}}\end{cases}}}
675:
Some jugglers, when noting 4-handed siteswap, divide the siteswap values by the number of jugglers. This leads to a fractional notation similar to the notation for social siteswaps, but the order of the notation can be different.
705:
and determines what number ball can be thrown next. For instance, we can look at the state just after our first throw in the diagram, it is xx--x. We can use the state to determine what can be thrown next. First we take the
2656:
Siteswaps may be considered either prime or composite. A siteswap is prime if the path created in its state diagram does not traverse any state more than once. Siteswaps that are not prime are called composite.
435:
because each hand is doing the same thing (although at different times). If the period is an even number then on every repeat of the pattern, each hand does the same thing it did last time and the pattern is
1948:
2085:
64:
number of objects, and has led to previously unknown patterns (such as 441). However, it does not describe body movements such as behind-the-back and under-the-leg. Siteswap assumes that "throws happen on
3026:"...Bruce Tiemann (Boppo) and the late Bengt Magnusson....Other contributors to the development of site swap theory include Jack Boyce, Allen Knutson, Ed Carstens, and jugglers on the computer network."
3017:"Invented independently around 1985 by Paul Klimek of the University of California at Santa Cruz, Bruce Tiemann of the California Institute of Technology and Michael Day of the University of Cambridge."
1212:
685:
1627:
1419:
1797:
1274:
to the third number and so on. Second, calculate the modulus (remainder) of each number with the period. If none of the numbers are duplicated in this final sequence, then the pattern is valid.
565:
is a 6 prop '2 count' passing pattern, where all left hand throws are passes and right hand throws are selves. This can also be used with synchronous patterns; a two-person 'shower' is then
1057:
2209:
667:
In practice, this is most easily obtained if the jugglers throw by turns, one sequence being (Right hand of juggler A, right hand of juggler B, left hand of A, left hand of B).
2147:
2783:
1533:
1325:
3032:"To give credit where it is due, the notation as presented here was independently (and previously) invented by Paul Klimek, with whom we have had helpful discussions."
3451:
490:
and separated by a comma. Since synchronous throws are only thrown on even beats, only even numbers are allowed. Throws that move to the other hand are marked by an
1659:
1565:
1483:
1451:
1357:
95:, and the late Bengt Magnusson at the California Institute of Technology in 1985, and by Mike Day, mathematician Colin Wright, and mathematician Adam Chalcraft in
2722:
2702:
2682:
1817:
1272:
1252:
1232:
1125:
1101:
1077:
119:
715:
can only throw a 2, 3, or a 5. In this diagram, the juggler threw a 3, so an x goes in the third spot, replacing the -, and we have x-xx- as the new state.
719:
this way. The diagram quickly becomes bigger when more balls or higher throws are introduced as there are more possible states and more possible throws.
1962:
A valid synchronous sequence can be converted to a valid asynchronous sequence and vice versa using the slide property. Given the synchronous sequence
3282:
588:
If both juggle the same pattern (although shifted in time), the pattern is called a social siteswap and only half of the pattern needs to be written:
140:
the blue ball again and then finally the purple ball is caught and thrown by the left hand on the fifth throw, this gives the first throw a count of
486:
Siteswap notation can be extended to denote patterns containing synchronous throws from both hands. The numbers for the two throws are combined in
1130:
3088:
3023:"Invented around 1985 by three people independently: Bruce "Boppo" Tiemann at Caltech, Paul Klimek in Santa Cruz, and Mike Day in Cambridge."
664:
Valid siteswaps can be juggled by a 4-handed juggler, or for 2 jugglers coordinating 4 hands, on the condition that hands throw alternately.
2725:
462:
Some hold to a convention in that a siteswap is written with its highest numbers first. One drawback to doing so is evident in the pattern
1822:
1965:
3401:
530:
A further extension allows siteswap to notate patterns involving multiple throws from either or both hands at the same time in a
4026:
4256:
3147:
1570:
1362:
3659:
1695:
561:
notation means one juggler does 'xxx' while another does 'yyy'. 'p' is used to represent a passing throw. For example,
3718:
3352:
3029:"Jack Boyce (also at Caltech) came up with the juggling state model to explain the phenomenon of excited-state tricks."
534:
pattern. The numbers for multiple throws from a single hand are written together inside square brackets. For example,
3590:
3327:
3480:
111:. For example, swapping the landing times of throws "5" and "1" in the siteswap "51" generates the siteswap "24".
107:
comes from the ability to generate patterns by "swapping" landing times of any 2 "sites" in a siteswap using the
710:
off the left hand side (that's the ball that's landing next) and shift everything else to the left filling in a
3224:
990:
152:-numbered throws send the ball to the other hand, while even-numbered throws send the ball to the same hand. A
3551:
Knutson, Allen; Lam, Thomas; Speyer, David (November 15, 2011). "Positroid
Varieties: Juggling and Geometry".
3992:
4031:
4002:
3279:
3254:
3119:
3653:
3020:"...site swap patterns...in the form invented by some of Bruce Tiemann, Bengt Magnusson, and Joel Hamkins"
2152:
443:
The number of balls used for the pattern is the average of the throw numbers in the pattern. For example,
4007:
3997:
2954:
977:
on the right. Since the transitions in this sequence create a cycle in the graph, this pattern is valid.
466:, a 3-ball pattern which cannot be inserted into the middle of a string of 3-throws, unlike its rotation
3393:
100:
in the most common juggling patterns. Siteswap has been described as, "perhaps the most popular" name.
636:
hands, on the condition that the hands throw after each other. Commonly used multi-hand siteswaps are
2752:
2090:
4266:
4208:
4048:
3096:
2407:
2236:
478:
451:
is a seven-object pattern. All patterns must therefore have a siteswap sequence that averages to an
3921:
3319:
2983:
2885:
A subset of these siteswap patterns naturally label strata in the positroid stratification of the
4198:
3828:
1488:
1280:
4235:
4036:
3711:
3056:
2742:
1359:. Since the pattern 531 has a period of 3, the results from the previous example would produce
980:
There are other methods of determining a sequence's validity based on the flavor of siteswap.
952:
580:
the juggler after the bar is supposed to be half a count later, and all fractions are passes.
88:
3580:
546:
136:. In this diagram, three balls are being juggled. Time progresses from the top to the bottom.
4158:
3214:
60:
3307:
4230:
4014:
3887:
3802:
3164:
1632:
1538:
1485:
are all unique. Another example, 513 is an invalid pattern because the first step produces
1456:
1424:
1330:
1104:
531:
149:
8:
4220:
4203:
3987:
3979:
3777:
3731:
3315:
3308:
3155:
1661:, and the final sequence contains at least a duplicate of one number, in this case a 2.
362:
328:
294:
260:
165:
3671:
3168:
3857:
3823:
3767:
3757:
3552:
3188:
2707:
2687:
2667:
1802:
1257:
1237:
1217:
1110:
1086:
1062:
379:
345:
311:
277:
243:
157:
92:
72:
3528:
3511:
3422:
3176:
4261:
4143:
4096:
4058:
4019:
3833:
3816:
3807:
3782:
3744:
3704:
3586:
3533:
3333:
3323:
3230:
3220:
3180:
3061:
2996:
2746:
495:
76:
48:
148:
of numbers which denote the height of each throw to be made. Since hands alternate,
4215:
4121:
4101:
3961:
3936:
3772:
3739:
3696:
3523:
3303:
3172:
2935:
52:
3196:
459:
with integer average 4 but its three throws all land at the same time, colliding.
431:
sequence is repeated, the sequence starts with the other hand, and the pattern is
4163:
3762:
3663:
3576:
3405:
3286:
3114:
2664:
It has been shown empirically that the longest prime siteswaps bounded by height
4225:
4081:
3951:
3946:
3926:
3797:
3752:
3646:
723:
multiplex states, i.e. the number 2 represents that 2 balls land on that beat.
697:, and each point in time when there is not yet a ball scheduled to land with a
507:
28:
3614:
4250:
4091:
4043:
3956:
3931:
3892:
3865:
3838:
3537:
3337:
3184:
968:
684:
487:
22:
3360:
3234:
1214:
To find if a pattern is valid, first create a new sequence formed by adding
4170:
4153:
4148:
4131:
4106:
4076:
3966:
3904:
3249:
2886:
65:
59:, possible patterns transcribed using siteswap. Throws are represented by
3897:
3875:
3848:
3682:
2950:
1080:
538:
is a normal 3-ball cascade, with a pair of balls always thrown together.
156:
represents a throw to the opposite hand at the height of the basic three-
3192:
2966:
2905:
Chevrons and vertical bar <|>: Simultaneous and passing patterns.
4175:
4116:
3880:
3488:
2939:
123:
Diagram of someone "juggling" with the siteswap notation and the state
91:
in 1981, and later developed by undergraduates Bruce "Boppo" Tiemann,
3916:
3792:
2975:- A web based, open source, 3d juggling animator and pattern library.
2786:
627:
Note that some jugglers use fractions to note multi-handed patterns.
426:
96:
3640:
3624:
2972:
4111:
4071:
4063:
3870:
3787:
3727:
2741:
Vanilla siteswap patterns may be viewed as certain elements of the
145:
3557:
2919:*: Synchronous pattern that switches sides. (4,2x)(2x,4) = (4,2x)*
522:
424:
Each pattern repeats after a certain number of throws, called the
4126:
4086:
3909:
2963:- 3d animator capable of displaying multihand (passing) patterns.
653:
452:
2849:
corresponds to the (infinitely repeated) siteswap pattern whose
1943:{\displaystyle a_{0}a_{1}a_{2}...(a_{i+1}+1)(a_{i}-1)...a_{n-1}}
4138:
3941:
3649:(Helpful Vanilla, Multiplex and Synchronous siteswap validator)
2080:{\displaystyle (a_{0},a_{1})(a_{2},a_{3})...(a_{n-2},a_{n-1})}
164:
represents a throw to the same hand at the height of the four-
4182:
1610:
1594:
1578:
1402:
1386:
1370:
1166:
1675:
it can be converted into a valid vanilla siteswap using the
3634:
2946:
2633:
2368:
1207:{\displaystyle S=\{(a_{i}+i){\bmod {n}}|0\leq i\leq n-1\}}
3280:
The
Academic Juggler: The Invention Of Juggling Notations
2911:
Fraction: Pass 1/y beats later. <4.5 3 3 | 3 4 3.5>
409:
is a quick pass straight across to the other hand, and a
3510:
Ehrenborg, Richard; Readdy, Margaret (October 1, 1996).
3089:
A Notation for
Juggling Tricks, A LOT of Juggling Tricks
2899:
Number: Relative duration (height) of a toss. 1, 2, 3...
455:. Not all such sequences describe patterns; for example
2960:
2922:
x: Toss to the other hand during a synchronous pattern.
2093:
1698:
1622:{\displaystyle 5{\bmod {3}},2{\bmod {3}},5{\bmod {3}}}
1414:{\displaystyle 5{\bmod {3}},4{\bmod {3}},3{\bmod {3}}}
951:
A siteswap state diagram can also be represented as a
447:
is a three-object pattern because (4+4+1)/3 is 3, and
2755:
2710:
2690:
2670:
2388:
2217:
2155:
1968:
1825:
1805:
1792:{\textstyle a_{0}a_{1}a_{2}...a_{i}a_{i+1}...a_{n-1}}
1635:
1573:
1541:
1491:
1459:
1427:
1365:
1333:
1283:
1260:
1240:
1220:
1133:
1113:
1089:
1065:
993:
413:
is a momentary hold of an object. Throws longer than
3726:
494:
following the number. Thus a synchronous three-prop
3654:
Staggered
Symmetric Passing Patterns for 2 jugglers
3219:. Mathematical Association of America. p. 99.
2785:). One presentation of this group is as the set of
2777:
2736:
2716:
2696:
2676:
2639:
2374:
2203:
2141:
2079:
1942:
1811:
1791:
1653:
1621:
1559:
1527:
1477:
1445:
1413:
1351:
1319:
1266:
1246:
1226:
1206:
1119:
1095:
1071:
1051:
3216:Mathematical Adventures for Students and Amateurs
2979:There are also some games to play with siteswap:
1079:is the period of the siteswap, is valid when the
688:State Diagram for 3 balls with a max throw of '5'
652:The siteswap is performed by a single hand, or a
4248:
3550:
3146:Beek, Peter J.; Lewbel, Arthur (November 1995).
2969:works on Windows, Macintosh and Wine (For Linux)
3509:
1453:. In this case, 531 is valid since the numbers
656:player throwing diabolos at different heights.
21:"Juggling beat" redirects here. For music, see
3712:
3087:Tiemann, Bruce and Magnusson, Bengt (1991). "
1201:
1140:
87:The notation was invented by Paul Klimek in
3212:
2731:
2087:, two new vanilla sequences can be formed:
1277:For example, the pattern 531 would produce
16:Notation used to describe juggling patterns
3719:
3705:
3512:"Juggling and applications to q-analogues"
3145:
2957:and interprets nearly all siteswap syntax.
2880:
958:
3556:
3527:
3394:Siteswap Ben's Guide to Juggling Patterns
3213:Hayes, David F.; Shubin, Tatiana (2004).
3141:
3139:
3137:
3083:
3081:
3079:
1052:{\displaystyle a_{0}a_{1}a_{2}...a_{n-1}}
3302:
3274:
3272:
967:
683:
576:If the pattern contains fractions, e.g.
545:
521:
477:
55:. The term may also be used to describe
27:
3575:
3296:
3054:
2793:on the integers such that, for a fixed
32:Siteswap beats shown as relative height
4249:
3669:
3134:
3076:
2986:developed by Sebi Haushofer (for Java)
571:
417:beats are given letters starting with
3700:
3478:
3452:"The Longest Prime Siteswap Patterns"
3416:
3414:
3388:
3386:
3384:
3382:
3380:
3378:
3350:
3269:
3108:
3106:
3104:
2916:Parentheses (): Synchronous pattern.
2829:satisfies the further condition that
1819:, to generate the new valid sequence
3670:Wright, Colin (September 29, 2017).
3656:" by Sean Gandini (social siteswaps)
3625:DSSS: The Diabolo Siteswap Simulator
2204:{\displaystyle c_{0}c_{1}...c_{n-1}}
482:Ladder diagram for box: (4,2x)(2x,4)
3420:
3247:
3112:
2865:: that is, the ball thrown at time
1676:
13:
3569:
3411:
3375:
3353:"Patterns from Lodi 1997 Workshop"
3241:
3101:
3055:Donahue, Bill (December 3, 2004).
2892:
401:There are three special throws: a
118:
108:
68:that are equally spaced in time."
14:
4278:
3607:
3479:Boyce, Jack (February 17, 1999).
3449:
3177:10.1038/scientificamerican1195-92
2651:
2142:{\textstyle b_{0}b_{1}...b_{n-1}}
1957:
1672:it only contains even numbers and
679:
614:. Then each juggler should start
405:is a pause with an empty hand, a
214:Throw from one hand to the other
3847:
3408: (archived August 10, 2015).
3351:Boyce, Jack (October 11, 1997).
3057:"The Mathematics of... Juggling"
2778:{\displaystyle {\tilde {A}}_{n}}
1686:
510:pattern), can be abbreviated to
3544:
3503:
3472:
3443:
3344:
2737:Connections to abstract algebra
630:
624:count after the previous one.)
550:Four-count, or "Every others":
526:3-ball Cascade with triplex: 33
517:
3666: (archived August 6, 2003)
3206:
3048:
3009:
2953:animator which was written in
2763:
2074:
2036:
2024:
1998:
1995:
1969:
1912:
1893:
1890:
1865:
1176:
1162:
1143:
473:
51:used to describe or represent
1:
3660:Smith, H.J. "Juggler Numbers"
3529:10.1016/S0012-365X(96)83010-X
3423:"The Mathematics of Juggling"
3041:
701:. This describes the current
75:may be notated "3 ", while a
4257:Juggling patterns and tricks
7:
3582:The Mathematics of Juggling
2990:
2929:
1567:, the second step produces
1528:{\displaystyle 5+0,1+1,3+2}
1320:{\displaystyle 5+0,3+1,1+2}
963:
725:
642:4-handed (passing) siteswap
638:1-handed (diabolo) siteswap
10:
4283:
3615:Symmetric Passing Patterns
2908:P: Pass. <333P|333P>
2684:contain mostly the throws
972:Siteswap 531 state diagram
541:
114:
71:For example, a three-ball
20:
4191:
4057:
3975:
3856:
3845:
3738:
3148:"The Science of Juggling"
3095:. Accessed July 8, 2014.
2902:Brackets : Multiplex. 33.
1683:otherwise it is invalid.
1107:) is equal to the period
578:<4.5 3 3 | 3 4 3.5>
567:<(4xp,2x)|(4xp,2x)>
82:
3647:TWJC Siteswap Calculator
3304:Sethares, William Arthur
3278:Lewbel, Arthur (1996). "
3002:
2732:Mathematical connections
598:<4.5 3 3| 3 4.5 3>
3637:(Downloadable animator)
2881:Connections to topology
2559: is even and
2447: is even and
2270: is even and
959:Mathematical properties
557:Simultaneous juggling:
180:Beats object is in Air
3585:. New York: Springer.
3285:July 14, 2014, at the
3250:"(in) IBM-PC Programs"
2779:
2743:affine symmetric group
2718:
2698:
2678:
2641:
2612: is odd and
2506: is odd and
2376:
2323: is odd and
2205:
2143:
2081:
1944:
1813:
1793:
1668:siteswap is valid if
1655:
1623:
1561:
1529:
1479:
1447:
1415:
1353:
1321:
1268:
1254:to the second number,
1248:
1228:
1208:
1121:
1097:
1073:
1053:
973:
953:state-transition table
689:
554:
527:
483:
124:
89:Santa Cruz, California
79:may be notated "5 1".
33:
3672:"Juggling by Numbers"
3392:Beever, Ben (2001). "
3310:Rhythm and Transforms
2947:Juggling Lab animator
2780:
2719:
2699:
2679:
2642:
2377:
2206:
2144:
2082:
1945:
1814:
1799:with arbitrary value
1794:
1656:
1654:{\displaystyle 2,2,2}
1624:
1562:
1560:{\displaystyle 5,2,5}
1530:
1480:
1478:{\displaystyle 2,1,0}
1448:
1446:{\displaystyle 2,1,0}
1416:
1354:
1352:{\displaystyle 5,4,3}
1322:
1269:
1249:
1234:to the first number,
1229:
1209:
1122:
1098:
1074:
1054:
971:
687:
549:
525:
481:
378:Throw from a 11 ball
361:Throw from a 10 ball
172:Siteswap Throw Names
122:
61:non-negative integers
31:
3988:Juggling Conventions
3829:Rubenstein's Revenge
3516:Discrete Mathematics
2942:juggling patterns.
2934:There are many free
2753:
2708:
2688:
2668:
2627: does not cross
2574: does not cross
2386:
2215:
2153:
2091:
1966:
1823:
1803:
1696:
1633:
1571:
1539:
1489:
1457:
1425:
1363:
1331:
1281:
1258:
1238:
1218:
1131:
1111:
1105:Set-builder notation
1087:
1063:
991:
344:Throw from a 9 ball
327:Throw from a 8 ball
310:Throw from a 7 ball
293:Throw from a 6 ball
276:Throw from a 5 ball
259:Throw from a 4 ball
242:Throw from a 3 ball
4221:Rhythmic gymnastics
3732:object manipulation
3491:on December 7, 2004
3363:on December 7, 2004
3169:1995SciAm.273e..92B
3156:Scientific American
2521: crosses hands
2462: crosses hands
2338: crosses hands
2285: crosses hands
648:1-handed (diabolo)
572:Fractional notation
173:
3421:Polster, Burkard.
2869:will land at time
2825:. If the element
2775:
2714:
2694:
2674:
2637:
2632:
2372:
2367:
2201:
2139:
2077:
1940:
1809:
1789:
1651:
1619:
1557:
1525:
1475:
1443:
1411:
1349:
1317:
1264:
1244:
1224:
1204:
1117:
1093:
1069:
1049:
987:siteswap sequence
974:
690:
590:<4p 3| 3 4p>
555:
528:
484:
171:
144:. This produces a
130:space-time diagram
125:
97:Cambridge, England
93:Joel David Hamkins
45:Cambridge notation
34:
4244:
4243:
4097:Coin manipulation
3643:(Online animator)
3202:on March 4, 2016.
3062:Discover Magazine
2997:List of siteswaps
2936:computer programs
2821:for all integers
2766:
2747:affine Weyl group
2717:{\displaystyle h}
2697:{\displaystyle 0}
2677:{\displaystyle h}
2628:
2613:
2605:
2575:
2560:
2552:
2522:
2507:
2499:
2463:
2448:
2440:
2363:
2339:
2324:
2316:
2286:
2271:
2263:
1812:{\displaystyle i}
1267:{\displaystyle 2}
1247:{\displaystyle 1}
1227:{\displaystyle 0}
1120:{\displaystyle n}
1096:{\displaystyle S}
1072:{\displaystyle n}
949:
948:
671:mixed-up notation
563:<3p 3|3p 3>
552:<333P|333P>
399:
398:
57:siteswap patterns
53:juggling patterns
49:juggling notation
4274:
4267:Rhythm and meter
4122:Flair bartending
4102:Contact juggling
3851:
3721:
3714:
3707:
3698:
3697:
3693:
3691:
3689:
3676:
3641:Gunswap Juggling
3629:ArtofDiabolo.com
3603:
3601:
3599:
3577:Polster, Burkard
3563:
3562:
3560:
3548:
3542:
3541:
3531:
3507:
3501:
3500:
3498:
3496:
3487:. Archived from
3476:
3470:
3469:
3467:
3465:
3456:
3447:
3441:
3440:
3438:
3436:
3427:
3418:
3409:
3398:JugglingEdge.com
3390:
3373:
3372:
3370:
3368:
3359:. Archived from
3348:
3342:
3341:
3313:
3300:
3294:
3276:
3267:
3266:
3264:
3262:
3248:Knutson, Allen.
3245:
3239:
3238:
3210:
3204:
3203:
3201:
3195:. Archived from
3152:
3143:
3132:
3131:
3129:
3127:
3113:Knutson, Allen.
3110:
3099:
3085:
3074:
3073:
3071:
3069:
3052:
3035:
3013:
2938:available which
2784:
2782:
2781:
2776:
2774:
2773:
2768:
2767:
2759:
2723:
2721:
2720:
2715:
2703:
2701:
2700:
2695:
2683:
2681:
2680:
2675:
2646:
2644:
2643:
2638:
2636:
2635:
2629:
2626:
2624:
2623:
2614:
2611:
2606:
2603:
2596:
2595:
2576:
2573:
2571:
2570:
2561:
2558:
2553:
2550:
2543:
2542:
2523:
2520:
2518:
2517:
2508:
2505:
2500:
2497:
2484:
2483:
2464:
2461:
2459:
2458:
2449:
2446:
2441:
2438:
2425:
2424:
2398:
2397:
2381:
2379:
2378:
2373:
2371:
2370:
2364:
2361:
2354:
2353:
2340:
2337:
2335:
2334:
2325:
2322:
2317:
2314:
2301:
2300:
2287:
2284:
2282:
2281:
2272:
2269:
2264:
2261:
2248:
2247:
2227:
2226:
2210:
2208:
2207:
2202:
2200:
2199:
2175:
2174:
2165:
2164:
2148:
2146:
2145:
2140:
2138:
2137:
2113:
2112:
2103:
2102:
2086:
2084:
2083:
2078:
2073:
2072:
2054:
2053:
2023:
2022:
2010:
2009:
1994:
1993:
1981:
1980:
1949:
1947:
1946:
1941:
1939:
1938:
1905:
1904:
1883:
1882:
1855:
1854:
1845:
1844:
1835:
1834:
1818:
1816:
1815:
1810:
1798:
1796:
1795:
1790:
1788:
1787:
1763:
1762:
1747:
1746:
1728:
1727:
1718:
1717:
1708:
1707:
1660:
1658:
1657:
1652:
1628:
1626:
1625:
1620:
1618:
1617:
1602:
1601:
1586:
1585:
1566:
1564:
1563:
1558:
1534:
1532:
1531:
1526:
1484:
1482:
1481:
1476:
1452:
1450:
1449:
1444:
1420:
1418:
1417:
1412:
1410:
1409:
1394:
1393:
1378:
1377:
1358:
1356:
1355:
1350:
1326:
1324:
1323:
1318:
1273:
1271:
1270:
1265:
1253:
1251:
1250:
1245:
1233:
1231:
1230:
1225:
1213:
1211:
1210:
1205:
1179:
1174:
1173:
1155:
1154:
1126:
1124:
1123:
1118:
1102:
1100:
1099:
1094:
1078:
1076:
1075:
1070:
1058:
1056:
1055:
1050:
1048:
1047:
1023:
1022:
1013:
1012:
1003:
1002:
726:
623:
622:
618:
613:
612:
608:
553:
174:
170:
41:quantum juggling
4282:
4281:
4277:
4276:
4275:
4273:
4272:
4271:
4247:
4246:
4245:
4240:
4187:
4061:
4053:
4020:Combat juggling
3977:
3971:
3852:
3843:
3742:
3734:
3725:
3687:
3685:
3674:
3664:Wayback Machine
3610:
3597:
3595:
3593:
3572:
3570:Further reading
3567:
3566:
3549:
3545:
3508:
3504:
3494:
3492:
3477:
3473:
3463:
3461:
3454:
3448:
3444:
3434:
3432:
3425:
3419:
3412:
3406:Wayback Machine
3391:
3376:
3366:
3364:
3349:
3345:
3330:
3301:
3297:
3287:Wayback Machine
3277:
3270:
3260:
3258:
3246:
3242:
3227:
3211:
3207:
3199:
3150:
3144:
3135:
3125:
3123:
3111:
3102:
3086:
3077:
3067:
3065:
3053:
3049:
3044:
3039:
3038:
3014:
3010:
3005:
2993:
2932:
2927:
2895:
2893:List of symbols
2883:
2769:
2758:
2757:
2756:
2754:
2751:
2750:
2739:
2734:
2709:
2706:
2705:
2689:
2686:
2685:
2669:
2666:
2665:
2654:
2631:
2630:
2625:
2619:
2615:
2610:
2602:
2600:
2585:
2581:
2578:
2577:
2572:
2566:
2562:
2557:
2549:
2547:
2532:
2528:
2525:
2524:
2519:
2513:
2509:
2504:
2496:
2494:
2473:
2469:
2466:
2465:
2460:
2454:
2450:
2445:
2437:
2435:
2414:
2410:
2403:
2402:
2393:
2389:
2387:
2384:
2383:
2366:
2365:
2360:
2358:
2349:
2345:
2342:
2341:
2336:
2330:
2326:
2321:
2313:
2311:
2296:
2292:
2289:
2288:
2283:
2277:
2273:
2268:
2260:
2258:
2243:
2239:
2232:
2231:
2222:
2218:
2216:
2213:
2212:
2189:
2185:
2170:
2166:
2160:
2156:
2154:
2151:
2150:
2127:
2123:
2108:
2104:
2098:
2094:
2092:
2089:
2088:
2062:
2058:
2043:
2039:
2018:
2014:
2005:
2001:
1989:
1985:
1976:
1972:
1967:
1964:
1963:
1960:
1928:
1924:
1900:
1896:
1872:
1868:
1850:
1846:
1840:
1836:
1830:
1826:
1824:
1821:
1820:
1804:
1801:
1800:
1777:
1773:
1752:
1748:
1742:
1738:
1723:
1719:
1713:
1709:
1703:
1699:
1697:
1694:
1693:
1689:
1634:
1631:
1630:
1613:
1609:
1597:
1593:
1581:
1577:
1572:
1569:
1568:
1540:
1537:
1536:
1490:
1487:
1486:
1458:
1455:
1454:
1426:
1423:
1422:
1405:
1401:
1389:
1385:
1373:
1369:
1364:
1361:
1360:
1332:
1329:
1328:
1282:
1279:
1278:
1259:
1256:
1255:
1239:
1236:
1235:
1219:
1216:
1215:
1175:
1169:
1165:
1150:
1146:
1132:
1129:
1128:
1112:
1109:
1108:
1088:
1085:
1084:
1064:
1061:
1060:
1037:
1033:
1018:
1014:
1008:
1004:
998:
994:
992:
989:
988:
966:
961:
734:
731:
682:
633:
620:
616:
615:
610:
606:
605:
584:social siteswap
574:
559:<xxx|yyy>
551:
544:
520:
476:
228:Momentary hold
183:Switches hands
117:
85:
47:, is a numeric
26:
17:
12:
11:
5:
4280:
4270:
4269:
4264:
4259:
4242:
4241:
4239:
4238:
4233:
4228:
4223:
4218:
4213:
4212:
4211:
4201:
4195:
4193:
4189:
4188:
4186:
4185:
4180:
4179:
4178:
4168:
4167:
4166:
4156:
4151:
4146:
4141:
4136:
4135:
4134:
4124:
4119:
4114:
4109:
4104:
4099:
4094:
4089:
4084:
4082:Baton twirling
4079:
4074:
4068:
4066:
4055:
4054:
4052:
4051:
4046:
4041:
4040:
4039:
4029:
4024:
4023:
4022:
4012:
4011:
4010:
4005:
4000:
3995:
3984:
3982:
3973:
3972:
3970:
3969:
3964:
3959:
3954:
3952:Plate spinning
3949:
3944:
3939:
3934:
3929:
3927:Flagging dance
3924:
3919:
3914:
3913:
3912:
3902:
3901:
3900:
3890:
3885:
3884:
3883:
3878:
3873:
3862:
3860:
3854:
3853:
3846:
3844:
3842:
3841:
3836:
3831:
3826:
3821:
3820:
3819:
3805:
3800:
3795:
3790:
3785:
3780:
3775:
3770:
3765:
3760:
3755:
3749:
3747:
3736:
3735:
3724:
3723:
3716:
3709:
3701:
3695:
3694:
3667:
3657:
3650:
3644:
3638:
3632:
3622:
3609:
3608:External links
3606:
3605:
3604:
3591:
3571:
3568:
3565:
3564:
3543:
3522:(1): 107–125.
3502:
3471:
3442:
3410:
3374:
3343:
3328:
3295:
3268:
3240:
3225:
3205:
3133:
3115:"Siteswap FAQ"
3100:
3075:
3046:
3045:
3043:
3040:
3037:
3036:
3034:
3033:
3030:
3027:
3024:
3021:
3018:
3007:
3006:
3004:
3001:
3000:
2999:
2992:
2989:
2988:
2987:
2977:
2976:
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2958:
2931:
2928:
2926:
2925:
2924:
2923:
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2912:
2909:
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2900:
2896:
2894:
2891:
2882:
2879:
2772:
2765:
2762:
2738:
2735:
2733:
2730:
2713:
2693:
2673:
2653:
2652:Prime patterns
2650:
2634:
2622:
2618:
2609:
2601:
2599:
2594:
2591:
2588:
2584:
2580:
2579:
2569:
2565:
2556:
2548:
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2538:
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2198:
2195:
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2188:
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2181:
2178:
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2159:
2136:
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2126:
2122:
2119:
2116:
2111:
2107:
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2065:
2061:
2057:
2052:
2049:
2046:
2042:
2038:
2035:
2032:
2029:
2026:
2021:
2017:
2013:
2008:
2004:
2000:
1997:
1992:
1988:
1984:
1979:
1975:
1971:
1959:
1958:Slide property
1956:
1937:
1934:
1931:
1927:
1923:
1920:
1917:
1914:
1911:
1908:
1903:
1899:
1895:
1892:
1889:
1886:
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1878:
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1731:
1726:
1722:
1716:
1712:
1706:
1702:
1688:
1685:
1681:
1680:
1677:slide property
1673:
1650:
1647:
1644:
1641:
1638:
1616:
1612:
1608:
1605:
1600:
1596:
1592:
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1584:
1580:
1576:
1556:
1553:
1550:
1547:
1544:
1524:
1521:
1518:
1515:
1512:
1509:
1506:
1503:
1500:
1497:
1494:
1474:
1471:
1468:
1465:
1462:
1442:
1439:
1436:
1433:
1430:
1408:
1404:
1400:
1397:
1392:
1388:
1384:
1381:
1376:
1372:
1368:
1348:
1345:
1342:
1339:
1336:
1316:
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1310:
1307:
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1301:
1298:
1295:
1292:
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1243:
1223:
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1200:
1197:
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1191:
1188:
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1182:
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1172:
1168:
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1161:
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1142:
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1136:
1116:
1092:
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1046:
1043:
1040:
1036:
1032:
1029:
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1021:
1017:
1011:
1007:
1001:
997:
965:
962:
960:
957:
947:
946:
943:
941:
938:
935:
933:
931:
927:
926:
923:
921:
918:
916:
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911:
907:
906:
903:
901:
899:
896:
893:
891:
887:
886:
884:
882:
880:
878:
876:
873:
869:
868:
866:
864:
862:
860:
858:
855:
851:
850:
848:
846:
844:
842:
840:
837:
833:
832:
829:
826:
824:
821:
819:
817:
813:
812:
809:
806:
804:
802:
799:
797:
793:
792:
790:
788:
786:
784:
782:
779:
775:
774:
771:
768:
765:
762:
760:
758:
754:
753:
750:
747:
744:
741:
738:
735:
732:
729:
681:
680:State diagrams
678:
632:
629:
573:
570:
543:
540:
519:
516:
475:
472:
397:
396:
393:
390:
387:
383:
382:
376:
373:
370:
366:
365:
359:
356:
353:
349:
348:
342:
339:
336:
332:
331:
325:
322:
319:
315:
314:
308:
305:
302:
298:
297:
291:
288:
285:
281:
280:
274:
271:
268:
264:
263:
257:
254:
251:
247:
246:
240:
237:
234:
230:
229:
226:
223:
220:
216:
215:
212:
209:
206:
202:
201:
198:
195:
192:
188:
187:
184:
181:
178:
134:ladder diagram
116:
113:
84:
81:
39:, also called
15:
9:
6:
4:
3:
2:
4279:
4268:
4265:
4263:
4260:
4258:
4255:
4254:
4252:
4237:
4236:World records
4234:
4232:
4229:
4227:
4224:
4222:
4219:
4217:
4214:
4210:
4209:Ancient China
4207:
4206:
4205:
4202:
4200:
4197:
4196:
4194:
4190:
4184:
4181:
4177:
4174:
4173:
4172:
4169:
4165:
4162:
4161:
4160:
4157:
4155:
4152:
4150:
4147:
4145:
4142:
4140:
4137:
4133:
4130:
4129:
4128:
4125:
4123:
4120:
4118:
4115:
4113:
4110:
4108:
4105:
4103:
4100:
4098:
4095:
4093:
4090:
4088:
4087:Chinese yo-yo
4085:
4083:
4080:
4078:
4077:Alaskan yo-yo
4075:
4073:
4070:
4069:
4067:
4065:
4060:
4056:
4050:
4047:
4045:
4044:Renegade show
4042:
4038:
4035:
4034:
4033:
4030:
4028:
4025:
4021:
4018:
4017:
4016:
4013:
4009:
4006:
4004:
4001:
3999:
3996:
3994:
3991:
3990:
3989:
3986:
3985:
3983:
3981:
3974:
3968:
3965:
3963:
3960:
3958:
3955:
3953:
3950:
3948:
3945:
3943:
3940:
3938:
3935:
3933:
3932:Flag throwing
3930:
3928:
3925:
3923:
3920:
3918:
3915:
3911:
3908:
3907:
3906:
3903:
3899:
3896:
3895:
3894:
3891:
3889:
3886:
3882:
3879:
3877:
3874:
3872:
3869:
3868:
3867:
3864:
3863:
3861:
3859:
3855:
3850:
3840:
3837:
3835:
3832:
3830:
3827:
3825:
3822:
3818:
3814:
3811:
3810:
3809:
3806:
3804:
3801:
3799:
3796:
3794:
3791:
3789:
3786:
3784:
3781:
3779:
3776:
3774:
3771:
3769:
3766:
3764:
3761:
3759:
3756:
3754:
3751:
3750:
3748:
3746:
3741:
3737:
3733:
3729:
3722:
3717:
3715:
3710:
3708:
3703:
3702:
3699:
3684:
3680:
3673:
3668:
3665:
3661:
3658:
3655:
3651:
3648:
3645:
3642:
3639:
3636:
3633:
3630:
3626:
3623:
3620:
3619:PassingDB.com
3616:
3612:
3611:
3594:
3592:0-387-95513-5
3588:
3584:
3583:
3578:
3574:
3573:
3559:
3554:
3547:
3539:
3535:
3530:
3525:
3521:
3517:
3513:
3506:
3490:
3486:
3482:
3475:
3460:
3453:
3450:Boyce, Jack.
3446:
3431:
3424:
3417:
3415:
3407:
3403:
3402:BenBeever.com
3399:
3395:
3389:
3387:
3385:
3383:
3381:
3379:
3362:
3358:
3354:
3347:
3339:
3335:
3331:
3329:9781846286407
3325:
3321:
3317:
3312:
3311:
3305:
3299:
3292:
3288:
3284:
3281:
3275:
3273:
3257:
3256:
3251:
3244:
3236:
3232:
3228:
3222:
3218:
3217:
3209:
3198:
3194:
3190:
3186:
3182:
3178:
3174:
3170:
3166:
3162:
3158:
3157:
3149:
3142:
3140:
3138:
3122:
3121:
3116:
3109:
3107:
3105:
3098:
3094:
3090:
3084:
3082:
3080:
3064:
3063:
3058:
3051:
3047:
3031:
3028:
3025:
3022:
3019:
3016:
3015:
3012:
3008:
2998:
2995:
2994:
2985:
2984:Siteswap Game
2982:
2981:
2980:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2952:
2948:
2945:
2944:
2943:
2941:
2937:
2921:
2918:
2917:
2915:
2910:
2907:
2906:
2904:
2901:
2898:
2897:
2890:
2888:
2878:
2876:
2872:
2868:
2864:
2860:
2856:
2853:th number is
2852:
2848:
2844:
2840:
2836:
2832:
2828:
2824:
2820:
2816:
2812:
2808:
2804:
2800:
2796:
2792:
2788:
2770:
2760:
2748:
2744:
2729:
2727:
2711:
2691:
2671:
2662:
2658:
2649:
2620:
2616:
2607:
2597:
2592:
2589:
2586:
2582:
2567:
2563:
2554:
2544:
2539:
2536:
2533:
2529:
2514:
2510:
2501:
2491:
2488:
2485:
2480:
2477:
2474:
2470:
2455:
2451:
2442:
2432:
2429:
2426:
2421:
2418:
2415:
2411:
2404:
2399:
2394:
2390:
2355:
2350:
2346:
2331:
2327:
2318:
2308:
2305:
2302:
2297:
2293:
2278:
2274:
2265:
2255:
2252:
2249:
2244:
2240:
2233:
2228:
2223:
2219:
2196:
2193:
2190:
2186:
2182:
2179:
2176:
2171:
2167:
2161:
2157:
2134:
2131:
2128:
2124:
2120:
2117:
2114:
2109:
2105:
2099:
2095:
2069:
2066:
2063:
2059:
2055:
2050:
2047:
2044:
2040:
2033:
2030:
2027:
2019:
2015:
2011:
2006:
2002:
1990:
1986:
1982:
1977:
1973:
1955:
1951:
1935:
1932:
1929:
1925:
1921:
1918:
1915:
1909:
1906:
1901:
1897:
1887:
1884:
1879:
1876:
1873:
1869:
1862:
1859:
1856:
1851:
1847:
1841:
1837:
1831:
1827:
1806:
1784:
1781:
1778:
1774:
1770:
1767:
1764:
1759:
1756:
1753:
1749:
1743:
1739:
1735:
1732:
1729:
1724:
1720:
1714:
1710:
1704:
1700:
1687:Swap property
1684:
1678:
1674:
1671:
1670:
1669:
1667:
1662:
1648:
1645:
1642:
1639:
1636:
1614:
1606:
1603:
1598:
1590:
1587:
1582:
1574:
1554:
1551:
1548:
1545:
1542:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1492:
1472:
1469:
1466:
1463:
1460:
1440:
1437:
1434:
1431:
1428:
1406:
1398:
1395:
1390:
1382:
1379:
1374:
1366:
1346:
1343:
1340:
1337:
1334:
1314:
1311:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1284:
1275:
1261:
1241:
1221:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1170:
1159:
1156:
1151:
1147:
1137:
1134:
1114:
1106:
1090:
1082:
1066:
1044:
1041:
1038:
1034:
1030:
1027:
1024:
1019:
1015:
1009:
1005:
999:
995:
986:
981:
978:
970:
956:
954:
944:
942:
939:
936:
934:
932:
929:
928:
924:
922:
919:
917:
914:
912:
909:
908:
904:
902:
900:
897:
894:
892:
889:
888:
885:
883:
881:
879:
877:
874:
871:
870:
867:
865:
863:
861:
859:
856:
853:
852:
849:
847:
845:
843:
841:
838:
835:
834:
830:
827:
825:
822:
820:
818:
815:
814:
810:
807:
805:
803:
800:
798:
795:
794:
791:
789:
787:
785:
783:
780:
777:
776:
772:
769:
766:
763:
761:
759:
756:
755:
751:
748:
745:
742:
739:
736:
728:
727:
724:
720:
716:
713:
709:
704:
700:
696:
693:land with an
686:
677:
673:
672:
668:
665:
662:
661:
657:
655:
650:
649:
645:
643:
639:
628:
625:
603:
599:
595:
591:
586:
585:
581:
579:
569:
568:
564:
560:
548:
539:
537:
533:
524:
515:
513:
509:
505:
501:
497:
493:
489:
480:
471:
469:
465:
460:
458:
454:
450:
446:
441:
439:
434:
429:
428:
422:
420:
416:
412:
408:
404:
394:
391:
388:
385:
384:
381:
377:
374:
371:
368:
367:
364:
360:
357:
354:
351:
350:
347:
343:
340:
337:
334:
333:
330:
326:
323:
320:
317:
316:
313:
309:
306:
303:
300:
299:
296:
292:
289:
286:
283:
282:
279:
275:
272:
269:
266:
265:
262:
258:
255:
252:
249:
248:
245:
241:
238:
235:
232:
231:
227:
224:
221:
218:
217:
213:
210:
207:
204:
203:
199:
196:
193:
190:
189:
185:
182:
179:
176:
175:
169:
168:, and so on.
167:
163:
159:
155:
151:
147:
143:
137:
135:
131:
121:
112:
110:
109:swap property
106:
101:
98:
94:
90:
80:
78:
74:
69:
67:
62:
58:
54:
50:
46:
42:
38:
30:
24:
23:beat juggling
19:
4154:Pen spinning
4149:Keepie uppie
4132:Hoop rolling
4107:Devil sticks
4015:Competitions
3967:Trick roping
3905:Cup-and-ball
3812:
3686:. Retrieved
3678:
3635:Juggling Lab
3628:
3618:
3596:. Retrieved
3581:
3546:
3519:
3515:
3505:
3493:. Retrieved
3489:the original
3484:
3474:
3462:. Retrieved
3459:jonglage.net
3458:
3445:
3433:. Retrieved
3429:
3397:
3365:. Retrieved
3361:the original
3356:
3346:
3309:
3298:
3290:
3259:. Retrieved
3255:Juggling.org
3253:
3243:
3215:
3208:
3197:the original
3163:(5): 92–97.
3160:
3154:
3124:. Retrieved
3120:Juggling.org
3118:
3097:original url
3092:
3066:. Retrieved
3060:
3050:
3011:
2978:
2933:
2887:Grassmannian
2884:
2874:
2870:
2866:
2862:
2858:
2854:
2850:
2846:
2842:
2838:
2834:
2830:
2826:
2822:
2818:
2814:
2810:
2806:
2802:
2798:
2794:
2790:
2740:
2663:
2659:
2655:
1961:
1952:
1690:
1682:
1665:
1663:
1276:
1103:(written in
984:
982:
979:
975:
950:
721:
717:
711:
707:
702:
698:
694:
691:
674:
670:
669:
666:
663:
659:
658:
651:
647:
646:
641:
637:
634:
631:Multi-handed
626:
601:
597:
593:
589:
587:
583:
582:
577:
575:
566:
562:
558:
556:
535:
529:
518:Multiplexing
511:
504:(4,2x)(2x,4)
503:
499:
491:
485:
467:
463:
461:
456:
448:
444:
442:
438:asymmetrical
437:
432:
425:
423:
418:
414:
410:
406:
402:
400:
186:Description
161:
153:
141:
138:
133:
129:
126:
104:
102:
86:
70:
56:
44:
40:
36:
35:
18:
4092:Chinese top
3922:Fingerboard
3898:Indian club
3876:Bouncy ball
3798:Mills' Mess
3683:Brady Haran
2951:open source
1666:synchronous
1083:of the set
1081:cardinality
498:is denoted
488:parentheses
474:Synchronous
470:which can.
433:symmetrical
200:Empty hand
177:Throw Name
4251:Categories
4176:Rattleback
4117:Fire staff
3881:Hacky sack
3688:October 4,
3598:August 23,
3430:qedcat.com
3318:. p.
3291:Juggle.org
3261:October 3,
3226:0883855488
3093:Juggle.org
3042:References
2861:) −
2789:functions
3917:Fan dance
3888:Cigar box
3803:Multiplex
3793:Jollyball
3558:1111.3660
3538:0012-365X
3495:April 27,
3485:sonic.net
3481:"jdeep.c"
3464:April 27,
3435:April 22,
3357:sonic.net
3338:261225487
3185:0036-8733
2787:bijective
2764:~
2590:−
2486:−
2478:−
2362:otherwise
2303:−
2194:−
2132:−
2067:−
2048:−
1933:−
1907:−
1782:−
1196:−
1190:≤
1184:≤
1042:−
660:4-handed
532:multiplex
103:The name
4262:Notation
4216:Jugglers
4112:Fire fan
4072:Astrojax
4064:twirling
3871:Bean bag
3813:Siteswap
3808:Notation
3788:Joggling
3778:Fountain
3740:Patterns
3728:Juggling
3579:(2003).
3396:", p.6,
3316:Springer
3306:(2007).
3283:Archived
3235:56020214
3193:24982089
3126:June 30,
3068:June 30,
2991:See also
2967:JoePass!
2940:simulate
2930:Programs
2841:for all
2749:of type
2604:if
2551:if
2498:if
2439:if
2315:if
2262:if
964:Validity
600:becomes
592:becomes
506:(3-ball
363:fountain
329:fountain
295:fountain
261:fountain
166:fountain
146:sequence
105:siteswap
37:Siteswap
4204:History
4199:Culture
4127:Hooping
4059:Balance
3910:Kendama
3824:Passing
3768:Columns
3758:Cascade
3679:YouTube
3675:(video)
3662:at the
3404:at the
3367:July 8,
3165:Bibcode
2973:Gunswap
2845:, then
2211:, where
985:vanilla
654:diabolo
619:⁄
609:⁄
602:4.5 3 3
542:Passing
512:(4,2x)*
500:(4x,2x)
453:integer
380:cascade
346:cascade
312:cascade
278:cascade
244:cascade
158:cascade
115:Vanilla
73:cascade
43:or the
4164:tricks
4144:Meteor
4139:Kemari
4037:JISCON
3980:events
3976:Groups
3942:Jianzi
3834:Shower
3783:Havana
3589:
3536:
3336:
3326:
3233:
3223:
3191:
3183:
1059:where
945:10011
930:11001
925:01011
910:10101
905:00111
890:10011
872:01101
854:01011
836:00111
831:10101
811:01101
773:11001
764:
640:, and
496:shower
427:period
83:Origin
77:shower
4231:Terms
4226:Robot
4192:Other
4183:Yo-yo
3993:BACAF
3962:Torch
3947:Knife
3858:Props
3773:Flash
3745:forms
3553:arXiv
3455:(PDF)
3426:(PDF)
3200:(PDF)
3189:JSTOR
3151:(PDF)
3003:Notes
2961:Jongl
2949:- An
2745:(the
1127:where
816:1101
796:1011
778:0111
733:State
730:Throw
703:state
468:45141
464:51414
66:beats
4062:and
3978:and
3957:Ring
3893:Club
3866:Ball
3839:Toss
3817:list
3763:Claw
3743:and
3730:and
3690:2017
3600:2012
3587:ISBN
3534:ISSN
3497:2020
3466:2020
3437:2020
3369:2020
3334:OCLC
3324:ISBN
3263:2023
3231:OCLC
3221:ISBN
3181:ISSN
3128:2017
3070:2017
2955:Java
2837:) ≥
2817:) +
2809:) =
2726:here
2704:and
2149:and
940:1011
937:1101
920:0111
915:1101
898:0111
895:1011
875:1101
857:1011
839:0111
828:1011
808:0111
770:1101
757:111
596:and
594:4p 3
395:...
392:...
389:...
386:...
375:Yes
341:Yes
307:Yes
273:Yes
239:Yes
211:Yes
160:; a
4171:Top
4159:Poi
4049:WJF
4032:JIS
4027:IJA
4008:IJC
4003:EJC
3998:BJC
3937:Hat
3753:Box
3617:",
3524:doi
3520:157
3289:",
3173:doi
3161:273
3091:",
2877:).
2797::
2382:and
1950:.
1629:or
1611:mod
1595:mod
1579:mod
1535:or
1421:or
1403:mod
1387:mod
1371:mod
1327:or
1167:mod
823:111
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