(As posted on rec.juggling, June 1999)

Many passing patterns are symmetric : 4 ct, Waltz, ultimate, triangles, ... . Feeds involving runarounds also enter this category.

Tired to show off in solo juggling, the passing partners engage in friendly cooperation to create beautiful multihand patterns in which improvisation, i.e. tricks and syncopations, is all the more interesting since there is more than one brain involved.

Anyway, in the basic form of a symmetric passing pattern, all partners do the same sequence of throws, either in phase or out of phase. Apparently, there does not exist a comprehensive description of these patterns, although digging the rec.juggling archive unearthed a somewhat related and definitely beautiful article by Tarim (March 94, A new class of passing patterns).

The purpose of this post is to describe all symmetric passing patterns in terms of equivalent solo patterns. As an application, I list at the end all 2 persons 7 objects 3 count patterns.

The analysis to follow makes heavy use of siteswaps. Causal diagrams are also invoked for geometric intuition. It is also rather lengthy ... I believe however that these theoretical considerations can be of practical use, therefore many examples are provided.

This case is well-known. All jugglers pass and self together the same throws, therefore when someone throws a pass, he is thrown a similar pass so that everything happens as if all jugglers had not passed at all and had instead thrown to themselfs.

These passing patterns reduce to independent identical solo siteswaps. For two people, the number of objects must be even.

Example : with two other waltzers, you can do a 4p 4p 1 PPS triangle pattern.

- Let us look first at the case of 2-persons Waltz PSS'.

J2 begins his PSS' sequence 1.5 beats after J1. When J1 passes and gets rid of a club, he is thrown back a P pass 1.5 beats later. Therefore everything happens as if J1 had thrown a P+1.5 self and J2 had thrown a P-1.5 self.

This is where causal diagrams provide intuition: translate J2's time axis backwards by 1.5 beats. As a result, selfs are unchanged, J1's passes are shortened, J2's passes are lenghtened, and the two jugglers are now passing in phase, swapping all passes then provides two independent solo patterns.

If n denotes the number of clubs of the valid solo siteswap P-1.5 S S' , the passing pattern PSS' must contain 2n+1 clubs, an odd number of clubs. Conversely, starting from any length 3 , n clubs siteswap abc , then a+1.5p bc will describe a valid 2-persons, 2n+1 clubs Waltz.

**Examples:**

222 --> 3.5p 2 2 : the slow 5 "ultimate" that Bruno and Hans brilliantly demonstrated in Edinburg.

333 --> 4.5p 3 3 : 7 Waltz which Tarim and Martin Frost denote by 966 considering it as a 4 hands siteswap (I find this description slightly akward and misleading as explained later).

- Similarly, for a two persons Pass Pass Self PP'S , P-1.5 P'-1.5 S must be a valid n objects solo pattern and the PP'S passing pattern will involve 2n + 2 objects, an even number. Sadly enough there is no 7 clubs symmetric PPS (for an asymmetric one, Martin Frost pointed out <4p 4p 3 / 3 3p 4p>).

**Examples:** 333 --> 4.5p 4.5p 3 , 423 --> 5.5p 3.5p 3 , two 8
clubs PPS patterns.

- More generally, for 2 persons, if a(1) ... a(L) denotes the sequence of throws, then b(1) ... b(L) must be a valid solo siteswap, where:

Conversely, given any n clubs solo pattern b(1)...b(L) , you may create a 2 persons 2n+k clubs passing pattern with k passes.

**Examples:**

33 --> 4p 3 : 7-shower

531333 --> 534p333 : 7-popcorn,

13141 --> 3.5p 3 3.5p 4 1 : why not?

- With more than 2 persons, a general description becomes slightly more complicated, though by no means impossible. Let L denote the length of the pattern and P denote the number of passers. I assume that the set of passers is connected through the passes (thus excluding the popular 4 count squares).

J0 starts first, J1 starts L/P beats later, ... , J(P-1) starts last, i.e. (P-1)L/P beats after J0. Let us denote the sequence of throws by a(1)pj(1) ... a(L)pj(L) : here a(i) denotes the "height" (siteswap value) of the ith throw and pj(i) means that when Juggler #k throws the ith throw, this throw will be a pass to Juggler #(k+j(i)) [mod P] . selfs are therefore these throws for which j(i) = 0 . This notation is essentially Ed Carstens' MHN notation. Let us now shift back by kL/P beats the time origin of Jk , for all k , as explained earlier. Then, all passers are juggling in phase and the ith throw of Juggler #k has become:

Now, swap all passes! I.e. have everyone throw selfs that are identical in height to the passes they are being thrown. This works because everyone is passing in phase. The passing pattern is then reduced to P independent solo patterns, in particular the pattern of the last juggler is:

By the average rule the number of objects of this last pattern is equal to (a(1) + ... a(L))/L - (j(1) + ... + j(L))/P

Conversely, starting from a solo siteswap with n objects, one may contruct a symmetric P-passers passing pattern involving Pn + (j(1) + ... + j(L)) objects, for any choice of the destination mapping j( ).

**Examples:** (they all involve triangles, P = 3)

- { j(1) = 1 , j(2) = j(3) = 0 , 333 } --> 4p1 3 3 : a 10 clubs triangle Waltz with passes always to the "next" partner.

- { j(1) = 2 , j(2) = j(3) = 0 , 333 or 144 } --> 5p2 3 3 or 3p2 4 4 : two 11 clubs triangle Waltzes with passes always to the "preceding" partner.

- Alternating passing partners in a triangle Waltz will require L = 6 , j(1) = 1 , j(2) = j(3) = 0 , j(4) = 2 , j(5) = j(6) = 0 , so that the number of objects involved will be Pn + (j(1) + ... + j(L)) = 3n + 3 which is impossible for 10 or 11 clubs! Ok, with 12 : 344133 --> 5p1 4 4 5p2 3 3 . Kind of ugly but who wants to juggle this anyway? (Of course passing in phase 4p44 is possible)

- { j(1) = 1 , 3 } --> 3.33p1 : 10 ultimate with passes to the next.

- { j(1) = 2 , 3 } --> 3.66p2 : 11 ultimate with passes the other way round.

- What about ultimate with alternating partners ? As above we need a multiple of 3 objects and there is nothing interesting apart from in phase patterns.

* Symmetric patterns can be used to create asymmetric patterns by shifting the time origins of one or more jugglers as explained above. Examples: In phase 7 ultimate < 4p / 3p > or Tarim's gallopped 7 shower with crossing passes < 4.5p 3 / 3.5p 3 (start 0.5 beats after J1) >

* The notation system used above, i.e. siteswap or mhn, does not say which passes cross. For that matter, causal diagrams do not say it either unless you have decided which hand each juggler uses first. So, do it.

* The reasons why I do not like very much Tarim's notation, e.g. 966 for the 7 clubs Waltz, are first that numbers in this system do not immediately reflect heights of throws and which throws are passes, and second that the sequence of numbers does not actually always denote what the jugglers have to do : as an example 4.5p 1 5 will be denoted in Tarim's system by 9 10 2 , so that dividing all numbers by 2 will yield 4.5 5 1 which is not the desired juggling sequence (and is also impossible since 351 is not a valid siteswap). Also I am more familiar with siteswaps involving 3 or 4 objects then 7 or more :)

5.5p 5 0 | 5.5p 4 1 |
5.5p 2 3 | 5.5p 1 4 |
4.5p 6 0 | 4.5p 4 2 |
4.5p 3 3

4.5p 1 5 | 4.5p 0 6 |
3.5p 6 1 | 3.5p 5 2 |
3.5p 3 4 | 3.5p 2 5