Date: Thu, 5 Sep 1991 17:38:29 PDT Reply-To: JuggleN - Juggling Notation & Computer List Sender: JuggleN - Juggling Notation & Computer List From: "Jack K. Boyce" Subject: Permutable tricks Message-ID: <910906.013829@indycms.bitnet> Hello Ed and anybody else interested, There is an interesting class of site-swap patterns that you might be interested in (Boppo alerted me to this). Consider a site-swap pattern such that every throw height difference within the pattern is a multiple of the trick's length. For example: 3: 441 (differences of 3, 0, and -3, all multiples of trick length) 4: 741, 714, 444 (trivially) 5: 744, 663, 8444, 771, 777171, 77722, 66661 These tricks all have the interesting property that you can permute any of their throws, yielding another valid trick. (You can also show that this class contains ALL site-swaps whose throws can be arbitrarily permuted.) As an example, the 4 ball trick 741 can be permuted to 714, which is also valid (any other permutation is just a rotation of one of these). A 5 ball example is the trio 9551, 9515, 9155, each of which is a different(and valid) excited state pattern. Or you can permute the 3-high flash, 77722, to get the excited state 72727, where each hand does 3 consecutive throws (similar to 552 for 4 balls). Anyway, there are several things that I've proven about the tricks in this class, some obvious and some not. One thing is that a ground state trick can have no more than 2 different kinds of throws, and no two throws can differ by more than the trick's length. Also, permuting the throws in a ground state trick gives either: 1) A rotation of the original trick, or 2) An excited state trick. (This is actually a special case of a more general theorem.) You can also prove that every ground state trick in this class is a simple loop, except for the cascade/fountain. (A trick is a simple loop if is has no repeatable subunits. For example, in the 3 ball ground state trick 45141 you can repeat the 51 as many times as you like (4515141 is also valid), so this trick is not a simple loop.) This class of tricks is neat from a theoretical point of view because you can actually derive some results for it. They are also fun to do, though; try 741 and 714 (4 balls) -- they look nearly identical, but feel very different. Anyway, if anyone is interested in the detailed results I can post them. Jack jboyce@tybalt.caltech.edu