Beatmap - A Complete Guide - Part One by Luke Burrage

Combat Rules by Luke Burrage

Endurance by Luke Burrage

Silly Juggling Games by Luke Burrage

Basic Team Combat Rules by Luke Burrage

Yesterday in the IJDb chatroom there was a discussion on combining siteswaps to make multiplex patterns. KlasA from Sweden had discovered that multiplex patterns could be broken down into layers.

So the multiplex siteswap [32] which is normally recognized as 5 ball splits is just a 3 ball cascade combined with a 2 ball fountain. The Gatto multiplex 26[67] is 267 mixed with a 006.

That is breaking them down but of course you can build new multiplex patterns with existing siteswaps. Mixing 501 with 441 makes [45]4[11] which looks pretty hard but probably jugglable. Maybe.

Siteswaps of different lengths can be combined too. 456 and 7531 will make a pattern 12 throws long (and totally impossible to juggle) [47][55][63][41][57][65][43][51][67][45][53][61]

A nice feature of this method is that you can combine the same siteswap, slightly out of phase. 501 can become [51]51. 423 can become [42][23][34].

You may be saying these siteswaps aren’t valid as two balls are being caught in the same hand . Well, I know that this can be done, not only because jugglemaster java can perform them but some of my own juggling creations feature balls caught at the same time in the same hand (lukes deal and lukes barrage). Even so, most people could probably manage [12][23][31].

Now this combination of the same siteswap can be done more than twice. Combining 534 three times can get you [534][345][453] that is, of course, [534]. This means you can take any siteswap, stick some brackets round it and it can be juggled. 12345 can become [12345][23451][34512][45123][51234] or simply [12345]. Looks great on jugglemaster, never to be juggled.

Ok, so that is one way you can combine two siteswaps or the same siteswap twice. Another way that I especially like is to juggle one siteswap in one hand and another in the other hand. To juggle a siteswap from only one site (hand) you simply double all the throw values.

So 234 would become 406080 if it were juggled in one hand. To juggle it in each hand would be 446688. This is a very possible juggling trick, though I prefer the synchronous version (4,4)(6,6)(8,8). The two ball shower 31 can be juggled in one hand as 6020. Might look a bit trivial when juggled, even when doubled up to (6,6)(2,2). But the whole point of this short experiment is to combine siteswaps into new patterns. A three ball cascade in one hand and a two ball shower in the other would result in the syncopated rhythmic pattern (6,6)(6,2), a popular pattern and a favorite of the Gandini jugglers.

My first two examples would combine into (4,6)(6,2)(8,6)(4,2)(6,6)(8,2). As a new pattern that is a nice enough looking pattern to juggle but the a great thing about combining siteswaps in this synchronous way is that even if you make every single ball crossing the patterns still stay separate. Try (4x,6x)(6x,2x)(8x,6x)(4x,2x)(6x,6x)(8x,2x) and you’ll see what I mean.

Just for completeness, lets have 12345 doubled up in this way and put out of phase. And why not, lets make it a crossing pattern. (2x,6x)(4x,8x)(6x,ax)(8x,2x)(ax,4x)

Now we have two different methods of combining siteswaps, be they the same or differing siteswaps. The next step is to combine these combining methods.

First I’ll take a single siteswap and juggle it four times in the same pattern. 423, as we know, can be juggled as the multiplex pattern [42][23][34]. Using the second technique we can juggle this siteswap in one hand by doubling the values. In one hand it would look like [84]0[46]0[86]0 or two hands synchronously as ([84],[84])([46],[46])([68],[68]). We can even put one hand out of synch with the other like in this pattern: ([84],[68])([46],[84])([68],[46])

As you see, we have the same three siteswap values repeated four times each. Even though we have mixed the pattern together, the separate patterns can still be seen as individual within the whole. This can be demonstrated by making two patterns crossing. ([84x],[68x])([46x],[84x])([68x],[46x])

Now those of you looking closely would have noticed that the same siteswap is repeated over in this kind of pattern. Without all the added synch and multiplex syntax we would be left with 846846864846. You can actually use any siteswap as a base to make up a pattern of this kind. 123 would become 246 and fit into the structure exactly the same way. ([24],[62])([46],[24])([62],[46])

456 could become ([8ax],[c8x])([acx],[8ax])([c8x],[acx]) if you so wanted.

Siteswaps of more or less than three values can be used too. A four long siteswap like 7531 fits in nicely like ([ea],[62]) ([ea],[62]) ([ea],[62]) ([ea],[62]). Or, reduced to it’s unrepeated form, ([ea],[62]). A five long siteswap would fit in slightly less comfortably. 97531 turned into a double combined semi-crossing synchronous multiplex would make ([ixe],[ax6])([2xi],[exa])([6x2],[ixe])([ax6],[2xi])([exa],[6x2]).

Looking at all these patterns makes my eyes go a bit funny though. They may look pretty and have some half-interesting maths behind them but are well beyond the possibility of anybody actually juggling them. That last one was using 20 balls! To find a jugglable pattern by this method requires us to go right back to the simplest low number siteswaps like 501. Doubling the values would get a02 and inserting that into the previous structure would produce ([a0],[2a])([02],[a0])([2a],[02]).That is actually just a very messy way of saying (a,[2a])(2,a)([2a],2) the same pattern but getting rid of all the dross.

This eight ball pattern, while being possible, is still a long way past most people’s abilities. How about making some four ball patterns from original 1 ball siteswaps?

012 is ultimately transformed into (2,4)([24],2)(4,[24]) by the method outlined, a nasty little brain teasing trick which I’m sure has been done before. There is no reason to use just one siteswap to make up these patterns but I just like the idea. I find it nice to make a four ball pattern from a single ball siteswap. If you wanted you could use one two ball siteswap and three one ball siteswaps to make up a new pattern of five balls.

But I’ll leave that up to you. If you have managed to read this far it shows you are far, far more interested in siteswaps than I am. So enjoy.

To conclude I’ll take a small step back yet look at some very basic impossible juggling patterns for people never to manage. At the beginning of this look into combining siteswaps we saw that you could take any length siteswap and make it into a multiplex pattern by putting it in brackets. Why not take the five long multiplex 12345, combine it with itself five times, double it with itself into two hands, have two of the resulting ten patterns crossing and see if we can juggle it?

([2x468xa],[4x68ax2])([4x68ax2],[6x8a2x4])([6x8a2x4],[8xa24x6])([8xa24x6],[ax246x8])([ax246x8],[2x468xa])

To get to the hard impossible patterns simply use 5 or 6 (prime numbers above 13) length siteswaps doubled and combined in one hand, ditto for the other hand, juggled with only half the throws crossing. Using building blocks like these we could make only slightly sub-infinity length siteswaps without a single ball going higher than a human being could throw comfortably.

Luke Burrage 21-12-02