# The mathematical soul of juggling

as you can see I'm quite serious about juggling have been serious about juggling for decades used to busk and do all sorts of things now what I want to show you today is how you can rip out mathematical so love

Juggling pattern and you to your own evil ends it's gonna be interesting no mathematical so love juggling pattern actually comes in the form of numbers so how do we get those numbers okay now when you juggle usually juggle to a beat

So the music might be our own you know and on every beat balls are being caught and thrown by the hands and the hands are taking turns at least was the basic juggling which we are going to stick to okay so i've tried to capture this

Graphically so here's a timeline and here we've got two beats and you see the hands like the left hand this one one sort of circular right hand is the other sort of circle okay now let me just juggle the basic pattern okay and count

Out the beats alright so here we go so it's one two three one two three one two three one two three so the yellow ball always goes up on the one right and I count to three so what does it mean for how many beats as the yellow balls stay

There three beats exactly and actually every single ball that I throw here stays three beats in the air okay and so we can actually put this in right so that's the red ball 1 3 3 3 3 3 and then the yellow ball and the green ball and

Really when you think about it when I juggle this pattern in the background you can hear that 3 3 3 3 3 3 3 okay and that's the mathematical sort of this very very basic pattern now if I were to do this with 4 balls

Looking for balls let's just have a look at the red ball a count out the beats there's 1 2 3 4 1 2 3 4 1 2 3 4 ok then you know this stays 4 beats in the air every ball stays four beats in the air

The mathematical soul of this pattern is just for yeah for you know you don't do the repeats okay that's for then the basic five ball pattern five the basic six for Peter and six we can also do too

So how do we do – well – is you know like this very impressive well usually you don't even bother kind of tossing them out – can't hold them like this even more impressive right then one just going back and forth like

This you can also do zeros so that's zero throws so in a 0 for there's nothing to both be in your hand because more important for later on you could also try it maybe – throws it's kind of throwing things back in time I'm still

Working on this and I don't know how you do fractions but anyway maybe someone can you suggest something okay now these are the basic patterns the basic pattern does correspond to the you know there's zero pattern there's a

1 pattern – pattern but it gets much more complicated so if you look at a really complicated pattern you can actually find numbers different numbers at the core so say some guys juggling something

Really complicated focus on what happens on one beat ok ok so what he throws there is a force a again ok for then next beat he throws another for next beat whole is throw the 1 and then a 3 and then things start repeating

For 4 1 3 4 4 1 3 4 4 1 3 you know that's a pattern and I just draw it for you right so now so the first speed was a 4 so we can draw in ball here then 4 then 1 then 3 and then things repeat for 4 1 3 and you

Know can it's kind of see at this stage that whole thing will kind of extend forever and that won't be any problems so I mean what problems can you expect anyway well what happens here on every beat is there's a ball kind of coming in

Being caught and being tossed immediately right so what do you think geez ever can go wrong here that's right so I mean so if you're not careful so if you try to juggle like 2 1 so cut it at 2

Here then on an XP to one the one goes there then they collide here and I mean there's juggling that takes care of this but in basic you know in the basic setup it's not possible okay so we don't want this to happen so that's one problem the

Second problem sometimes you would have like a zero in your pattern but something lands and then you actually hold something in your hand but you're supposed to do a zero oh just misses like this so that's also

Out okay you know as I said I mean there's you know extensions to all the stuff that we're doing but we do the basic thing so nothing should land on on a zero how many balls do you think are in here so how many balls does the

Juggler actually juggle so at the moment or so except for we give you a hint okay so now you have yeah so that's three trajectories right and so basically every trajectory stands for one ball so that's the red ball

The green ball and the proper ball okay and they'd you know they exhaust the pattern so it's just three three balls it'd been in play here it's the same ball yeah and that's actually a good thing it's

Kind of colored in like this because you can actually see how you're supposed to juggle the pattern so if we just look at the purple ball for example well what is that going to do well it's staying in one hand and it's just just doing force

So basically just across this here one over and over and then there's also the green ball it does the same thing on the other side and then in the middle we've got the red ball and that does one three one three okay so let's put it all

Together okay and when I put it together it looks like this four four one three four four one three four four one three four four one three four four one three and I'm so good okay now yeah that's just one of

The patterns that you can get there's like lots and lots so just show you a few of them right so these are all possible mathematical souls of juggling parents there's infinitely many and just kind of the left column here that's just

Juggling with two balls again so juggling with two balls and well there we've got the two which was this really fancy you know or just forget about it pattern then there was three one which is just our

What is three-one three-one three-one you can also do three one can't you put the other way and when you change directions then just continuously change directions you actually get this three one two okay that's three one two three

Three three is kind of interesting that's what you do this you just kind of juggle two three three three three three and three and now you get rid of one and then you know you just do the zero there that was three three zero three three

Zero good one and then what about that one four zero four zero what would that look like that's right yeah so four goes on up here then is zero then a fourth and a zero so it's just like this one which

Was there four zero four zeros it kind of interesting and I've just highlighted a couple of other patterns here actually there's one more thing that I can tell you about all this stuff which is interesting which kind of need if you

Want to figure out if you just somebody gives you like a sequence of numbers like this and says this is a mathematical sole you don't actually have to draw the picture to figure out what the number of balls is that you

Need the only thing I have to do is you have to do the average of the numbers and able to tell you so for example 4 4 1 3 if you add up the numbers we get 4 plus 4 is 8 plus 1 is 9 plus 3 is 12 divided by 4 numbers is 3 right or this

Guy here 4 4 plus 4 is 8 plus 1 is 9 divided by 3 is 3 Martin looks really really neat all right now I'm going to show you how you can generate all the mathematical sorts of all the juggling patterns in existence using maths well

Not all of them let's just focus on the ones that are really relevant so in a second video I'm actually going to teach you how to juggle three balls so if you're really keen watch the video and well then once you know how to juggle

Three balls you want to also figure out well what are the different juggling patterns that I can do with three balls and reasonable throats say throws to hide 0 1 2 3 of course and then 4 & 5 let's throw those two in to okay so the

Only thing you need to know is the simple diagram that's all this simple diagram here okay well if we just explain to you how how does it's made up and how you get the juggling patterns out of this thing okay so what

You see here is some oblong boxes and they're filled with all possible combinations of three ones and two zeroes obviously the three ones have something to do with the fact that we want to juggle three balls here okay

Then we've got arrows between those oblong boxes and they're labeled now the labels go from 0 to 5 so that obviously is something to do with the sort of flows we want to use and now how do you get a juggling sole out of this one here

Well it's very easy you just go to one of the oblong boxes and then you can follow the arrows and as soon as you get back to the box that you started with you've got a juggling so ok let's just do this

Let's start at this box here go over here so in this direction this double arrow counts as a four then we go up there and then we go back to where we started from and we get a 4 4 1 which is actually

Just a u-shape pattern but we don't have to stop here we could actually do this mini loop there 2 3 and then it's 4 4 1 3 and you start and end at the same place you can actually get go across the same place a couple of times doesn't

Matter and you've got your juggling pattern 4 4 1 3 2 1 that we've been talking about and we're beginning oh that's the error kind of going the other way so if you're going this way it's counts

As a tool again just to make it a bit more compact okay now actually this one here is kind of interesting you start from here you cannot just do this mini loop that gives you a 3 that's just the basic pattern 3 3 3 3 3 yard right now

Everything possible is in this diagram right so here's another one for example just going around here this actually is really really boring one so everything's in there but it's really the majority of these things are terribly boring right

So we don't even want to want to touch right but point is you have this diagram here and you just give it to the computer the computer can pick out like all the cycles no problem list them and you can kind of go through them and see

Which one you actually want to incorporate in your traveling routine now how do you actually get this diagram I also wanted to tell you about that okay now for that one I really really need you to be alert of

Brought some special juggling balls they actually look like bottled mathematical Souls themselves so here we go so just to wake you up I'm going to juggle these things and actually very very special hopefully everybody's awake

Now okay so here's how you actually get this thing here going to be juggling three balls okay so imagine you juggling three balls and what I've drawn here is the dots at which the beats occur so beat dots yeah now we're in between

Beats and it means that all the juggling balls are in the air okay and now there's three balls in the air and the red one is actually going to land on the next beat the yellow one is going to land on the beat after that then two

Beats nothing's happening and then five beats from now at the green balls supposed to have a land okay and I record this in the string of ones and zeroes okay so I say lands lands nothing nothing lands so it's 1 1 0 0 1

So you're in the state in the jaggery state 1 1 0 0 1 at the moment right and you can be obviously in many many different juggling States and this is what oblong box is in the diagram corresponds ok now let's fast-forward to

The next beat okay what happens then is of course that the red ball falls into your hand okay so the red balls here and everything kind of shifts forward by 1 so we're kind of going like this so we've got the red ball in our hand and

On the next beat the yellow ball is going to land and – nothing's then a land and then nothing planned for for a moment ok now we have to figure out what can we do with our red paw you know what throw can we do can we throw a 0 so I

Can't throw a 0 because that's supposed to look like this okay what about a one that's right so they're dead at that one day you see the 1 so if I throw a 1 that those two would come along together the red one and a yellow one so I can't do

That so can I do it too yep because there's nothing scheduled for 2:00 and then obviously also not for three what about for ya and then I could also go for five these already all the choices so two

Three and five work everything else doesn't work doesn't everything else we'll get us into trouble okay let's say we're going for a five okay what that means is that this zero here becomes a 1 and we're in a different juggling state

That one here right and we've basic come from this charming state to just are in state by juggling a 5 ok and now we can actually put the whole diagram together it's quite easy so what you do is you that was our first state here and these

Are the three different states to which we get by juggling a 5 a 3 & a 2 right and so we do the same sort of analysis for every single state and that gives us this this diagram here okay and that it's got the complete picture as far as

Three ball juggling up to height five is concerned now you know at some point in time you may want to add height six but then you need a different diagram and you know this gets a bit more complicated it's a big diagram it's not

A big deal for a computer okay alright so what's the point of all this well just in terms of recording stuff it's really good to kind of take out the soul and I just say juggling this thing which is basically a four four one three

And I'm wearing blindfold and I'm using chain saw and I throw them behind the back so it's pretty easy for somebody else to kind of reconstruct what I'm actually doing right another good reason would be to get some sequences or some

Souls that allow it to learn in stages so for example you want to practice the black belt five throw so you don't want to immediately do five five five five five is five balls so what you do is you first look for one that only involves

Two balls but has a five in it like this five zero one actually practice that one and then you maybe get to a three ball pattern with a five in it and so on right so that's good there's a couple of other reasons actually I've written

Few articles about dissolve stuff which I'll link in in in the blurb here so check it out is also linked to a book which I've written about this so I was pretty serious about this at some stage now what's the point of all this well

Point of juggling is to have fun and thought the point of doing this is austere fun so once you've got this analysis going you can actually now you just hit it hit a computer with this and let it show you all sorts of amazing

Things so for example here I've got a computer juggling 35 balls in the basic pattern okay and it would look exactly like this if anybody could actually juggle is now here we've got one 18 in a circle okay here is going 1 2 3 4 5 6 7

8 9 10 and actually for compactness we always use like a 4 10 B 4 11 and so on so you can see you know the balls getting higher and higher up and so now what else have we got 2 years this is a really nice pattern it's kind of a

U-shape what and actually I mean just talking about having fun obviously you know once you're talking about letters you can ask well it's my name jag level well sadly not and in fact it's actually quite unlikely that if you just take an

Ordinary word and it's jumbled but some words are talking to for example my wife's name is Jabbar her name is Anna and I can I've tried to juggle her but I've failed as you can see it's quite hard to juggle I know and

Just in general say how to juggle a wife you'll find out ok and I'm making a lot of fancier than these animations here so this is actually beyond the basic juggling so that's actually catching simultaneously five and catching

Simultaneously 500 throw this one here's kind of simultaneous throws so the hands are no longer taking turns there's also simultaneous ones and obviously nobody can really juggle this but you know now it gets a lot fancier

If you actually you know you can put unicycles in the blindfolds and the clubs and all these sorts of things and so you can actually check out quite a few juggling animators on the weapon I also have some links down there

Definitely worth doing here's another one's called the torture chamber it's a very nice one okay and well to finish people always ask me how many how many balls cannot juggle and you've already seen me juggle five what comes after

Five six so that's three in one hand three in the other hand that's I can do that sever I can do that well actually just to impress everybody here I'm going to go nine

All right ready for a night okay here we go so the first really tricky business about nine is you have to sit them in your hand okay so there's three here there's another three you know if you have to hold them like this now there's

Six pause and a hold room like this and I've got another three here okay are you ready yeah ready to be impressed go for it here we go okay very good and as always some people would need some extra if I

Cheat well I actually can't juggle either but it's that's it for today now what we just going to call here call oh I know God hello hello