The Beauty and Mathematics of Juggling | Alexander Leymann | TEDxDresden

Hello everyone, thank you, yes I am a theoretical physicist and a juggler and I am a big fan of Paul Dirac, he is one of the founding fathers of quantum mechanics and he once said that God used beautiful

mathematics

to create the universe, today we will do something . a little less epic, so we'll use simple math to create beautiful juggles. So how do we create our mathematical theory? our mathematical theory for

juggling

first, let's look at this pattern. This is a three-ball cascade, one of the first patterns every new juggler learns. Maybe we can understand the pattern a little better when we look at it from a different perspective, so imagine there is a camera on the ceiling and it would film me as I juggle and walk in this direction, then the green ball would leave a trail that looks like a little to this so let's see how it stacks up so I start with the green ball in my hand and then I walk in this direction and throw from left to right to left to right to left and to the right and the other two balls would leave a trail like this, and you can hear that

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has a very different rhythm every time I catch a ball, there is a Boop rhythm and now we will count these times to do our mathematical theory, so let's focus on the green ball and we will realize that always lands on the third beat three one two three one two three one two three so now we will call the shot with which I threw the green ball a three shot and when when I have thrown a three shot, I have time to throw two more balls , that's why I'm doing the three-ball cascade with three tosses, so now, like good mathematicians, we'll generalize this concept to the concept of n tosses, so an N throw lands on the In other words, after you've made N throws, I can throw n minus 1 extra balls and when I have n jumps and only do n throws, this is the basic n ball juggling pattern.

Great, yeah, now that no one is confused, I can show it. some examples, so here comes the basic pattern of juggling with a ball, these are a throw and you see how low I throw the ball and I have 1 -1 time to throw another ball, yes I can feel your amazement, okay, like this which we continue quickly. to two balls, so I juggle two balls. I have a ball in my right hand and another in my left and they throw them very little and since it is a basic pattern I throw them as little as I can, so I don't throw them at all.

This is your basic two-ball juggling pattern, almost as amazing as one-ball juggling, and we've already seen three-ball juggling, so I move on to four-ball juggling. It's actually the first real trick I'm doing, so it's two balls. in my right hand and two in my left hand, then I combine them and you can see how the green balls always stick to my right hand and the red balls always stick to my left hand. Well, now comes the five-ball juggling and I asked the organizers of this conference, if it's okay, if I brag a little and they said yes, of course, this is what TEDx is about, so for your entertainment, before juggling with the juggling black belt, I will balance a ball on my foot and then I will balance one. ball on my head, then I'll juggle three balls, I'll let this ball fall into the pattern, then I'll kick this ball from down here and finally I'll juggle five balls and if this works, you'll go crazy for a second, okay , so I'm just increasing the tension.

Ah, don't worry, don't worry, every good juggler has three tries, so here comes my first try again, the tension and on the first try I nailed it on the first try, so let's look at the pattern one more time and focus in the green. ball and you can see how similar this is to juggling one ball and three balls, the ball is crossed and while the green ball is in the air I have time to throw the other four balls, yeah right, otherwise I couldn't juggle five balls, so We can summarize and even and throw lands in the same hand as to juggle two balls or four ball throws or let's say six throws and an odd throw crosses and the higher the number, the higher it will be the throw, so now that we have defined n throws, we can do something cool now we can combine different n throws with a new pattern now we will show an example, so here is a nice four four one goes like this: I throw a four here, then a four here and then I deliver a ball with a one then it goes for four one once again four for one and it continually looks like this and the three balls here are doing the same thing so let's look at the green ball it's doing it one four four one four four one then maybe look at another example five three one five three one I throw a five here then I go down a little three here and then I deliver a one so it looks like this five three one once again five three one and you can see how the green The ball now is the chest doing three and the other two balls are doing five and ones, so we could say that we can represent the juggling trick by a sequence of n tosses or by a sequence of numbers and we call this lateral exchange, we will solve it in a minute while we call it lateral swapping, but there are some rules on how we should interpret such a string, so when I say 531 what I actually mean is five three one five three one five three one, so I might as well have said three one five or one five three, but it is convenient to start with one of the highest throws and then to keep our notation for this top point very simple, we have the rule that we catch and throw only alternately, so we say we exclude synchronous throws , so there are no tricks like this. and we have another rule, we say that each of our hands catches and throws only one ball at a time, so there are no throws like this or tricks like this or let's say like this because here we see him at one point he was throwing two balls. at the time and then I said that a side swap represents a juggling trick, it actually represents a whole class of juggling tricks because this notation ignores any specific body movement or hand position, so the basic three ball cascade is 3 3 3, but also this is 3 3 3 or 4 the skinny jugglers this is 3 3 3 and this is 3 3 3 let's say this and this even this is 3 3 3 thank you because I did not change the rhythm with which I catch and throw the balls so these site wolves have a cool property if you want to know how many balls you need, yeah, we'll see how clear this is.

If you want to know how many balls you need to juggle in a site swap, you just take the average so that it adds up all the numbers of the N toss and then divide it by the length of the pattern and you will get the number of balls, so let's check this for an example , no one is five, three, one anymore, we add five plus three plus one, it gives us nine divided. times three makes three and yes five three one was a 5 volt trick a three ball trick sorry and in the last part of my talk I will now show you how we can prove that this central theorem of site swapping juggling is actually true and Along the way we will generate a lot of new sites with books for us to juggle because it happens that you can't juggle any random combination of numbers so if you want me to juggle your phone number this probably won't work but I'll show you . now a method to generate lateral exchanges that can be juggled in the way that I showed you and we will demonstrate it in a very interesting example: the lateral exchange seven one three one and seven one three one is a combination of the lateral exchange one three that is seen like this and this is one of the first side trades that I was doing was in the winter in 1997 and that was in the schoolyard with two snowballs and I was doing one three one three one three without knowing and the other part of this trick it's seven one she's a four ball trick and it looks like this and this is the trick that Krusty the Clown would normally do when juggling four balls so in combination seven one three one it looks like this and you can see how the ball green is just making a three and the two red balls are making seven and ones, okay we know this trick is working so this is a good starting point to come up with a new trick based on this, so what are we doing to achieve it? with the new trick we exchange the seven and the one on the side we exchange seven one three one what are we really doing?

We swap the times when we throw the red ball and the green ball but we want to make sure that the ball still lands in the same position and in the same most important time at the same time because when the balls fall at the same time as they did in 7 1 3 1 we know that the new trick is also working, so what do we have to do now? We threw the red ball a little earlier a little later so it has to spend a little less time in the air just to land in the same time before we have thrown it with the 7 now we have to throw it with the 6 so that the 7 red has to become a red 6 and the green ball, we launch the green ball now some time earlier, so when we want it to land in the same time as before, it has to spend a little more time in the air, so the green one has to become a green: Maybe you can see this in this little animation that they still land on the same beat.

Okay, so what does the new site exchange look like now? since I changed the red and green ball now that the red and green loop are intertwined then the green ball wanders around the pattern oh it's up here it's down here ok cool this worked and it's more important for our theorem central juggling site sharing. We don't change the number of balls we are juggling because we simply swap the times when we throw two balls and we don't change the average because we add one to a number and subtract one from the other number so now we do what mathematicians always do when They discover that something is working, we repeat the process, so we start with seven one three one we swap the 7 and the 1 and we get to 6 3 1, which is the same as 6 3 1 2 and now we do it again. we swap the 6 and the 3 and we get 4 5 1 2 which looks like this here the green ball just goes forward and the red balls make 5 1 2 most people would miss the 2 because I basically don't do anything for a while so try to emphasize it, okay and now we change again now we change the 5 for 1 and we get 4 2 4 2 which looks like this yesterday - yes this one is boring but you can still do fun things, there is a yo-yo with the orbit the factory , thank you and now we're almost done, we just change the 4 to 2 and go back to the whips for both and go back to the beginning, go back to the simple three ball cascade, it's 3. 3 3 and we know we can juggle three balls because it only consists of three throws, so now we have finally shown that we can also juggle all the other tricks with 3 balls and we have generated them along the way to juggle. for us, so now everyone knows the

mathematics

and I would like to leave you with another quote from Paul Dirac.

When you are receptive and humble, mathematics will take you by the hand, so grab some balls, start juggling and feel how mathematics literally. take your hands thank you very much