The Beauty and Mathematics of Juggling | Alexander Leymann | TEDxDresden

thank you yeah I’m a theoretical

thank you yeah I’m a theoretical physicist and a juggler and I’m a huge

physicist and a juggler and I’m a huge

physicist and a juggler and I’m a huge fan of Paul Dirac he’s one of the

fan of Paul Dirac he’s one of the

fan of Paul Dirac he’s one of the founding fathers of quantum mechanics

founding fathers of quantum mechanics

founding fathers of quantum mechanics and he once said God used beautiful

and he once said God used beautiful

and he once said God used beautiful mathematics in creating the universe

mathematics in creating the universe

mathematics in creating the universe today we will do something a little bit

today we will do something a little bit

today we will do something a little bit less epic so we will use simple

less epic so we will use simple

less epic so we will use simple mathematics to create beautiful juggling

mathematics to create beautiful juggling

mathematics to create beautiful juggling so how do we create our theory for

so how do we create our theory for

so how do we create our theory for mathematics our mathematical theory for

mathematics our mathematical theory for

mathematics our mathematical theory for juggling first let’s look at this

juggling first let’s look at this

juggling first let’s look at this pattern this is a three ball cascade one

pattern this is a three ball cascade one

pattern this is a three ball cascade one of the first patterns every new juggler

of the first patterns every new juggler

of the first patterns every new juggler learns maybe we can understand the

learns maybe we can understand the

learns maybe we can understand the pattern a little bit better when we look

pattern a little bit better when we look

pattern a little bit better when we look at it from a different perspective so

at it from a different perspective so

at it from a different perspective so imagine there was a camera at the

imagine there was a camera at the

imagine there was a camera at the ceiling and it would film me while I’m

ceiling and it would film me while I’m

ceiling and it would film me while I’m juggling and walking along this

juggling and walking along this

juggling and walking along this direction then the green ball would

direction then the green ball would

direction then the green ball would leave a trace that looks a little bit

leave a trace that looks a little bit

leave a trace that looks a little bit like this so let’s see how this builds

like this so let’s see how this builds

like this so let’s see how this builds up so I start with the green ball in my

up so I start with the green ball in my

up so I start with the green ball in my hand and then I walk this direction and

hand and then I walk this direction and

hand and then I walk this direction and I throw from the left to the right to

I throw from the left to the right to

I throw from the left to the right to the left to the right to the left and to

the left to the right to the left and to

the left to the right to the left and to the right and the two other balls would

the right and the two other balls would

the right and the two other balls would leave a trace like this so and you can

leave a trace like this so and you can

leave a trace like this so and you can hear juggling has a very distinct rhythm

hear juggling has a very distinct rhythm

hear juggling has a very distinct rhythm whenever I catch a ball there’s a beat

Boop and we will now count these beats

Boop and we will now count these beats to make our mathematical theory so let’s

to make our mathematical theory so let’s

to make our mathematical theory so let’s focus on the green ball and we will

focus on the green ball and we will

focus on the green ball and we will realize it always lands on the third

realize it always lands on the third

realize it always lands on the third beat three one two three one two three

beat three one two three one two three

beat three one two three one two three one two three so now we will call the

one two three so now we will call the

one two three so now we will call the throw with which I’ve thrown the green

throw with which I’ve thrown the green

throw with which I’ve thrown the green green ball a three throw and when the

green ball a three throw and when the

green ball a three throw and when the when I’ve thrown a three throw I have

when I’ve thrown a three throw I have

when I’ve thrown a three throw I have time to throw two more balls that’s why

time to throw two more balls that’s why

time to throw two more balls that’s why I’m doing the three ball cascade with

I’m doing the three ball cascade with

I’m doing the three ball cascade with three throws so now as good

three throws so now as good

three throws so now as good mathematicians we will now generalize

mathematicians we will now generalize

mathematicians we will now generalize this concept to the concept of n throws

this concept to the concept of n throws

this concept to the concept of n throws so an N throw lands on the nth beat in

so an N throw lands on the nth beat in

so an N throw lands on the nth beat in other words after I have thrown an N

other words after I have thrown an N

other words after I have thrown an N throw I can throw n minus 1 additional

throw I can throw n minus 1 additional

throw I can throw n minus 1 additional balls and when I have n vaults and only

balls and when I have n vaults and only

balls and when I have n vaults and only do n throws then this is the basic n

do n throws then this is the basic n

do n throws then this is the basic n ball juggling pattern great yeah now

ball juggling pattern great yeah now

ball juggling pattern great yeah now that nobody is confused I can show some

that nobody is confused I can show some

that nobody is confused I can show some examples so here comes the basic one

examples so here comes the basic one

examples so here comes the basic one ball juggling pattern these are one

ball juggling pattern these are one

ball juggling pattern these are one throws and you see how low I throw the

throws and you see how low I throw the

throws and you see how low I throw the ball and I have 1 -1 beats time to throw

ball and I have 1 -1 beats time to throw

ball and I have 1 -1 beats time to throw another ball yeah I can feel your

another ball yeah I can feel your

another ball yeah I can feel your amazement okay so we so we quickly go on

amazement okay so we so we quickly go on

amazement okay so we so we quickly go on to two balls so two ball juggling I have

to two balls so two ball juggling I have

to two balls so two ball juggling I have one ball in my right hand one of my left

one ball in my right hand one of my left

one ball in my right hand one of my left and they throw them very little and

and they throw them very little and

and they throw them very little and since it’s a basic pattern I throw them

since it’s a basic pattern I throw them

since it’s a basic pattern I throw them as little as I can this is I don’t throw

as little as I can this is I don’t throw

as little as I can this is I don’t throw them at all so this is your basic two

them at all so this is your basic two

them at all so this is your basic two ball juggling pattern almost as is

ball juggling pattern almost as is

ball juggling pattern almost as is amazing as one ball juggling so and

amazing as one ball juggling so and

amazing as one ball juggling so and we’ve already seen three ball juggling

we’ve already seen three ball juggling

we’ve already seen three ball juggling so I move on to four ball juggling

so I move on to four ball juggling

so I move on to four ball juggling it’s actually the first real trick I’m

it’s actually the first real trick I’m

it’s actually the first real trick I’m doing so this is two balls in my right

doing so this is two balls in my right

doing so this is two balls in my right hand and two in my left hand then I

hand and two in my left hand then I

hand and two in my left hand then I combine it and you can see how the green

combine it and you can see how the green

combine it and you can see how the green balls always stick to my right hand and

balls always stick to my right hand and

balls always stick to my right hand and the red balls always stick to my left

the red balls always stick to my left

the red balls always stick to my left hand okay so now comes five ball

hand okay so now comes five ball

hand okay so now comes five ball juggling and I asked the organizers of

juggling and I asked the organizers of

juggling and I asked the organizers of this conference if it’s okay if I show

this conference if it’s okay if I show

this conference if it’s okay if I show off a little bit and they said yeah of

off a little bit and they said yeah of

off a little bit and they said yeah of course this is what TEDx is about so for

course this is what TEDx is about so for

course this is what TEDx is about so for your entertainment before I will juggle

your entertainment before I will juggle

your entertainment before I will juggle the black belt of juggling I will

the black belt of juggling I will

the black belt of juggling I will balance one ball on my foot

balance one ball on my foot

balance one ball on my foot then I will balance one ball on my head

then I will balance one ball on my head

then I will balance one ball on my head then I will juggle three balls let this

then I will juggle three balls let this

then I will juggle three balls let this ball fall into the pattern then kick up

ball fall into the pattern then kick up

ball fall into the pattern then kick up this ball from down here and finally

this ball from down here and finally

this ball from down here and finally juggle five balls and if this works you

juggle five balls and if this works you

juggle five balls and if this works you go crazy for a second

go crazy for a second

go crazy for a second okay so I’m just building up the tension

okay so I’m just building up the tension

okay so I’m just building up the tension ah don’t worry don’t worry

ah don’t worry don’t worry

ah don’t worry don’t worry every good juggler has three tries so

every good juggler has three tries so

every good juggler has three tries so here comes my first try again the

here comes my first try again the

here comes my first try again the tension

and on the first try I made it on the

and on the first try I made it on the first try so let’s look at the pattern

first try so let’s look at the pattern

first try so let’s look at the pattern once more and let’s focus on the green

once more and let’s focus on the green

once more and let’s focus on the green ball and you can see how similar this is

ball and you can see how similar this is

ball and you can see how similar this is to one ball and three ball juggling the

to one ball and three ball juggling the

to one ball and three ball juggling the ball is crossed and while the green ball

ball is crossed and while the green ball

ball is crossed and while the green ball is in the air I have time to throw the

is in the air I have time to throw the

is in the air I have time to throw the four other balls yeah sure otherwise I

four other balls yeah sure otherwise I

four other balls yeah sure otherwise I couldn’t juggle five balls so we can

couldn’t juggle five balls so we can

couldn’t juggle five balls so we can summarize and even and throw lands on

summarize and even and throw lands on

summarize and even and throw lands on the same hand like for two ball juggling

the same hand like for two ball juggling

the same hand like for two ball juggling or four ball throws or let’s say six

or four ball throws or let’s say six

or four ball throws or let’s say six throws and an odd an throw crosses and

throws and an odd an throw crosses and

throws and an odd an throw crosses and the higher the number the higher the

the higher the number the higher the

the higher the number the higher the throw so now that we’ve defined n throws

throw so now that we’ve defined n throws

throw so now that we’ve defined n throws we can do a cool thing now we can

we can do a cool thing now we can

we can do a cool thing now we can combine different n throws to a new

combine different n throws to a new

combine different n throws to a new pattern now we’ll show an example so

pattern now we’ll show an example so

pattern now we’ll show an example so here’s a nice one four four one goes

here’s a nice one four four one goes

here’s a nice one four four one goes like this I throw a four here then a

like this I throw a four here then a

like this I throw a four here then a four here and then I hand over a ball

four here and then I hand over a ball

four here and then I hand over a ball with a one so it goes for four one once

with a one so it goes for four one once

with a one so it goes for four one once more four for one and continuously looks

more four for one and continuously looks

more four for one and continuously looks like this and all the three balls here

like this and all the three balls here

like this and all the three balls here are doing the same thing so let’s look

are doing the same thing so let’s look

are doing the same thing so let’s look at the green ball it’s doing it one four

at the green ball it’s doing it one four

at the green ball it’s doing it one four four one four four one so maybe we look

four one four four one so maybe we look

four one four four one so maybe we look at another example five three one five

at another example five three one five

at another example five three one five three one I throw a five here then a

three one I throw a five here then a

three one I throw a five here then a little bit lower three here and then I

little bit lower three here and then I

little bit lower three here and then I hand over a one so it goes like this

hand over a one so it goes like this

hand over a one so it goes like this five three one once more five three one

five three one once more five three one

five three one once more five three one so and you can see how the green ball

so and you can see how the green ball

so and you can see how the green ball now is chest doing threes and the two

now is chest doing threes and the two

now is chest doing threes and the two other balls are doing the fives and ones

so we could say we can represent the

so we could say we can represent the juggling trick by a sequence of n throws

juggling trick by a sequence of n throws

juggling trick by a sequence of n throws or by sequence of numbers and we call

or by sequence of numbers and we call

or by sequence of numbers and we call this a side swap we will figure out in a

this a side swap we will figure out in a

this a side swap we will figure out in a minute while we call it a side swap but

minute while we call it a side swap but

minute while we call it a side swap but there are some rules how we have to

there are some rules how we have to

there are some rules how we have to interpret such a string so when I say

interpret such a string so when I say

interpret such a string so when I say 531 what I actually mean is five three

531 what I actually mean is five three

531 what I actually mean is five three one five three one five three one so I

one five three one five three one so I

one five three one five three one so I could have well said three one five or

could have well said three one five or

could have well said three one five or one five three but it’s just convenient

one five three but it’s just convenient

one five three but it’s just convenient to start with one of the higher throws

to start with one of the higher throws

to start with one of the higher throws and then to keep our notation for this

and then to keep our notation for this

and then to keep our notation for this top

top

top very simple we have the rule that we

very simple we have the rule that we

very simple we have the rule that we catch and throw only alternating so we

catch and throw only alternating so we

catch and throw only alternating so we say we exclude synchronous throws so no

say we exclude synchronous throws so no

say we exclude synchronous throws so no tricks like this and we have another

tricks like this and we have another

tricks like this and we have another rule we say each of our hands catches

rule we say each of our hands catches

rule we say each of our hands catches and throws only one ball at a time so no

and throws only one ball at a time so no

and throws only one ball at a time so no throws like this and no tricks like this

throws like this and no tricks like this

throws like this and no tricks like this or let’s say like this like this because

or let’s say like this like this because

or let’s say like this like this because here we see it at one point I was

here we see it at one point I was

here we see it at one point I was throwing two balls at the time and then

throwing two balls at the time and then

throwing two balls at the time and then I said a side swap stands for a juggling

I said a side swap stands for a juggling

I said a side swap stands for a juggling trick actually it stands for a whole

trick actually it stands for a whole

trick actually it stands for a whole class of juggling trick tricks because

class of juggling trick tricks because

class of juggling trick tricks because this notation ignores any specific body

this notation ignores any specific body

this notation ignores any specific body movement or hand position so the basic

movement or hand position so the basic

movement or hand position so the basic three ball cascade is 3 3 3 but also

three ball cascade is 3 3 3 but also

three ball cascade is 3 3 3 but also this is 3 3 3 or 4 the skinny jugglers

this is 3 3 3 or 4 the skinny jugglers

this is 3 3 3 or 4 the skinny jugglers this is 3 3 3 and this is 3 3 3 let’s

this is 3 3 3 and this is 3 3 3 let’s

this is 3 3 3 and this is 3 3 3 let’s say this and this even this is 3 3 3

say this and this even this is 3 3 3

say this and this even this is 3 3 3 thank you

because I did not change the rhythm at

because I did not change the rhythm at which I catch and throw the balls so in

which I catch and throw the balls so in

which I catch and throw the balls so in these sites wolves they have a very cool

these sites wolves they have a very cool

these sites wolves they have a very cool property if you want to know how many

property if you want to know how many

property if you want to know how many balls you need yeah I will we will see

balls you need yeah I will we will see

balls you need yeah I will we will see how clear this is if you want to know

how clear this is if you want to know

how clear this is if you want to know how many balls you need to juggle a

how many balls you need to juggle a

how many balls you need to juggle a siteswap you just take the average so

siteswap you just take the average so

siteswap you just take the average so you sum up all the numbers of the N

you sum up all the numbers of the N

you sum up all the numbers of the N throw and then divide it by the pattern

throw and then divide it by the pattern

throw and then divide it by the pattern length and you get the number of balls

length and you get the number of balls

length and you get the number of balls so let’s check this for one example no

so let’s check this for one example no

so let’s check this for one example no one already five-three-one we sum up

one already five-three-one we sum up

one already five-three-one we sum up five plus three plus one gives nine

five plus three plus one gives nine

five plus three plus one gives nine divided by three gives three and yeah

divided by three gives three and yeah

divided by three gives three and yeah five three one was a 5-volt trick a

five three one was a 5-volt trick a

five three one was a 5-volt trick a three ball trick sorry so and in the

three ball trick sorry so and in the

three ball trick sorry so and in the last part of my talk I will now show you

last part of my talk I will now show you

last part of my talk I will now show you how we can actually prove that this

how we can actually prove that this

how we can actually prove that this central theorem of siteswap juggling is

central theorem of siteswap juggling is

central theorem of siteswap juggling is actually true and on the way we will

actually true and on the way we will

actually true and on the way we will generate a lot of new sites books for us

generate a lot of new sites books for us

generate a lot of new sites books for us to juggle because it happens that you

to juggle because it happens that you

to juggle because it happens that you cannot juggle any any random combination

cannot juggle any any random combination

cannot juggle any any random combination of numbers so if you would want me to

of numbers so if you would want me to

of numbers so if you would want me to juggle your telephone number this will

juggle your telephone number this will

juggle your telephone number this will probably not work but I will show you

probably not work but I will show you

probably not work but I will show you now a method how we can generate side

now a method how we can generate side

now a method how we can generate side swaps that can be juggled in the way I

swaps that can be juggled in the way I

swaps that can be juggled in the way I showed you and we will demonstrate this

showed you and we will demonstrate this

showed you and we will demonstrate this on a very cool example it’s the side

on a very cool example it’s the side

on a very cool example it’s the side swap seven one three one and seven one

swap seven one three one and seven one

swap seven one three one and seven one three one is a combination of the side

three one is a combination of the side

three one is a combination of the side swap one three which looks like this and

swap one three which looks like this and

swap one three which looks like this and this is one of the first side swaps that

this is one of the first side swaps that

this is one of the first side swaps that I was doing it was in winter in 1997 and

I was doing it was in winter in 1997 and

I was doing it was in winter in 1997 and that was on the schoolyard with two

that was on the schoolyard with two

that was on the schoolyard with two snowballs and I was doing one three one

snowballs and I was doing one three one

snowballs and I was doing one three one three one three without knowing and the

three one three without knowing and the

three one three without knowing and the other part of this trick is seven one

other part of this trick is seven one

other part of this trick is seven one she’s a four ball trick and it looks

she’s a four ball trick and it looks

she’s a four ball trick and it looks like this and this is the trick that

like this and this is the trick that

like this and this is the trick that Krusty the Klown would usually do when

Krusty the Klown would usually do when

Krusty the Klown would usually do when he juggle four balls so in combination

he juggle four balls so in combination

he juggle four balls so in combination seven one three one looks like this and

seven one three one looks like this and

seven one three one looks like this and you can see how the green ball is just

you can see how the green ball is just

you can see how the green ball is just doing one threes and the two red balls

doing one threes and the two red balls

doing one threes and the two red balls are doing sevens and ones okay fine we

are doing sevens and ones okay fine we

are doing sevens and ones okay fine we know that this trick is working so this

know that this trick is working so this

know that this trick is working so this is a good starting point to come up with

is a good starting point to come up with

is a good starting point to come up with a new trick on this basis so and what

a new trick on this basis so and what

a new trick on this basis so and what are we doing to come up with the new

are we doing to come up with the new

are we doing to come up with the new trick

trick

trick we swap the seven and the one in the

we swap the seven and the one in the

we swap the seven and the one in the side swap

side swap

side swap seven one three one what are we actually

seven one three one what are we actually

seven one three one what are we actually doing we swapped the moments at which we

doing we swapped the moments at which we

doing we swapped the moments at which we throw the red and the green ball but we

throw the red and the green ball but we

throw the red and the green ball but we want to make sure that the ball still

want to make sure that the ball still

want to make sure that the ball still land on the same position and on the

land on the same position and on the

land on the same position and on the same more important on the same beat

same more important on the same beat

same more important on the same beat because when the balls land on the same

because when the balls land on the same

because when the balls land on the same beat that they did in 7 1 3 1 we know

beat that they did in 7 1 3 1 we know

beat that they did in 7 1 3 1 we know that the new trick is working as well so

that the new trick is working as well so

that the new trick is working as well so what do we have to do now we throw the

what do we have to do now we throw the

what do we have to do now we throw the red ball one bit earlier one bit later

red ball one bit earlier one bit later

red ball one bit earlier one bit later so it has to spend one beat less time in

so it has to spend one beat less time in

so it has to spend one beat less time in the air just to land on the same beat

the air just to land on the same beat

the air just to land on the same beat before we have thrown it with the 7 now

before we have thrown it with the 7 now

before we have thrown it with the 7 now we have to throw it with the 6 so the

we have to throw it with the 6 so the

we have to throw it with the 6 so the red 7 has to become a red 6 and the

red 7 has to become a red 6 and the

red 7 has to become a red 6 and the green ball we throw the green ball now

green ball we throw the green ball now

green ball we throw the green ball now one beat earlier so when we want them to

one beat earlier so when we want them to

one beat earlier so when we want them to land on the same beat as before it has

land on the same beat as before it has

land on the same beat as before it has to spend one bit longer time in the air

to spend one bit longer time in the air

to spend one bit longer time in the air so the green one has to become a green –

so the green one has to become a green –

so the green one has to become a green – maybe you can see this on this little

maybe you can see this on this little

maybe you can see this on this little animation that they still land on the

animation that they still land on the

animation that they still land on the same beat ok so how does the new site

same beat ok so how does the new site

same beat ok so how does the new site swap now look like it’s 2 6 3 1 she

swap now look like it’s 2 6 3 1 she

swap now look like it’s 2 6 3 1 she looks like this that was a drop that’s

looks like this that was a drop that’s

looks like this that was a drop that’s not this goes like this and since I

not this goes like this and since I

not this goes like this and since I swapped the red and the green ball now

swapped the red and the green ball now

swapped the red and the green ball now that the red and the green loop are

that the red and the green loop are

that the red and the green loop are intertwine then the green ball is

intertwine then the green ball is

intertwine then the green ball is wandering through the pattern oh it’s up

wandering through the pattern oh it’s up

wandering through the pattern oh it’s up here it’s down here ok great this worked

here it’s down here ok great this worked

here it’s down here ok great this worked and more important for our central

and more important for our central

and more important for our central theorem of siteswap juggling we did not

theorem of siteswap juggling we did not

theorem of siteswap juggling we did not change the number of balls with which we

change the number of balls with which we

change the number of balls with which we are juggling because we just swapped the

are juggling because we just swapped the

are juggling because we just swapped the moments at which we throw two balls and

moments at which we throw two balls and

moments at which we throw two balls and we did not change the average because we

we did not change the average because we

we did not change the average because we added one to one number and subtracted

added one to one number and subtracted

added one to one number and subtracted one from the other number so now we do

one from the other number so now we do

one from the other number so now we do what mathematicians always do when they

what mathematicians always do when they

what mathematicians always do when they figure out that something is working we

figure out that something is working we

figure out that something is working we iterate the process so we started with

iterate the process so we started with

iterate the process so we started with seven one three one swap the 7 and the 1

seven one three one swap the 7 and the 1

seven one three one swap the 7 and the 1 and got to 6 3 1 which is the same as 6

and got to 6 3 1 which is the same as 6

and got to 6 3 1 which is the same as 6 3 1 2 and now we do it again we swap the

3 1 2 and now we do it again we swap the

3 1 2 and now we do it again we swap the 6 and 3 and we get 4 5 1 2 which looks

6 and 3 and we get 4 5 1 2 which looks

6 and 3 and we get 4 5 1 2 which looks like this here the green ball it’s just

like this here the green ball it’s just

like this here the green ball it’s just doing forth and the red balls are doing

doing forth and the red balls are doing

doing forth and the red balls are doing 5 1 2 most people would miss the 2

5 1 2 most people would miss the 2

5 1 2 most people would miss the 2 because I basically do nothing for one

because I basically do nothing for one

because I basically do nothing for one beat so I try to emphasize it ok and now

beat so I try to emphasize it ok and now

beat so I try to emphasize it ok and now we swap again now we swap the 5 into 1

we swap again now we swap the 5 into 1

we swap again now we swap the 5 into 1 and we get 4 2 4 2 which looks like this

and we get 4 2 4 2 which looks like this

and we get 4 2 4 2 which looks like this yester – yeah this one is lame but you

yester – yeah this one is lame but you

yester – yeah this one is lame but you still can do fun stuff there’s a yo-yo

still can do fun stuff there’s a yo-yo

still can do fun stuff there’s a yo-yo with the orbit the factory thank you and

with the orbit the factory thank you and

with the orbit the factory thank you and now now we’re almost done now we just

now now we’re almost done now we just

now now we’re almost done now we just swap the 4 into 2 and we’re back whips

swap the 4 into 2 and we’re back whips

swap the 4 into 2 and we’re back whips for both of them and we’re back at the

for both of them and we’re back at the

for both of them and we’re back at the beginning we’re back at the simple three

beginning we’re back at the simple three

beginning we’re back at the simple three ball cascade it’s 3 3 3 and we know that

ball cascade it’s 3 3 3 and we know that

ball cascade it’s 3 3 3 and we know that we can juggle this with three balls

we can juggle this with three balls

we can juggle this with three balls because it just consists of three throws

because it just consists of three throws

because it just consists of three throws so now we have finally proven that we

so now we have finally proven that we

so now we have finally proven that we can juggle all the other tricks as well

can juggle all the other tricks as well

can juggle all the other tricks as well with 3 balls and we have generated them

with 3 balls and we have generated them

with 3 balls and we have generated them on the way to to juggle them for us so

on the way to to juggle them for us so

on the way to to juggle them for us so now you all know the math and I’d like

now you all know the math and I’d like

now you all know the math and I’d like to leave you with another quote from

to leave you with another quote from

to leave you with another quote from Paul Dirac when you’re receptive and

Paul Dirac when you’re receptive and

Paul Dirac when you’re receptive and humble mathematics will lead you by the

humble mathematics will lead you by the

humble mathematics will lead you by the hand so grab some balls start to juggle

hand so grab some balls start to juggle

hand so grab some balls start to juggle and feel how mathematics literally leads

and feel how mathematics literally leads

and feel how mathematics literally leads your hands thank you very much

your hands thank you very much

your hands thank you very much [Applause]

[Applause]

[Applause] [Music]