hi everyone
thank you yeah I’m a theoretical
physicist and a juggler and I’m a huge
fan of Paul Dirac he’s one of the
founding fathers of quantum mechanics
and he once said God used beautiful
mathematics in creating the universe
today we will do something a little bit
less epic so we will use simple
mathematics to create beautiful juggling
so how do we create our theory for
mathematics 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 bit better when we look
at it from a different perspective so
imagine there was a camera at the
ceiling and it would film me while I’m
juggling and walking along this
direction then the green ball would
leave a trace that looks a little bit
like this so let’s see how this builds
up so I start with the green ball in my
hand and then I walk this direction and
I throw from the left to the right to
the left to the right to the left and to
the right and the two other balls would
leave a trace like this so and you can
hear juggling has a very distinct rhythm
whenever I catch a ball there’s a beat
Boop and we will now count these beats
to make our mathematical theory so let’s
focus on the green ball and we will
realize it always lands on the third
beat three one two three one two three
one two three so now we will call the
throw with which I’ve thrown the green
green ball a three throw and when the
when I’ve thrown a three throw I have
time to throw two more balls that’s why
I’m doing the three ball cascade with
three throws so now as good
mathematicians we will now generalize
this concept to the concept of n throws
so an N throw lands on the nth beat in
other words after I have thrown an N
throw I can throw n minus 1 additional
balls and when I have n vaults and only
do n throws then this is the basic n
ball juggling pattern great yeah now
that nobody is confused I can show some
examples so here comes the basic one
ball juggling pattern these are one
throws and you see how low I throw the
ball and I have 1 -1 beats time to throw
another ball yeah I can feel your
amazement okay so we so we quickly go on
to two balls so two ball juggling I have
one ball in my right hand one of my left
and they throw them very little and
since it’s a basic pattern I throw them
as little as I can this is I don’t throw
them at all so this is your basic two
ball juggling pattern almost as is
amazing as one ball juggling so 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 this is two balls in my right
hand and two in my left hand then I
combine it and you can see how the green
balls always stick to my right hand and
the red balls always stick to my left
hand okay so now comes five ball
juggling and I asked the organizers of
this conference if it’s okay if I show
off a little bit and they said yeah of
course this is what TEDx is about so for
your entertainment before I will juggle
the black belt of juggling I will
balance one ball on my foot
then I will balance one ball on my head
then I will juggle three balls let this
ball fall into the pattern then kick up
this ball from down here and finally
juggle five balls and if this works you
go crazy for a second
okay so I’m just building up 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 made it on the
first try so let’s look at the pattern
once more and let’s focus on the green
ball and you can see how similar this is
to one ball and three ball juggling the
ball is crossed and while the green ball
is in the air I have time to throw the
four other balls yeah sure otherwise I
couldn’t juggle five balls so we can
summarize and even and throw lands on
the same hand like for two ball juggling
or four ball throws or let’s say six
throws and an odd an throw crosses and
the higher the number the higher the
throw so now that we’ve defined n throws
we can do a cool thing now we can
combine different n throws to a new
pattern now we’ll show an example so
here’s a nice one four four one goes
like this I throw a four here then a
four here and then I hand over a ball
with a one so it goes for four one once
more four for one and continuously looks
like this and all 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 so maybe we look
at another example five three one five
three one I throw a five here then a
little bit lower three here and then I
hand over a one so it goes like this
five three one once more five three one
so and you can see how the green ball
now is chest doing threes and the two
other balls are doing the fives and ones
so we could say we can represent the
juggling trick by a sequence of n throws
or by sequence of numbers and we call
this a side swap we will figure out in a
minute while we call it a side swap but
there are some rules how we have to
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
could have well said three one five or
one five three but it’s just convenient
to start with one of the higher throws
and then to keep our notation for this
top
very simple we have the rule that we
catch and throw only alternating so we
say we exclude synchronous throws so no
tricks like this and we have another
rule we say each of our hands catches
and throws only one ball at a time so no
throws like this and no tricks like this
or let’s say like this like this because
here we see it at one point I was
throwing two balls at the time and then
I said a side swap stands for a juggling
trick actually it stands for a whole
class of juggling trick 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 at
which I catch and throw the balls so in
these sites wolves they have a very cool
property if you want to know how many
balls you need yeah I will we will see
how clear this is if you want to know
how many balls you need to juggle a
siteswap you just take the average so
you sum up all the numbers of the N
throw and then divide it by the pattern
length and you get the number of balls
so let’s check this for one example no
one already five-three-one we sum up
five plus three plus one gives nine
divided by three gives three and yeah
five three one was a 5-volt trick a
three ball trick sorry so and in the
last part of my talk I will now show you
how we can actually prove that this
central theorem of siteswap juggling is
actually true and on the way we will
generate a lot of new sites books for us
to juggle because it happens that you
cannot juggle any any random combination
of numbers so if you would want me to
juggle your telephone number this will
probably not work but I will show you
now a method how we can generate side
swaps that can be juggled in the way I
showed you and we will demonstrate this
on a very cool example it’s the side
swap seven one three one and seven one
three one is a combination of the side
swap one three which looks like this and
this is one of the first side swaps that
I was doing it was in winter in 1997 and
that was on the schoolyard with two
snowballs and I was doing one three one
three one three without knowing and the
other part of this trick is seven one
she’s a four ball trick and it looks
like this and this is the trick that
Krusty the Klown would usually do when
he juggle four balls so in combination
seven one three one looks like this and
you can see how the green ball is just
doing one threes and the two red balls
are doing sevens and ones okay fine we
know that this trick is working so this
is a good starting point to come up with
a new trick on this basis so and what
are we doing to come up with the new
trick
we swap the seven and the one in the
side swap
seven one three one what are we actually
doing we swapped the moments at which we
throw the red and the green ball but we
want to make sure that the ball still
land on the same position and on the
same more important on the same beat
because when the balls land on the same
beat that they did in 7 1 3 1 we know
that the new trick is working as well so
what do we have to do now we throw the
red ball one bit earlier one bit later
so it has to spend one beat less time in
the air just to land on the same beat
before we have thrown it with the 7 now
we have to throw it with the 6 so the
red 7 has to become a red 6 and the
green ball we throw the green ball now
one beat earlier so when we want them to
land on the same beat as before it has
to spend one bit longer time in the air
so the green one has to become a green –
maybe you can see this on this little
animation that they still land on the
same beat ok so how does the new site
swap now look like it’s 2 6 3 1 she
looks like this that was a drop that’s
not this goes like this and since I
swapped the red and the green ball now
that the red and the green loop are
intertwine then the green ball is
wandering through the pattern oh it’s up
here it’s down here ok great this worked
and more important for our central
theorem of siteswap juggling we did not
change the number of balls with which we
are juggling because we just swapped the
moments at which we throw two balls and
we did not change the average because we
added one to one number and subtracted
one from the other number so now we do
what mathematicians always do when they
figure out that something is working we
iterate the process so we started with
seven one three one swap the 7 and the 1
and got to 6 3 1 which is the same as 6
3 1 2 and now we do it again we swap the
6 and 3 and we get 4 5 1 2 which looks
like this here the green ball it’s just
doing forth and the red balls are doing
5 1 2 most people would miss the 2
because I basically do nothing for one
beat so I try to emphasize it ok and now
we swap again now we swap the 5 into 1
and we get 4 2 4 2 which looks like this
yester – yeah this one is lame but you
still can do fun stuff there’s a yo-yo
with the orbit the factory thank you and
now now we’re almost done now we just
swap the 4 into 2 and we’re back whips
for both of them and we’re back at the
beginning we’re back at the simple three
ball cascade it’s 3 3 3 and we know that
we can juggle this with three balls
because it just consists of three throws
so now we have finally proven that we
can juggle all the other tricks as well
with 3 balls and we have generated them
on the way to to juggle them for us so
now you all know the math and I’d like
to leave you with another quote from
Paul Dirac when you’re receptive and
humble mathematics will lead you by the
hand so grab some balls start to juggle
and feel how mathematics literally leads
your hands thank you very much
[Applause]
[Music]
thank you yeah I’m a theoretical
physicist and a juggler and I’m a huge
fan of Paul Dirac he’s one of the
founding fathers of quantum mechanics
and he once said God used beautiful
mathematics in creating the universe
today we will do something a little bit
less epic so we will use simple
mathematics to create beautiful juggling
so how do we create our theory for
mathematics 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 bit better when we look
at it from a different perspective so
imagine there was a camera at the
ceiling and it would film me while I’m
juggling and walking along this
direction then the green ball would
leave a trace that looks a little bit
like this so let’s see how this builds
up so I start with the green ball in my
hand and then I walk this direction and
I throw from the left to the right to
the left to the right to the left and to
the right and the two other balls would
leave a trace like this so and you can
hear juggling has a very distinct rhythm
whenever I catch a ball there’s a beat
Boop and we will now count these beats
to make our mathematical theory so let’s
focus on the green ball and we will
realize it always lands on the third
beat three one two three one two three
one two three so now we will call the
throw with which I’ve thrown the green
green ball a three throw and when the
when I’ve thrown a three throw I have
time to throw two more balls that’s why
I’m doing the three ball cascade with
three throws so now as good
mathematicians we will now generalize
this concept to the concept of n throws
so an N throw lands on the nth beat in
other words after I have thrown an N
throw I can throw n minus 1 additional
balls and when I have n vaults and only
do n throws then this is the basic n
ball juggling pattern great yeah now
that nobody is confused I can show some
examples so here comes the basic one
ball juggling pattern these are one
throws and you see how low I throw the
ball and I have 1 -1 beats time to throw
another ball yeah I can feel your
amazement okay so we so we quickly go on
to two balls so two ball juggling I have
one ball in my right hand one of my left
and they throw them very little and
since it’s a basic pattern I throw them
as little as I can this is I don’t throw
them at all so this is your basic two
ball juggling pattern almost as is
amazing as one ball juggling so 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 this is two balls in my right
hand and two in my left hand then I
combine it and you can see how the green
balls always stick to my right hand and
the red balls always stick to my left
hand okay so now comes five ball
juggling and I asked the organizers of
this conference if it’s okay if I show
off a little bit and they said yeah of
course this is what TEDx is about so for
your entertainment before I will juggle
the black belt of juggling I will
balance one ball on my foot
then I will balance one ball on my head
then I will juggle three balls let this
ball fall into the pattern then kick up
this ball from down here and finally
juggle five balls and if this works you
go crazy for a second
okay so I’m just building up 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 made it on the
first try so let’s look at the pattern
once more and let’s focus on the green
ball and you can see how similar this is
to one ball and three ball juggling the
ball is crossed and while the green ball
is in the air I have time to throw the
four other balls yeah sure otherwise I
couldn’t juggle five balls so we can
summarize and even and throw lands on
the same hand like for two ball juggling
or four ball throws or let’s say six
throws and an odd an throw crosses and
the higher the number the higher the
throw so now that we’ve defined n throws
we can do a cool thing now we can
combine different n throws to a new
pattern now we’ll show an example so
here’s a nice one four four one goes
like this I throw a four here then a
four here and then I hand over a ball
with a one so it goes for four one once
more four for one and continuously looks
like this and all 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 so maybe we look
at another example five three one five
three one I throw a five here then a
little bit lower three here and then I
hand over a one so it goes like this
five three one once more five three one
so and you can see how the green ball
now is chest doing threes and the two
other balls are doing the fives and ones
so we could say we can represent the
juggling trick by a sequence of n throws
or by sequence of numbers and we call
this a side swap we will figure out in a
minute while we call it a side swap but
there are some rules how we have to
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
could have well said three one five or
one five three but it’s just convenient
to start with one of the higher throws
and then to keep our notation for this
top
very simple we have the rule that we
catch and throw only alternating so we
say we exclude synchronous throws so no
tricks like this and we have another
rule we say each of our hands catches
and throws only one ball at a time so no
throws like this and no tricks like this
or let’s say like this like this because
here we see it at one point I was
throwing two balls at the time and then
I said a side swap stands for a juggling
trick actually it stands for a whole
class of juggling trick 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 at
which I catch and throw the balls so in
these sites wolves they have a very cool
property if you want to know how many
balls you need yeah I will we will see
how clear this is if you want to know
how many balls you need to juggle a
siteswap you just take the average so
you sum up all the numbers of the N
throw and then divide it by the pattern
length and you get the number of balls
so let’s check this for one example no
one already five-three-one we sum up
five plus three plus one gives nine
divided by three gives three and yeah
five three one was a 5-volt trick a
three ball trick sorry so and in the
last part of my talk I will now show you
how we can actually prove that this
central theorem of siteswap juggling is
actually true and on the way we will
generate a lot of new sites books for us
to juggle because it happens that you
cannot juggle any any random combination
of numbers so if you would want me to
juggle your telephone number this will
probably not work but I will show you
now a method how we can generate side
swaps that can be juggled in the way I
showed you and we will demonstrate this
on a very cool example it’s the side
swap seven one three one and seven one
three one is a combination of the side
swap one three which looks like this and
this is one of the first side swaps that
I was doing it was in winter in 1997 and
that was on the schoolyard with two
snowballs and I was doing one three one
three one three without knowing and the
other part of this trick is seven one
she’s a four ball trick and it looks
like this and this is the trick that
Krusty the Klown would usually do when
he juggle four balls so in combination
seven one three one looks like this and
you can see how the green ball is just
doing one threes and the two red balls
are doing sevens and ones okay fine we
know that this trick is working so this
is a good starting point to come up with
a new trick on this basis so and what
are we doing to come up with the new
trick
we swap the seven and the one in the
side swap
seven one three one what are we actually
doing we swapped the moments at which we
throw the red and the green ball but we
want to make sure that the ball still
land on the same position and on the
same more important on the same beat
because when the balls land on the same
beat that they did in 7 1 3 1 we know
that the new trick is working as well so
what do we have to do now we throw the
red ball one bit earlier one bit later
so it has to spend one beat less time in
the air just to land on the same beat
before we have thrown it with the 7 now
we have to throw it with the 6 so the
red 7 has to become a red 6 and the
green ball we throw the green ball now
one beat earlier so when we want them to
land on the same beat as before it has
to spend one bit longer time in the air
so the green one has to become a green –
maybe you can see this on this little
animation that they still land on the
same beat ok so how does the new site
swap now look like it’s 2 6 3 1 she
looks like this that was a drop that’s
not this goes like this and since I
swapped the red and the green ball now
that the red and the green loop are
intertwine then the green ball is
wandering through the pattern oh it’s up
here it’s down here ok great this worked
and more important for our central
theorem of siteswap juggling we did not
change the number of balls with which we
are juggling because we just swapped the
moments at which we throw two balls and
we did not change the average because we
added one to one number and subtracted
one from the other number so now we do
what mathematicians always do when they
figure out that something is working we
iterate the process so we started with
seven one three one swap the 7 and the 1
and got to 6 3 1 which is the same as 6
3 1 2 and now we do it again we swap the
6 and 3 and we get 4 5 1 2 which looks
like this here the green ball it’s just
doing forth and the red balls are doing
5 1 2 most people would miss the 2
because I basically do nothing for one
beat so I try to emphasize it ok and now
we swap again now we swap the 5 into 1
and we get 4 2 4 2 which looks like this
yester – yeah this one is lame but you
still can do fun stuff there’s a yo-yo
with the orbit the factory thank you and
now now we’re almost done now we just
swap the 4 into 2 and we’re back whips
for both of them and we’re back at the
beginning we’re back at the simple three
ball cascade it’s 3 3 3 and we know that
we can juggle this with three balls
because it just consists of three throws
so now we have finally proven that we
can juggle all the other tricks as well
with 3 balls and we have generated them
on the way to to juggle them for us so
now you all know the math and I’d like
to leave you with another quote from
Paul Dirac when you’re receptive and
humble mathematics will lead you by the
hand so grab some balls start to juggle
and feel how mathematics literally leads
your hands thank you very much
[Applause]
[Music]
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