Welcome to another Mathologer video. As a gentle intro to what I'll do today.

Here's a bit of a warm-up exercise. Here two circles the smaller one half the

radius of the larger one. The red dot marks a point on the circumference of the

smaller circle. Now imagine that the smaller circle rolls around inside the

larger one. What curve will be traced by the red dot? Think about it for a second.

You think you've got it? Well let's see. As you can see the dot traces a

straight line. Pretty cool, huh? So you can draw a line using circles. Who

would have thought. This little miracle is called the Tusi couple and is

named after the 13th century Persian astronomer and mathematician Nasir

al-din al-Tusi. I probably mucked that one up, right. Okay, so today's video is all about the

mathematics of combining circular motions and it's rich history, from early

models of the universe to the modern theory of Fourier analysis. A special

highlight will be the reconstruction of the mysterious Homer Simpsons orbit

discovered in 2008. Just wait. Okay let's really get going by considering the

orbit of our Moon around the Sun. Well we all know that the Earth travels around

the Sun on a roughly circular orbit, that the Moon travels around the Earth also

on a roughly circular orbit, and that there are 12 and a bit lunar months in a

year. Based on these three facts try to picture the orbit of the Moon around the

Sun. You've got it? Well let's see. In this picture the Moon is between the Sun and

Earth and so we're dealing with a new moon.

Fast forward one lunar months and we

have a new moon again. There are 11 more lunar months left in the year until

Earth completes its trip around the Sun. So in total we get this very pretty

picture of the orbit of the moon, with 12 distinctive loops corresponding to the

twelve months of the year. Right? Wrong! The moon's orbit looks nothing like this.

Yes, yes we should be using ellipses not circles, the year's not exactly 12 lunar

months long, the two orbits are not exactly in the

same plane, and so on. But none of that gets to why our picture is really

misleading. What has really mucked up our picture is that relatively the

earth-moon distance is much much, much smaller than we've indicated. Okay

so let's be honest, stop cheating and shrink our moon's orbit. Shrink, shrink

shrink and there the loops are gone and have turned into sharp cusps. So is

that what our orbit looks like? Nope, we have to keep shrinking. As you can see

the orbit now has shallow waves but we're still not there. Let's keep on

shrinking. So what we've got finally is a convex curve with no indentations that

looks like a regular 12-gon with rounded off corners and that really is what the

orbit of the moon around the Sun looks like, just even more rounded and circle

like than the curve in our picture. Pretty surprising isn't it. Well it gets

even more surprising. It turns out that our moon is the only moon in the solar

system with a convex orbit. The orbits of all the other moons are of the wavy and

loopy types. In fact because of this planet like orbit some experts consider

the Earth and the Moon as a double planet and not as a planet moon system.

Okay, let's now have a closer look at our simplified Sun-Earth-Moon setup. In our

system the Earth and the Moon circle around at constant speeds but what if we

vary those speeds (and the radii of the orbits)? What other orbit shapes are possible? For

a really extreme example, the orbit can be a line segment. Yes that coin rolling

business that I showed you earlier can also occur in this setting. Here the moon

is the red point on the rim of the rolling circle and the planet is the

center of this little circle. But this rolling coin setup also may remind you of

some very famous toy. Starts with an S rhymes with a why-rograph. Well, yes it's

a spirograph. Wow, that was bad :)

All spirograph curves are possible moon orbits and you're all probably aware of

the amazing variety of such curves. Here are some particularly remarkable ones.

You get perfect ellipses when you keep the size of the little circle the same

but move the moon to the inside of the circle. In fact, any shape ellipse can be

obtained in this way. Now change the rolling circle to be one-third the size

of the big circle and we get a pretty good approximation of an equilateral

triangle. Then with the rolling circle one fourth the size of the big circle we

get a pretty squarish orbit. So, conceivably, there's a moon somewhere in

the universe that moves around its Sun on an essentially square orbit. That's

pretty amazing, isn't it? Now what if the moon itself has a mini moon? Astronomers

seem pretty certain that at least in our solar system there's no such thing.

But, of course, mathematicians don't have to worry about reality. If we want a

mini moon of a moon we just create a virtual one. What then is possible in

terms of orbits of mini moons of moons or mini mini moons of mini moons of

moons? Well let's have a look at this. Here's a chain of 1,000 nested mini

moons at work, each one moving at some constant speed around its maxi moon and

the minniest of the moons follows an orbit in the shape of Homer Simpson. This

classic animation is the work of Santiago Ginnobili. Now that is really,

really, really amazing, don't you agree? You'd figure if we get a mini moon to

travel on a Homer path, then we can get it to do pretty much anything, right? And

that's true, any reasonably nice closed curve can be

traced by an orbit of a mini moon. Actually, we can do even more. Any such

loopy curve can be travelled along in infinitely many ways, slow in some parts

and fast in others, going back and forth over the same part of the curve, and so

on. And every one of these travel variations can be replicated by one of

our moon systems. Usually to get things mathematically spot-on

you'll need infinitely many moons. But even with finitely many moons you can

get as close to spot-on as you want. The orbits of moons and mini moons are

usually called epicycles and epicycle mathematics has been used for millennia

to describe complicated motions. It dates back to Ptolemy and the other ancient

Greek astronomers who used epicycles and other tricks to describe the apparently

crazy motions of the planets as observed from Earth. Most of us picture these old

geocentric models of the solar system as being very simple like this. So Earth in

the center with the moon, the planets and the Sun moving around it on concentric

circles. However every one of the planet circles really stands for a much more

complicated orbit. For example, Mars's orbit in the Ptolemaic system looks

something like this. Essentially this is an epicycle orbit

with a little extra tinkering thrown in to actually make this into a reasonably

predictive model. Copernicus's revolutionary system supposedly swept

away much of the complication of the geocentric models by placing the Sun at

the center of the solar system. This part of the story is also usually illustrated

with a simple picture of concentric circles. However, just like Ptolemy and

his greek mates, Copernicus also resorted to epicycle

acrobatics to fine-tune orbits of the planets. In fact Copernicus's model of

planetary motion wound up being at least as complicated as the Greeks' geocentric

systems, and no more accurate. Just to give you an idea, here's a drawing that

illustrates the motion of Mars according to Copernicus. Without going

into details, doesn't it look even more complicated

than the model for Mars in the Ptolemaic system that we looked at before. Well

does to me :) Of course what was really needed was

replacing all the circles within circles by fundamentally different geometry: the

elliptical orbits introduced by Johannes Kepler 60-odd years after Copernicus.

Anyway epicycle mathematics is ages old, even if it's not the simplest approach

to studying the heavens. However it was only at the beginning of the 19th

century that the great French mathematician Joseph Fourier published a

paper that led to a proper understanding of the possibilities of epicycle

mathematics and it's incredible usefulness outside astronomy. Today this

branch of mathematics is called Fourier analysis. For the more technically

inclined among you I will now show you how you can make your own epicycle

drawings of whatever takes your fancy and I'll show you how the epicycle

systems underlying these drawings can be reconstructed using complex Fourier

series. It gets a little detailed later, so feel free to bail out at any point.

Okay so down to work on our very important question: how to draw Homer

Simpson just using epicycles? And how would you naturally go about finding the

answer to this question? We all know, right? Google Homer and epicycle. If

you do, you won't be disappointed. It turns out that this exact question was

recently posted on the Mathematica stackexchange by someone writing under

the pseudonym Anderstood. Then Anderstood followed up with an answer

including the Mathematica code and explanations. Really brilliant stuff.

Let's have a look. As a warm-up Anderstood describes how to use

epicycles to draw the outline of an elephant. Starting with just a jpeg of a

silhouette to pin down the curve Anderstood first generates a set of

roughly equally spaced points along the perimeter of the silhouette. Then he

interprets the coordinates of these points as complex numbers and uses the

so called discrete Fourier transform to calculate epicycle approximations of the

silhouettes in terms of complex Fourier series. Very fancy and very scary words

but it's all pretty standard stuff. Nothing to panic about yet, promise. The

blob at the top was drawn using 20 epicycles. Maybe not so impressive though

if you squint a little it has a vague elephantine feel to it. But the bottom

picture drawn using 100 epicycles is most definitely an elephant. It all works

really well and I've gone on to use Anderstood's

code to generate epicycle approximations to various silhouettes. So have a guess

what's being drawn here. (music playing)

Back to Anderstood. After elephanting he tackles Bart Simpson. So he starts with

the drawing over there and his final output looks like ... wait for it ... this :) So

really pretty good. Unfortunately the bits of code

Anderstood provides for also capturing strokes inside the outline don't work

straight off the page. There's also a bit of a systemic problem with the centers

of the epicycles in the animations being wrongly placed. Having said that all the

ingredients are basically there and it wasn't too difficult to adapt Anderstood's

code and ideas to also reverse-engineer the Homer animation. Just really quick

here's what I did. So what I did it was I grabbed the screenshot of the original

video and cleaned it up in Photoshop and Illustrator. Then, again in Photoshop, I

made a low-res version of this drawing in which the outline is drawn as strings

of pixels. This is where Mathematica enters the picture for me and you can

have a close look at what exactly I do by inspecting my Mathematica notebook

linked in from the description. So using Mathematica I generate a list of the

coordinates of all the pixels in this picture, all the black ones. Now we

have to order the list of coordinates such that the corresponding points

appear in the order in which we want to trace them. Here Anderstood uses a very

neat trick. He unleashes the command FindShortestTour on the list. What this does

is it attempts to solve the Traveling Salesman problem for our set of points,

that is, it tries to create the shortest round trip that comprises all our points

and puts them into the corresponding order. Here's what this round trip looks

like. So this is one continuous loop which visits every pixel in our outline

exactly once. It's a somewhat quick and dirty solution

which includes some undesirable artefacts but it gets us there in finite

time which is great. Now from here we can just use exactly the same code as for

the elephant to generate the animation. Not bad, huh,

and if you're keen on something smoother, it's just a matter of more accurately

generating a string of points that indicates more exactly how you want the

original picture traced. Then you feed in the corresponding coordinates and the

program does its epicycle magic. Okay here's a challenge for you.

Get creative and come up with some epicycle animations of your own and link

them in via a comments below. The contribution I like best wins a copy of

Marty and my latest book, that one there. Okay, in the rest of the video I'd like

to really show you how these systems of epicycles are constructed and for that

let me just give you a crash course in Fourier series. Should be easy, right? Well

it's actually not bad and it's beautiful stuff. So to begin I have to remind you

of Euler's formula which has already appeared in about about a thousand

Mathologer videos. So e to the i t is equal to cos t plus i sine T. Euler's

formula amounts to a very compact way of tracing or in maths lingo parametrising

the unit circle in the complex plane. to see what I mean have a look at this

picture there so the red point on the unit circle is the complex number cost t

plus i sine t. That's the right side of Euler's formula, right? As we let the angle

t go from 0 to 2pi, the red point traces out the unit circle but now

Euler's formula tells us that the red point is also e to the i t. So, for

example, setting t equal to pi, that is, making a half turn, the red point moves

to ... -1, of course. So e^i pi =-1, most

mathematicians favourite identity. And, setting t equal to 0 we get e^0

is equal to 1 which of course is no surprise.

Just to say it again, as we let t go from 0 to 2 pi the red point travels around

the unit circle once in the counterclockwise direction. Now let's

write -t instead of t. So e^-it. This

corresponds to traversing the circle once in the opposite, the clockwise

direction. And if we write 2t instead of t this corresponds to traversing the

circle twice in the counterclockwise direction as t goes from 0 to 2 pi.

And if you write 3, three times, and so on.

Here's another important observation. Let's multiply our circle exponential by

4. What does this correspond to? Think about for a moment.... Easy, right? Here we

are traversing a circle of radius 4. Another way of expressing this is to say

that we traverse a circle through the complex number we multiply by, in this

case the number 4, starting and ending at this number. That sounds complicated

but what's nice about this way of looking at things is that this stays

true for all complex numbers, not just real numbers, like 4. So, for example, if

we multiply by the complex number 1+i we're now traversing the circle

starting and ending at 1+i. Okay, as we will see in a moment,

it's expressions of this form: complex number times e^i times some

integer times t that stand for the different epicycles produced by the

magical Fourier machine. And what is the source of the magic of the magical Fourier

machine? It's a very simple property of these exponential expressions. It turns

out that the integrals of these exponential's from 0 to 2 pi are always

equal to 0. Whoa, where did that one come from? And how can I possibly claim that this

is very simple? Well, it is actually easy if you know a little calculus. Basically

it's a straightforward symmetry argument: any complex number we come across as t

goes from 0 to 2 pi is canceled out by its negative which we'll also come across.

Maybe one of you can fill in the details in the comments.

Otherwise, just take my word for it for the moment. Okay, so now we are ready to

reveal the inner workings of the magical Fourier machine. Let's say what we want is

to trace this outline of the letter pi at a constant speed as shown.

So feeding this to the magical Fourier machine will create a chain of

infinitely many epicycles whose limit moon will do exactly the tracing we're

after. Here we go. If we only use the first few epicycles, then the last moon in

this finite chain will trace an approximation of our pi outline and the

more epicycles we use the better our approximation will be. For example, just

using the first four epicycles gives this orbit here. Using six epicycles will give

this. Here's the output from eight epicycles, and so on. Let's now have a

close look at the tracing produced by the first four epicycles, this one here.

Here we go in slow motion. Mathematically what you see here is represented by this

sum here. There's one term for each of the four circles which I've colour-coded,

plus the term at the top. So what we have here is a straightforward sum of the

four individual circular motions. The extra term on top is the complex number

that represents the black anchor point.

Now let's have a closer look at the exponent of e in these terms. For the

blue circle, the exponent is 1 times i t Then -1 i t for the

purple circle, 2 i t for the orange circle and, finally,

-2 i t for the red circle. The pattern's obvious next in the infinite

sum would be 3, -3, 4, -4, and so on, every integer appearing

exactly once. Except, ... well there is no 0, right? Well, actually, the 0 is

there, just in disguise. Here it is. e to the power of 0 is 1. So the infinite

sum that exactly captures the motion we're after is of this form here.

So we're producing a doubly infinite sum. Nifty if also a little scary, hmm.

Just to reiterate, the t in everything we're doing ranges from 0 to 2 pi and

because we're tracing closed curves starting and ending at the same complex

number the value of the sum is the same at 0 and 2 pi. This means our sums are

periodic functions from the interval 0 to 2pi to the complex numbers and so

here's another way of expressing what Fourier's magical machine does. The Fourier

machine rewrites any sufficiently nice periodic function f(t) from the

interval 0 to 2pi to the complex numbers as this sort of two-tailed infinite sum.

That's great but how do we find those infinitely many complex coefficients

that precisely pin down our epicycles. Well that's where the real magic is

hidden. Let me show you how you can find one of these coefficients, let's say c_2,

the coefficient in front of e^2it. Let's zoom in on the three

terms around this coefficient. To get rid of the e^2it next to the

c_2 we multiply our equation by e^ -2it. Now let's switch

to algebra autopilot. (music playing)

Okay, at this stage every single term of our

infinite sum except for the one we are focusing on features an exponential

factor. Now we can use the key integration factor that I mentioned earlier

to obliterate all the terms except for c_2. Ready for the integral magic? Here we

go.

There it is. Remember I told you earlier that all these integrals are

equal to 0 and so what's left on the right is this. But the green integral is

just equal to a 2 pi. Dividing through we find an expression that allows us to

calculate c_2. And, of course, the exact same trick works for all the other

coefficients as well. And so, on input of a tracing f(t) Fourier's magical machine

calculates these coefficients, each one exactly specifying one of our epicycles.

And here I take a bow to professor Fourier. Absolutely amazing stuff.

Of course I've glossed over a heap of details here, but really only details.

Maybe some of you in the know can supplement a bit of a discussion of when

exactly this works and what can go wrong. There's also still the question of how

you actually use this magic machine in practice. How you use it depends very

much on how you're tracing, the function f(t) is given to you. If it comes in a

nice form, you may be able to evaluate all those integrals. On the other hand, if,

as in the case of our Homer drawing, we are given an approximation of our

tracing by a sequence of points in this case there are specialised tools that

take these sequences of points as input and produce very good approximations of

these integrals. In particular, Anderstood in his Mathematica program uses the so

called discrete Fourier transform. A different approach is illustrated in a

very nice video by GoldPlatedGoof. Also very much worth checking out. Okay so all

under control in terms of epicycle mathematics. But what has all is to do

with all those real-life applications of Fourier series that you may have heard

of. Everything :) Just as a bit of a teaser, the animation below illustrates that the

famous representation of a square wave as a sum of sine functions is really

nothing but the imaginary part of a very pretty system of epicycles. Hopefully

I'll eventually get around to dedicating a few more videos to this incredibly

beautiful circle of ideas. Anyway I hope you enjoyed this video as much as I

enjoyed making it. And that's it for today.