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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
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.