Creativity and ownership Many math students feel very little ownership for what's going on.
This was soon shown to be wrong. The problem was that if you watch the planets carefully, sometimes they move backwards in the sky. So Ptolemy came up with a new idea - the planets move around in one big circle, but then move around a little circle at the same time.
Think of holding out a long stick and spinning around, and at the same time on the end of the stick there's a wheel that's spinning. The planet moves like a point on the edge of the wheel.
Well, once they started watching really closely, they realized that even this didn't work, so they put circles on circles on circles Eventually, they had a map of the solar system that looked like this: This "epicycles" idea turns out to be a bad theory.
One reason it's bad is that we know now that planets orbit in ellipses around the sun. The ellipses are not perfect because they're perturbed by the influence of other gravitating bodies, and by relativistic effects.
But it's wrong for an even worse reason that that, as illustrated in this wonderful youtube video.
In the video, by adding up enough circles, they made a planet trace out Homer Simpson's face. It turns out we can make any orbit at all by adding up enough circles, as long as we get to vary their size and speeds. So the epicycle theory of planetary orbits is a bad one not because it's wrong, but because it doesn't say anything at all about orbits.
Claiming "planets move around in epicycles" is mathematically equivalent to saying "planets move around in two dimensions". Well, that's not saying nothing, but it's not saying much, either!
A simple mathematical way to represent "moving around in a circle" is to say that positions in a plane are represented by complex numbers, so a point moving in the plane is represented by a complex function of time. If you start by tracing any time-dependent path you want through two-dimensions, your path can be perfectly-emulated by infinitely many circles of different frequencies, all added up, and the radii of those circles is the Fourier transform of your path.
This isn't weird, though. It's the same thing as saying the circles have real radii, but they do not all have to start at the same place.
At time zero, you can start however far you want around each circle. If your path closes on itself, as it does in the video, the Fourier transform turns out to simplify to a Fourier series. The only circles we need are the slowest circle, then one twice as fast as that, then one three times as fast as the slowest one, etc.
There are still infinitely-many circles if you want to reproduce a repeating path perfectly, but they are countably-infinite now. If you take the first twenty or so and drop the rest, you should get close to your desired answer.
In this way, you can use Fourier analysis to create your own epicycle video of your favorite cartoon character. That's what Fourier analysis says.
The questions that remain are how to do it, what it's for, and why it works. I think I will mostly leave those alone. Why it works is a rather deep question. It's a consequence of the spectral theorem. What it's for has a huge range.When you use a browser, like Chrome, it saves some information from websites in its cache and cookies.
Clearing them fixes certain problems, like loading or formatting issues on sites. Scribd est le plus grand site social de lecture et publication au monde. Mathematics HL (3rd edn) - /4/29 - page Chapter 15 VECTOR APPLICATIONS EXERCISE 15A 1a If the athlete is assisted by a wind of 1 ms¡1 his speed will be 7 ms¡1.
b. Activity book plus all the old work in film study unit. Tomorrow I will collect all the books and see who has not done well and who has. Activity book is a good indicator of.
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