Epicycles can be thought of as correlated 2-D Fourier sums.
As such, they are complete.
Any connected, delimited, periodic curve in the plane can be reconstructed to arbitary precision with high enough order epicycles and requisite deferents.
Below is an illustration of this. It is impressive.
Doh!
That, apparently, took 1000 nested epicycles.
It is awesome.
Generalization to arbitary (centered, connected) figures is left as an exercise for the reader.
For extra credit, construct the system of N-bodies which would generate this periodic orbit. Discuss its stability to small perturbations and whether perturbed orbits have strange states.
For total übergeek credit, actually construct the N-bodies as an exercise in SETI transmission.
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i am so using that the next i teach epicycles.
I want to see the RV curve for that orbit!
Sweet...
How to do it with N bodies:
When one body travelling in a straight line is approached by a light, fast-moving body, they scatter, and the velocity and direction of each is changed by an amount that depends on the velocity, impact parameter, and relative masses, but the net result is that each body winds up travelling at a constant velocity, different from what it started with.
Suppose we have a large mass plodding along. To make this simple, suppose we fire in a small "bender" mass with such a small impact parameter that it emerges almost along the trajectory it went in. Then it changes the momentum of the large mass by about 2mv in the direction you fired it. In this way you can introduce a quite sharp "corner" in the plodding mass' trajectory; by using a very light, fast test mass, you can be sure it's well gone by the time you need another corner.
Making Homer's picture, then, is just a question of approximating the trajectory with line segments and firing in a bender mass at each corner.
Now, if you want it *periodic*, you'll have to come up with some scheme for returning your bender masses; good luck. The whole scheme is also, of course, wildly unstable. As are, arguably, people who sit around dreaming up such schemes.
One of the professors at my university once gave a talk at a congress about the history of applied mathematics, where he used a similar example, but with the Bat-sign. It was a riot.
Great idea, but seems a bit ARBITRARY. I admit I am not into science, so I need some more TRIVIAL explanations. Thanks!
Enrico