According to Claudius Ptolemaeus, who proposed the Ptolemaic System, the Earth was the center of the universe, and each planet revolved around it. However some observations (such as retrograde motion, where a planet starts to go 'backwards' in the sky for a time) didn't match with a theory based on perfectly circular orbits. To account for this, Ptolemaic astronomers came up with the notion of an epicycle - each planet revolved about a point, which in turn revolved around the Earth. A bit like a spirograph, in fact.
Last night I started to wonder what it would look like if you had epicycles around epicycles around epicycles - fractal epicycles, if you like. Fifty lines of Python or so later, I had some crude pictures of the orbits planets might have in this strange alternate universe:
A few more examples here.