Take a polygon with random vertices, find the mid-points of its sides, and use these to create a new polygon, replacing the original. Do this repeatedly and an ellipse will eventually form!
Note that in the above demonstration, the scales are automatically adjusted with each step.
I discovered this while reading The Mathematical Experience, shown as an example of "order out of chaos". In fact, Elmachtoub and van Loan show using eigenanalysis that any random polygon will tend towards an ellipse tilted at 45° when undergoing this transformation.