Solving Puzzles over a Cup of Coffee

By Burkard Polster and Marty Ross

The Age, 11 February 2008

Next time you are in a sunny café, have a closer look at the surface of your coffee. Can you see bright and dark areas separated by a curiously pointy curve?

This complicated curve is produced by the rays of the sun reflecting off the walls of the cup.

There is another simple way to conjure this. Roll a circle around a second circle twice its size. Then a point on the perimeter of the smaller circle traces two copies of the coffee cup curve.

This curve is an example of an epicycle. It has been made famous by astronomers, from Ptolemy to Copernicus, who used them to model the motions of the planets. For the coffee cup curve there is just one pair of circles. By comparison, the astronomical models had circles upon circles upon circles: up to 80 epicycles in an intricate dance. Whenever their astronomical predictions deviated too much, they could simple include more epicycles!

Now, everybody knows that epicycles are not the way to go and that in fact the planets move on ellipses. But it so happens that everybody is not quite correct. In fact, ellipses also can be seen to be epicycles of a sort. Start with the two coffee cup circles, but now have the smaller circle run inside the larger one. Then, fix the tracing point to be inside the smaller circle rather than on the perimeter: the result is an ellipse! Of course, this means that to describe the orbits of all nine planets (we still count Pluto!) in our solar system we only need nine epicycles. So, the ancients were on the right track after all.

Here is a final puzzle: what curve do you get if the tracing point of the inner circle is chosen to be on the perimeter? Email us for the rather surprising answer.