The White/Yolk Theorem

A module of Egg Math for Chickscope: in which we see that any two regions in the plane can be equally divided with a single cut.

Suppose you and a friend want to split an egg. Neither of you wants to be cheated out of your fair share, and it's not enough to split the overall egg into two equal halves. You want to be sure you get exactly half of the yolk and exactly half of the white.
If the egg has the nice symmetric shape of a surface of revolution, then any straight cut along the axis of symmetry will do the trick. A knife blade can cut along any plane including this axis, and exactly half of the yolk and half of the white will be on each side. And a two-dimensional picture is enough to show what is going on.
But suppose the yolk has settled to one side. Now there is no axis of symmetry. For simplicitly, we consider first a two-dimensional version of the problem, though now this is not exactly equivalent. Can we still find a straight line (along which to cut) that divides each region (the white and the yolk) exactly in half? In the example shown, the yolk is still a round disk, so we know the cut has to go through its center. As we sweep possible cut-lines around this center, we start out with more white above the cut, and end up with more white below, so somewhere in the middle we could find the perfect cut.
The same argument won't work if the yolk isn't perfectly round. But a surprising theorem says that we can still find a straight-line cut which divides the yolk and the white exactly in half, no matter what shape they have. In fact, any two areas in the plane (even if one or both are split into several pieces) can be both divided exactly in half by a properly chosen straight cut. So even a scrambled egg can be divided fairly with a single cut!
Here are some examples of pairs of regions; using the java program, you should try to divide each one fairly.
In three dimensions, we can do even better. We have greater freedom to choose the direction of a single straight cut (now along some flat plane), so in fact we can arrange to simultaneously divide each of three different volumes exactly in half. For instance, we could fairly divide not just the yolk and white, but also the shell of an egg. This amazing fact is often called the ham sandwich theorem since it shows you can fairly divide the ham, the cheese, and the bread when splitting a ham sandwich (even if the ham and cheese are not laid out nicely). This is a consequence of the Borsuk-Ulam theorem in topology.