The White/Yolk Theorem
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
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