To understand the Ham Sandwich Theorem, one has to start with the Pancake Theorem. This theorem states that if there is a pancake on a plate, there is at least one cut that would divide the pancake cleanly in two. If another pancake is added, there will still be at least one cut that divides both pancakes in half. Fairly simple, but that’s due to the fact that the Pancake Theorem only deals with 2-dimensional space. The Ham Sandwich Theorem takes the concept of the Pancake Theorem and moves it into 3-dimensional space.
The Pancake Theorem states that given n measurable objects in n-dimensional Euclidean space, it is possible to divide all of the objects in half (with respect to their volume) with a single (n-1)-dimensional hyperplane. This convoluted statement actually expresses something very simple — if there is a ham sandwich with three parts (2 slices of bread and 1 slice of ham), it is possible to use a single cut to divide the sandwich in half, no matter where the 3 items are. This doesn’t just mean that the sandwich as a whole can be halved but also means that if the sandwich was dropped and the individual pieces were scattered around, there still would be a slice that divided the sandwich equally in two. While this theorem seems silly and fun, it actually is helping research the very nature of the way the universe behaves, assisting in computer simulation of everything from atomic phenomena to the movement of celestial bodies.