Consider lunch. Perhaps a nice ham sandwich. A slice of a knife neatly halves the ham and its two bread slices. But what if you slip? Oops, the ham now rests folded under a flipped plate, with one slice of bread on the floor and the other stuck to the ceiling. Here’s some solace: geometry ensures that a single straight cut, perhaps using a room-sized machete, can still perfectly bisect all three pieces of your tumbled lunch, leaving exactly half of the ham and half of each slice of bread on either side of the cut. That’s because math’s “ham sandwich theorem” promises that for any three (potentially asymmetric) objects in any orientation, there is always some straight cut that simultaneously bisects them all. This fact has some bizarre implications as well as some sobering ones as it relates to gerrymandering in politics.
The theorem generalizes to other dimensions as well. A more mathematical phrasing says that n objects in n-dimensional space can be simultaneously bisected by an (n – 1)–dimensional cut. That ham sandwich is a bit of a mouthful, but we’ll make it more digestible. On a two-dimensional piece of paper, you can draw whatever two shapes you want, and there will always be a (one-dimensional) straight line that cuts both perfectly in half. To guarantee an equal cut for three objects, we need to graduate to three dimensions and cut them with a two-dimensional plane: think of that room-ravaging machete as slipping a thin piece of paper between the two halves of the room. In three dimensions, the machete has three degrees of freedom: you can scan it back and forth across the room, then stop and rotate it to different angles, and then also rock the machete from side to side (like how carrots are often cut obliquely, and not straight).
If you can imagine a four-dimensional ham sandwich, as mathematicians like to do, then you could also bisect a fourth ingredient with a three-dimensional cut.
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