How Not to Be Wrong: The Power of Mathematical Thinking

by Jordan Ellenberg

For a code to be an error-correcting code, no string—no point, if we’re to take this geometric analogy seriously—can be within distance 1 of two different code words; in other words, we ask that no two of the Hamming spheres centered at the code words share any points. So the problem of constructing error-correcting codes has the same structure as a classical geometric problem, that of sphere packing: how do we fit a bunch of equal-sized spheres as tightly as possible into a small space, in such a way that no two spheres overlap? More succinctly, how many oranges can you stuff into a box? The sphere-packing problem is a lot older than error-correcting codes; it goes back to the astronomer Johannes Kepler, who wrote a short booklet in 1611 called Strena Seu De Nive Sexangula, or “The Six-Cornered Snowflake.” Despite the rather specific title, Kepler’s book contemplates the general question of the origin of natural form. Why do snowflakes and the chambers of honeycombs form hexagons, while the seed chambers of an apple are more apt to come in fives? Most relevantly for us right now: why do the seeds of pomegranates tend to have twelve flat sides? Here’s Kepler’s explanation. The pomegranate wants to fit as many seeds as possible inside its skin; in other words, it is carrying out a sphere-packing problem. If we believe nature does as good a job as can be done, then these spheres ought to be arranged in the densest possible fashion. Kepler argued that the tightest possible pa...