These arrangements are periodic, meaning that they repeat.Īn irregular arrangement is one in which the centers of the spheres do not form a lattice in some sense, they appear to be "randomly" spread about.īecause of the symmetry and regularity, lattice arrangements are much easier to analyze than irregular ones. More formally, a lattice arrangement (or a regular arrangement) is one where the centers of the spheres require only \(n\) vectors to define completely (in the \(n\)-dimensional case in the case of spheres, \(n=3\)). As the last section showed, this can lead to strange optimal tilings when the space is of a specific size, so the more interesting question arises when considering how the density changes as the space gets bigger and bigger in a formal sense, the goal is to find the limit of the densities of the optimal sphere packing of an \(n \times n \times n\) box, as \(n\) tends to infinity.Ī natural strategy is to choose a "small" arrangement and repeat it over and over again, attempting to do something similar to tessellating the plane. More formally, the density of a sphere packing in some finite space is the fraction of the space that can be filled with spheres. The problem of sphere packing is best understood in terms of density: rather than trying to determine how many spheres can fit into a specifically sized box, the more interesting question is how much of 3-D space can be filled with spheres (in terms of volume). A 250-page paper and over 3 gigabytes of data later, Kepler's conjecture was finally proved. Still, coming up with the strategy to perform the computation took another 41 years when Hales developed an algorithm that he spent four more years executing. This meant that, like the four color theorem, it was possible to prove the theorem with dedicated use of a computer. The next major breakthrough came in 1953 when Laszlo Toth reduced the problem to a (very) large number of specific cases. However, there are non-lattice packings that are even optimal in some cases, so this didn't prove the orange-stacking method was the best possible amongst all packings.Ī (very) irregular, but optimal, packing of 15 circles into a square Gauss's major result was that the orange-stacking method was the best possible amongst lattice packings, meaning intuitively packings that repeat in a highly regular way. However, approaching the problem proved difficult, and it was not until Gauss that major progress was made. Kepler conjectured that the "obvious" packing-the orange-stacking method-was in fact the best possible, and supported this by comparing it to some other basic strategies. The mathematician realized he couldn't answer the question, and passed it along to Johannes Kepler (better known for his work in astronomy). In fact, the origin of the problem is quite similar: Sir Walter Raleigh, on one of his expeditions, asked a mathematician friend what the most efficient way of stacking the cannonballs he had on the ship was. The problem of sphere packing is notable for being quite visible in everyday life many grocery stores, for instance, stack fruit (oranges in particular) in a pyramidal pattern that illustrates a possible sphere packing.
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