Background
Imagine filling a large container with small equal-sized spheres. The density of the arrangement is the proportion of the volume of the container that is taken up by the spheres. In order to maximize the number of spheres in the container, you need to find an arrangement with the highest possible density, so that the spheres are packed together as closely as possible.
Experiment shows that dropping the spheres in randomly will achieve a density of around 65%. However, a higher density can be achieved by carefully arranging the spheres as follows. Start with a layer of spheres in a hexagonal lattice, then put the next layer of spheres in the lowest points you can find above the first layer, and so on – this is just the way you see oranges stacked in a shop. At each step there are two choices of where to put the next layer, so this natural method of stacking the spheres creates an uncountably infinite number of equally dense packings, the best known of which are called cubic close packing and hexagonal close packing. Each of these arrangements has an average density of
The Kepler conjecture says that this is the best that can be done—no other arrangement of spheres has a higher average density.
Read more about this topic: Kepler Conjecture
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