Kepler's conjecture

If you pour unit spheres randomly into a large container, you will fill only some 55 to 60 percent of the space. If you shake the box while you are filling it, you will get a denser packing - something like 64 percent. What is the densest packing possible? In a little booklet which he published in 1611, Johannes Kepler claimed that the hexagonal close packing did the trick: pack one horizontal layer so densely that each sphere is surrounded by six spheres, then add the next layer by placing spheres into the dimples formed by the fnrst layer, etc. The density is now 74.05 percent. And so Kepler's conjecture was born...

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