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Concept
Kepler conjecture
Summary
The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements. The density of these arrangements is around 74.05%.
In 1998, Thomas Hales, following an approach suggested by , announced that he had a proof of the Kepler conjecture. Hales' proof is a proof by exhaustion involving the checking of many individual cases using complex computer calculations. Referees said that they were "99% certain" of the correctness of Hales' proof, and the Kepler conjecture was accepted as a theorem. In 2014, the Flyspeck project team, headed by Hales, announced the completion of a formal proof of the Kepler conjecture using a combination of the Isabelle and HOL Light proof assistants. In 2017, the formal proof was accepted by the journal Forum of Mathematics, Pi.
Imagine filling a large container with small equal-sized spheres: Say a porcelain gallon jug with identical marbles. The "density" of the arrangement is equal to the total volume of all the marbles, divided by the volume of the jug. To maximize the number of marbles in the jug means to create an arrangement of marbles stacked between the sides and bottom of the jug, that has the highest possible density, so that the marbles are packed together as closely as possible.
Experiment shows that dropping the marbles in randomly, with no effort to arrange them tightly, will achieve a density of around 65%. However, a higher density can be achieved by carefully arranging the marbles as follows:
For the first layer of marbles, arrange them in a hexagonal lattice (the honeycomb pattern)
Put the next layer of marbles in the lowest lying gaps you can find above and between the marbles in the first layer, regardless of pattern
Continue with the same procedure of filling in the lowest gaps in the prior layer, for the third and remaining layers, until the marbles reach the top edge of the jug.
Official source
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