Johannes Kepler was a German mathematician, astrologer and astronomer who presented one of the greatest mathematical challenges in 1661. The problem that has confused mathematicians for nearly 400 years seemed simple in practice, but impossible to explain in theory: the face-centered cubic packing, preferred for arranging fruit by sellers, is the densest possible arrangement, with density slightly greater than 74%. Thomas Callister Hales, an American mathematician, presented a computer-aided solution to this mathematical mystery in 1998.

When explained in simpler terms, the conjecture of Kepler means that the standard way we pack sphere units is the most effective manner of utilizing space. When Kepler observed the way greengrocers packed oranges, he noticed that they didn’t place the fruits in one layer on top of an identical one. They instinctively found a more efficient way to pack by placing the orange in the hollows created by the lower level of fruits. Kepler imposed the mystery of discovering the densest sphere packing through mathematical equations.

Although the proportion seems natural to fruit sellers and all other people, mathematicians didn’t find a way to prove it for nearly 4 centuries. David Hilbert, R. Buckminster Fuller, C.F. Gauss, and many other math geniuses struggled with the problem without finding the solution. The geometric methods used by Wu-Yi Hsiang in 1993 and 2001 sparked a lot of controversy. The explanation attracted a lot of attention, but did not provide reliable proof for the most important statements.

Thomas Callister Hales first announced his theory in 1998, saying that computer calculations can help mathematicians prove the Kepler conjecture. Hales derived his methods from interval arithmetic, linear programming, and the theory of global optimization. It took five years for the jury of twelve referees to examine the proof, but their efforts were not successful. In 2003, the jury announced that it wasn’t possible to verify the proof.

Thomas Hales founded the “Flyspeck project” with the intention of constructing a clearer manuscript of the proof, which could be verified by automatic software for proof checking. This was a great challenge that would require 20 years of constant work to result with a successful outcome. Fortunately, Hales’ predictions were unnecessarily pessimistic; the Flyspeck project was successfully completed in 2014. Hales managed to provide a formal proof of the same theory he presented in 1997, but this manuscript included modifications in the geometric partition of space. Since the proof was verified, we can now rest assured that we are packing oranges in a mathematically correct manner.