Editing Kepler conjecture (section)

==Nineteenth century==
Kepler did not have a proof of the conjecture, and the next step was taken by {{harvs|txt|author-link=Carl Friedrich Gauss|first=Carl Friedrich|last=Gauss|year=1831}}, who proved that the Kepler conjecture is true if the spheres have to be arranged in a regular [[lattice (group)|lattice]].

This meant that any packing arrangement that disproved the Kepler conjecture would have to be an irregular one. But eliminating all possible irregular arrangements is very difficult, and this is what made the Kepler conjecture so hard to prove. In fact, there are irregular arrangements that are denser than the cubic close packing arrangement over a small enough volume, but any attempt to extend these arrangements to fill a larger volume is now known to always reduce their density.

After Gauss, no further progress was made towards proving the Kepler conjecture in the nineteenth century. In 1900 [[David Hilbert]] included it in his list of [[Hilbert's problems|twenty three unsolved problems of mathematics]]&mdash;it forms part of [[Hilbert's eighteenth problem]].