The sphere packing problem is the three-dimensional version of a class of ball-packing problems in arbitrary dimensions. In two dimensions, the equivalent problem is packing circles on a plane. In one dimension it is packing line segments into a linear universe.
In dimensions higher than three, the densest lattice packings of hyperspheres are known up to 8 dimensions. Very little is known about irregular hypersphere packings; it is possible that in some dimensions the densest packing may be irregular. Some support for this conjecture comes from the fact that in certain dimensions (e.g. 10) the densest known irregular packing is denser than the densest known regular packing.
In 2016, Maryna Viazovska announced a proof that the E lattice provides the optimal packing (regardless of regularity) in eight-dimensional space, and soon afterwards she and a group of collaborators announced a similar proof that the Leech lattice is optimal in 24 dimensions. This result built on and improved previous methods which showed that these two lattices are very close to optimal.
The new proofs involve using the Laplace transform of a carefully chosen modular function to construct a radially symmetric function  such that  and its Fourier transform  both equal 1 at the origin, and both vanish at all other points of the optimal lattice, with  negative outside the central sphere of the packing and  positive. Then, the Poisson summation formula for  is used to compare the density of the optimal lattice with that of any other packing. Before the proof had been formally refereed and published, mathematician Peter Sarnak called the proof "stunningly simple" and wrote that "You just start reading the paper and you know this is correct."
Another line of research in high dimensions is trying to find asymptotic bounds for the density of the densest packings. It is known that for large , the densest lattice in dimension  has density between  (for some constant ) and . to