sphere packing
How do you cram identical balls into space as tightly as possible without overlap — the way a grocer stacks oranges? The sphere-packing problem asks for the arrangement of non-overlapping equal balls that fills the largest fraction of space, and for the value of that fraction, the packing density. It sounds elementary and grocer-obvious, yet in most dimensions the answer is unknown, and the cases that ARE solved required some of the most striking mathematics of the last century.
Formally, a packing is a collection of equal-radius balls with disjoint interiors; its density is the limiting fraction of space their union occupies. Lattice packings, where centers sit at the points of a lattice, are the most studied: density is then the ball volume divided by the lattice covolume. In dimension 2 the hexagonal lattice gives the optimal density pi/sqrt(12) ~ 0.9069 (Thue, Toth). In dimension 3 the face-centered-cubic stacking (the grocer's pyramid) gives pi/sqrt(18) ~ 0.7405 — the Kepler conjecture, proved by Hales in 1998 with massive computer assistance and formally verified in 2014. Closely tied is the kissing number, the most balls that can simultaneously touch one central ball: 6 in the plane, 12 in dimension 3, and famously 240 in dimension 8 and 196560 in dimension 24.
The landmark modern results are dimensions 8 and 24: in 2016 Viazovska proved the E8 lattice is the densest packing in dimension 8, and with collaborators that the Leech lattice is optimal in dimension 24, using astonishingly clean 'magic functions' built from modular forms that exactly certify optimality. These dimensions are special because their exceptional lattices are so symmetric that the linear-programming bounds become tight. An honest caveat: outside dimensions 1, 2, 3, 8, and 24 the optimal packing density is unknown, and it is not even known in general whether the densest packing is a lattice packing — in high dimensions the best known packings are often disordered, and the gap between upper and lower density bounds is enormous. Sphere packing also has a concrete payoff: dense packings give the best error-correcting codes, linking the problem to information theory.
Stack oranges the grocer's way: a flat layer in a hexagonal grid, then each upper orange nestled into a dimple of the three below. This face-centered-cubic packing fills pi/sqrt(18) ~ 74.05 percent of space, and Hales proved no arrangement does better — the Kepler conjecture, open since 1611. Each orange touches 12 others, so the kissing number in dimension 3 is 12, just shy of the 13 that naive volume counting might tempt you to hope for.
Grocer's orange stack (FCC): density pi/sqrt(18), kissing number 12 — the Kepler conjecture.
Optimality is known only in dimensions 1, 2, 3, 8, 24. Elsewhere the best packing is open, and it is not even established in general that an optimal packing must be a lattice — high-dimensional record packings are frequently disordered, so 'just use the best lattice' is not known to be right.