The ** hexagonal packing ** of discs in the plane is obtained by laying out a row of discs in a line, then successively adding rows on either side packed in as closely as possible. This coincides with what you get by fitting discs tightly inside a honeycomb pattern of hexagons. **Thue's theorem.** **No packing of non-overlapping discsof equal size in the plane has density higher than that of the hexagonalpacking**.
It is really impossible to imagine how it could be otherwise.We can also build the hexagonal packing in this way: we start with a single discin the plane, and then place around it six others. In contrast tothe similar construction in 3D, where spheres are placedaround a sphere, it is clear that no more than six can be so placed. Furthermore this continues on for each of the newdiscs etc. to give a global packing, which *has* to be optimal - doesn't it?But no straightforward proof of the Theorem has yet been found. | The hexagonal packing is obtained by layingout rows of discs as close to one another as possible. |