Packing Pennies in the Plane

2. Statement of Thue's theorem

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.