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Packing Pennies in the Plane |
Thue's theorem.
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 disc
in the plane, and then place around it six others. In contrast to
the similar construction in 3D, where spheres are placed
around a sphere, it is clear that no more than six can be so placed.
Furthermore this continues on for each of the new
discs 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.
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.
2. Statement of Thue's Theorem