Packings in The Plane
In two dimensional Euclidean space, Joseph Louis Lagrange proved in 1773 that the lattice arrangement of circles with the highest density is the hexagonal packing arrangement, in which the centres of the circles are arranged in a hexagonal lattice (staggered rows, like a honeycomb), and each circle is surrounded by 6 other circles. The density of this arrangement is
Axel Thue provided the first proof that this was optimal in 1890, showing that the hexagonal lattice is the densest of all possible circle packings, both regular and irregular. However, his proof was considered by some to be incomplete. The first rigorous proof is attributed to László Fejes Tóth in 1940.
At the other extreme, very low density arrangements of rigidly packed circles have been identified.
Read more about this topic: Circle Packing
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