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Concept
Circle packing
Summary
In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap. The associated packing density, η, of an arrangement is the proportion of the surface covered by the circles. Generalisations can be made to higher dimensions – this is called sphere packing, which usually deals only with identical spheres.
The branch of mathematics generally known as "circle packing" is concerned with the geometry and combinatorics of packings of arbitrarily-sized circles: these give rise to discrete analogs of conformal mapping, Riemann surfaces and the like.
In the two-dimensional Euclidean plane, Joseph Louis Lagrange proved in 1773 that the highest-density lattice packing of circles 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 six other circles. For circles of diameter D and hexagons of side length D, the hexagon area and the circle area are, respectively:
The area covered within each hexagon by circles is:
Finally, the packing density is:
In 1890, Axel Thue published a proof that this same density is optimal among all packings, not just lattice packings, but his proof was considered by some to be incomplete. The first rigorous proof is attributed to László Fejes Tóth in 1942.
While the circle has a relatively low maximum packing density, it does not have the lowest possible, even among centrally-symmetric convex shapes: the smoothed octagon has a packing density of about 0.902414, the smallest known for centrally-symmetric convex shapes and conjectured to be the smallest possible. (Packing densities of concave shapes such as star polygons can be arbitrarily small.)
At the other extreme, Böröczky demonstrated that arbitrarily low density arrangements of rigidly packed circles exist.
There are eleven circle packings based on the eleven uniform tilings of the plane.
Official source
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