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. 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.ĭensest packing Identical circles in a hexagonal packing arrangement, the densest packing possible Hexagonal packing through natural arrangement of equal circles with transitions to an irregular arrangement of unequal circles Generalisations can be made to higher dimensions – this is called sphere packing, which usually deals only with identical spheres. The associated packing density, η, of an arrangement is the proportion of the surface covered by the circles.
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 most efficient way to pack different-sized circles together is not obvious. For circle packing with a prescribed intersection graph, see Circle packing theorem.
This article is about the packing of circles on surfaces.
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