Circle packing

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 all circles touch one another. 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. 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 all circles touch one another. 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. While the circle has a relatively low maximum packing density of 0.9069 on the Euclidean plane, it does not have the lowest possible. The 'worst' shape to pack onto a plane is not known, but the smoothed octagon has a packing density of about 0.902414, which is the lowest maximum packing density known of any centrally-symmetric convex shape.Packing densities of concave shapes such as star polygons can be arbitrarily small. 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 two dimensional Euclidean space, Joseph Louis Lagrange proved in 1773 that the highest-density lattice arrangement 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 6 other circles. The density of this arrangement for circles of diameter D, is D is also the side of the hexagon in the first figure. The first term (3*Pi*D^2 / 4) in the ratio above is the sum total of the area of all the circles and partial circles enclosed by the hexagon. The second term (3*Sqrt(3)*D^2 / 2) is the area of the hexagon itself. Hexagonal packing of equal circles was found to fill a fraction Pi/Sqrt(12) ≃ 0.91 of area—which was proved maximal for periodic packings by Carl Friedrich Gauss in 1831. Later, 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, Böröczky demonstrated that arbitrarily low density arrangements of rigidly packed circles exist. There are 11 circle packings based on the 11 uniform tilings of the plane. In these packings, every circle can be mapped to every other circle by reflections and rotations. The hexagonal gaps can be filled by one circle and the dodecagonal gaps can be filled with 7 circles, creating 3-uniform packings. The truncated trihexagonal tiling with both types of gaps can be filled as a 4-uniform packing. The snub hexagonal tiling has two mirror-image forms.