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 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.

Read more about Circle Packing:  Packings in The Plane, Packings On The Sphere, Packings in Bounded Areas, Unequal Circles

Other articles related to "circle packing, packing, circles, circle, packings":

Circle Packing Theorem - A Uniqueness Statement
... Any triangulated planar graph G is connected and simple, so the circle packing theorem guarantees the existence of a circle packing whose intersection graph is (isomorphic to) G ... Such a G must also be finite, so its packing will have a finite number of circles ... more formally, every maximal planar graph can have at most one packing ...
Paper Folding - Mathematics and Technical Origami - Technical Origami
... as 'uniaxial bases', the pattern of allocations is referred to as the 'circle-packing' ... Using optimization algorithms, a circle-packing figure can be computed for any uniaxial base of arbitrary complexity ... process, hence it is possible for two designs to have the same circle-packing, and yet different crease pattern structures ...
Circle Packing - Unequal Circles
... are also a range of problems which permit the sizes of the circles to be non-uniform ... extension is to find the maximum possible density of a system with two specific sizes of circle (a binary system) ... For six of these radius ratios it is also known that the densest packing must be one of the compact packings ...
Circle Packing Theorem
... Circle packing theorem For every connected simple planar graph G there is a circle packing in the plane whose intersection graph is (isomorphic to) G ...

Famous quotes containing the words packing and/or circle:

    The good husband finds method as efficient in the packing of fire-wood in a shed, or in the harvesting of fruits in the cellar, as in Peninsular campaigns or the files of the Department of State.
    Ralph Waldo Emerson (1803–1882)

    There is all the difference in the world between departure from recognised rules by one who has learned to obey them, and neglect of them through want of training or want of skill or want of understanding. Before you can be eccentric you must know where the circle is.
    Ellen Terry (1847–1928)