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The most efficient way to pack different-sized circles together is not obvious. 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.
Circle packing in a circle is a two-dimensional packing problem with the objective of packing unit circles into the smallest possible larger circle. Table of solutions, 1 ≤ n ≤ 20 [ edit ]
The circle packing theorem states that a circle packing exists if and only if the pattern of adjacencies forms a planar graph; it was originally proved by Paul Koebe in the 1930s, and popularized by William Thurston, who rediscovered it in the 1970s and connected it with the theory of conformal maps and conformal geometry. [1]
A circle packing for a five-vertex planar graph. The circle packing theorem (also known as the Koebe–Andreev–Thurston theorem) describes the possible tangency relations between circles in the plane whose interiors are disjoint. A circle packing is a connected collection of circles (in general, on any Riemann surface) whose interiors are ...
A diagram showing circles packing in a square packing arrangement. Date: 25 February 2009: Source: Own work: Author: Inductiveload: Permission (Reusing this file) Own work, all rights released (Public domain) Other versions
The related circle packing problem deals with packing circles, possibly of different sizes, on a surface, for instance the plane or a sphere. The counterparts of a circle in other dimensions can never be packed with complete efficiency in dimensions larger than one (in a one-dimensional universe, the circle analogue is just two points). That is ...
This category groups articles relating to the packing of circles in planes, on spheres, and on other types of surfaces, both with the aim of high packing density (circle packing) and with specified combinatorial patterns of tangencies (circle packing theorem).
Circle packing in a square is a packing problem in recreational mathematics, where the aim is to pack n unit circles into the smallest possible square. Equivalently, the problem is to arrange n points in a unit square aiming to get the greatest minimal separation, d n , between points. [ 1 ]