Circle packing

Circle packing studies configurations of circles arranged so that specified pairs are tangent, often within a bounded region. It connects geometry, graph theory, and complex analysis.

Tangency graph: A graph whose vertices represent circles and whose edges represent tangencies between them. It records the prescribed contacts that define a packing.
Euclidean geometry: The geometry of points, lines, distances, and angles in flat space. Ordinary circle tangency is defined using Euclidean distance.
Discrete conformal geometry: The study of discrete structures that approximate conformal geometry and its transformations. Circle packings serve as a principal model of discrete conformality.
Frederick Soddy: An English chemist who published a 1936 poem presenting a formula for four mutually tangent circles. His popularization of the curvature relation helped make the subject visible.
Circle packing constant: The fraction of the plane covered by equal circles in the densest possible arrangement, equal to π divided by the square root of 12. It gives a benchmark for the density of equal-circle arrangements.
Descartes' theorem: A relation among four mutually tangent circles' signed curvatures, where curvature is the reciprocal of radius. It computes a circle tangent to three given circles.
Planar graph: A graph that can be drawn in the plane with no edges crossing except at shared vertices. Circle tangency patterns correspond to planar graphs.
Riemann mapping theorem: A theorem stating that every simply connected proper open subset of the complex plane is conformally equivalent to the unit disk. Circle-packing approximations give discrete routes to conformal maps.
Philip Beecroft: A nineteenth-century mathematician who studied configurations of tangent circles, including an early form of the four-circle curvature relation. His work preceded Soddy's popular presentation of the same relation.
Conformal dimension: An invariant describing the smallest Hausdorff dimension attainable within a suitable conformal gauge. Packing methods help estimate dimensions of certain fractal sets.

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