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 associated packing density, η, of an arrangement is the proportion of the surface covered by the circles.

Circle packing in a circle is a two-dimensional packing problem with the objective of packing unit circles into the smallest possible larger circle. If more than one optimal solution exists, all are shown. [1] 1.0 000... Trivially optimal. 0.5 000... Trivially optimal. 2.155... 0.6466... Trivially optimal. 2.414... 0.6864... Trivially optimal.

A circle packing is an arrangement of circles inside a given boundary such that no two overlap and some (or all) of them are mutually tangent. The generalization to spheres is called a sphere packing. Tessellations of regular polygons correspond to particular circle …

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 …

Our story traces its origin to William Thurston’s famous Notes. In constructing 3-manifolds, Thurston proves that associated with any triangulation of a sphere is a “circle packing”, that is, a configuration of circles which are tangent with one another in the pattern of the triangulation.

This review is dedicated to the memory of Oded Schramm, who worked in circle packing before his discovery of stochastic Loewner evolution and its applications to critical phenomena. This extraordinary mathematician's untimely death on 01 September 2008 in a hiking accident was a great loss for our community. Abstract.

A circle packing is a configuration of circles having a specified pattern of tangencies, as introduced by William Thurston in 1985. This book lays out their study, from first definitions to the latest theory, computations, and applications.

The Apollonian packing used as an example previously is integral, making the tetrahedron an integral polyhedron. How can we verify that a given packing is indeed integral? Thanks to DIMACS, the Rutgers Math Department, the NSF, and Professor Kontorovich.