Mathematical analysis of 2D packing of circles on bounded and unbounded planes

TL;DR

This paper develops mathematical formulas and recurrence relations for packing circles in bounded and unbounded planes, verifying results with MATLAB and discussing potential extensions to 3D sphere packing and spherical surfaces.

Contribution

It introduces analytic formulas and recurrence relations for circle packing in various plane regions, providing a deterministic approach distinct from heuristic methods.

Findings

Formulated packing density for tangent circles on infinite planes.

Verified analytic formulas with MATLAB simulations.

Discussed potential generalizations to 3D sphere packing and spherical surfaces.

Abstract

This paper encompasses the mathematical derivations of the analytic and generalized formula and recurrence relations to find out the radii of n umber of circles inscribed or packed in the plane region bounded by circular arcs (including sectors, semi and quarter circles) & the straight lines. The values of radii obtained using analytic formula and recurrence relations have been verified by comparing with those obtained using MATLAB codes. The methods used in this paper for packing circles are deterministic unlike heuristic strategies and optimization techniques. The analytic formulae derived for plane packing of tangent circles can be generalized and used for packing of spheres in 3D space and packing of circles on the spherical surface which is analogous to distribution of non-point charges. The packing density of identical circles, externally tangent to each other, the most densely packed on the regular hexagonal and the infinite planes have been formulated and analysed. This study paves the way for mathematically solving the problems of dense packing of circles in 2D containers, the packing of spheres in the voids (tetrahedral and octahedral) and finding the planar density on crystallographic plane.