Chapter 7: Mechanisms in Programs and Nature

Section 8: The Problem of Satisfying Constraints

Discrete packings

The pictures below show a discrete analog of circle packing in which one arranges as many circles as possible with a given diameter on a grid. (The grid is assumed to wrap around.)

The pictures show all the distinct maximal cases that exist for a 7 ×7 grid, corresponding to possible circles with diameters Sqrt[m^2+n^2]. Already some of these are difficult to find. And in fact in general finding such packings is an NP-complete problem: it is equivalent to the problem of finding the maximum clique (completely connected set) in the graph whose vertices are joined whenever they correspond to grid points on which non-overlapping circles could be centered.

On large grids, optimal packings seem to approach rational approximations to hexagonal packings. But what happens if one generalizes to allow circles of different sizes is not clear.

From Stephen Wolfram: A New Kind of Science [citation]