Covering a square by equal circles
Results for the finite packing problem of \(n\) equal circles in the unit square \(S\) are well known for many numbers \(n\). Corresponding results for the dual finite covering problem seem to be unknown: What is the smallest radius \(r_n\) of \(n\) equal circles which can cover \(S\) and how must \(S\) be covered? The authors associate with a covering of \(S\) a bipartite graph which is interpreted as a network of straight bars. With an interesting physical ``cooling technique'' as developed by the authors [Math. Proc. Camb. Philos. Soc. 110, No. 1, 71-89 (1991; Zbl 0736.52009)] they have discovered locally optimal coverings for all \(n\leq 10\). The results for \(r_n\) and the conjectured minimum density are collected in a table without details of the calculations.
scientific article; zbMATH DE number 850642