Packing 16, 17 or 18 circles in an equilateral triangle
What is the smallest circle, or square, or equilateral triangle etc., that can contain \(n\) nonoverlapping unit circles? After giving a survey on results related to this problem, the authors consider the case of equilateral triangles for the special values \(n\in \{16, 17, 18\}\), filling a gap in the literature. They present efficient packings for these cases. Clearly, the problem of finding the largest common radius \(r_n\) of \(n\) congruent circles inside a unilateral triangle \(T\) is equivalent to maximizing the minimum pairwise distance \(t_n\) of \(n\) points in \(T\). By the homothety between \(T\) and the smallest equilateral triangle containing their centers, the relation \[ t_n= {{2r_n} \over {1-2 \sqrt {3r_n}}} \] between \(r_n\) and \(t_n\) is obtained. For the latter problem, two numerical methods generating near-optimal solutions are described.
scientific article; zbMATH DE number 810280
scientific article