Some new structures for the ‘equal circles packing in a square’ problem

To find the densest packing for n equal and non-overlapping circles in a square is a classical problem of geometry. Nurmela and Östergård determined repeated patterns and structures for the n = k², k² − 1, k² − 3, k(k + 1) and 4k² + k cases, while Graham and Lubachevsky did it for the n = k² − 2, k² + ⌊k/2⌋. After a short review of the previous patterns further on we will give some new and similar structures for n = k² − l (l = 4, 5, 6, 7). The found structures have suggested patterns, which improved those lower bounds of packing, which stem from patterns. These packings were found with the TAMSASS-PECS stochastic algorithm.