Thue's theorem

Thue's theorem states that the most efficient way to cover a plane with equal-sized circles so that they do not overlap is to arrange them in a hexagonal pattern. In this pattern, each circle is surrounded by six others, creating a tightly packed, repeating hexagon-shaped layout. This arrangement maximizes the area covered by the circles while minimizing unused space, making it the most efficient circle packing in a plane. Essentially, Thue's theorem identifies the optimal pattern for covering a surface with equal circles without overlapping.