The Shape of Tiles: Regular and Irregular, Hard and Soft

Tiling involves filling a plane or space with repeated elements, known as tiles. This simple concept is deeply embedded in the natural world and human design, appearing in structures as varied as the hexagonal wax cells of a beehive and decorative wallpapers. While regular hard tiles—geometric shapes with straight edges that fit together without gaps or overlaps—are common in human-made designs, nature often favours soft or irregular patterns, shaped by physical forces. In this lecture, I will explore how both regular and irregular tiling patterns, hard and soft, emerge in nature and the underlying mathematical principles that govern their formation.