Plane Filling

Plane filling refers to covering a two-dimensional surface completely with shapes such as shapes, patterns, or tiles, without any gaps or overlaps. It often involves creating a repeating or complex pattern that extends infinitely across the plane. Mathematicians study which kinds of shapes can fill a plane perfectly, focusing on their symmetry, angles, and how they fit together. This concept is fundamental in fields like tiling design, crystallography, and computational geometry, ensuring efficient and seamless coverage of surfaces—like flooring, wallpapers, or digital textures—while maintaining structural consistency and aesthetic appeal.