tessellation

Tessellation is the process of tiling a surface using one or more geometric shapes without overlaps or gaps. It is a fundamental concept in geometry, art, and design, where repeated patterns create visually appealing and often mathematically significant arrangements.

The basic requirement for tessellation is that the shapes fit together perfectly, covering a plane in a repeating pattern. Regular tessellations use congruent regular polygons, such as equilateral triangles, squares, or regular hexagons, which meet at vertices with identical arrangements. Semi-regular tessellations incorporate two or more types of regular polygons arranged in a repeating pattern, while irregular tessellations use shapes of different sizes and forms.

Tessellations have been studied and employed by various cultures throughout history. They appear in Islamic art, where intricate tile work features complex, repeating geometric patterns, and in ancient mosaics. In mathematics, tessellations relate to concepts like symmetry, tiling theory, and group theory. They are also used in fields such as architecture, computer graphics, and materials science for pattern design, space optimization, and structural analysis.

Mathematically, tessellations can be described through symmetries and transformations such as translations, rotations, reflections, and glide reflections. The study of tessellations contributes to understanding periodic patterns and the rules governing their formation. Beyond the plane, tessellation principles extend to three-dimensional structures in crystalline materials and biological tissues.

In summary, tessellation is the art and science of seamlessly covering surfaces with shapes, reflecting both aesthetic appeal and underlying mathematical principles.