tessellazione

Tessellation, also known as tiling, is the covering of a surface using one or more geometric shapes, called tiles, with no overlaps or gaps. The shapes must fit together perfectly to form a continuous pattern. These shapes are typically polygons, but can also be more complex forms. The most common and simplest tessellations are created using regular polygons such as squares, equilateral triangles, and regular hexagons. These are known as regular tessellations.

There are also semi-regular tessellations, which use two or more different regular polygons. For a tessellation to be considered semi-regular, each vertex where the tiles meet must be identical in terms of the arrangement of the polygons surrounding it. The sum of the angles of the polygons meeting at any vertex must equal 360 degrees. This principle is fundamental to how tessellations fit together without leaving any empty space.

Tessellations have been observed in nature, such as in honeycombs (hexagonal tessellation) and the scales of some reptiles. They are also widely used in art, architecture, and design, from ancient Roman mosaics to the works of M.C. Escher, who masterfully explored the mathematical and visual possibilities of tessellations, often incorporating repeating figures and symmetries. The concept extends beyond two dimensions, with space-filling polyhedra representing three-dimensional tessellations.