tessellationsrepeating

Tessellations are repeating patterns that cover a plane with no gaps or overlaps. These patterns are created using one or more geometric shapes, called tiles, which fit together perfectly. For a shape to be a valid tile in a tessellation, its angles must sum to 360 degrees at every vertex where multiple tiles meet. Regular polygons, such as equilateral triangles, squares, and regular hexagons, are common shapes used in tessellations because their angles divide evenly into 360 degrees. For example, six equilateral triangles, four squares, or three regular hexagons can meet at a single point without leaving any space.

Tessellations can be created with a single type of tile, known as a regular tessellation, or with multiple types of tiles, referred to as semi-regular or non-regular tessellations. Semi-regular tessellations use two or more different regular polygons, arranged in a repeating pattern around each vertex. Non-regular tessellations can involve irregular shapes, but these shapes must also fit together without gaps or overlaps in a repeating manner. The concept of tessellation is found in nature, such as in the patterns of honeycomb or the scales of a fish, and has been extensively used in art and architecture throughout history. M.C. Escher is a famous artist known for his intricate and often complex tessellated designs.