Tesselaation

Tesselaation, sometimes spelled tessellation (and occasionally written as tesselaation), is the arrangement of shapes in a plane so that there are no gaps or overlaps. The term covers monohedral tilings, formed from a single shape, as well as tilings made from multiple shapes.

In the Euclidean plane, there are exactly three regular tilings by congruent shapes: by equilateral triangles, squares, and regular hexagons. These tilings are edge-to-edge and have identical vertex configurations, meaning the same pattern of meeting shapes at every corner.

Beyond these regular cases, numerous tiling patterns exist. Archimedean or semi-regular tilings use more than one regular polygon but maintain a uniform arrangement around each vertex. Other tilings combine irregular shapes or variable vertex configurations to produce diverse geometric textures, often used in art and design.

Aperiodic tilings are patterns that cover the plane without repetition or translational symmetry, though they can exhibit long-range order. The best known example is the Penrose tiling, discovered in the 1970s, which uses nonperiodic shapes to achieve non-repeating but highly structured patterns. Such tilings have influenced mathematics, physics, and materials science, including studies of quasi-crystals.

Historically, tessellations appear in many cultures, notably in ancient mosaics and Islamic geometric art, where repeating patterns create complex visual rhythms. In modern times, artists such as M. C. Escher explored mathematically inspired tessellations of shapes that morph into one another, expanding the public sense of what tilings can do.

Applications of tesselaation include architectural and decorative design, computer graphics and texture mapping, and the modeling of crystalline and quasi-crystalline structures in science. See also tiling, tessellation, Penrose tiling.