Tessellierung

Tessellierung is a mathematical concept that involves the division of a two-dimensional plane into relatively regular, non-overlapping shapes called tiles, such that there are no gaps or overlaps between the tiles. The word "tessellation" comes from the Latin word "tessella," meaning a small square tile.

Mathematically, a tessellation is a covering of the plane by a set of shapes, called tiles, such that each point in the plane belongs to exactly one tile. To be a tessellation, the tiles must fit together perfectly, with no gaps or overlaps. This requirement means that each tile must be able to be translated by its position, or any integer multiple of its position, without overlapping or leaving any gaps.

Tessellations have many applications in mathematics, art, and architecture. They can be used to construct periodic and aperiodic patterns in geometry, and to analyze congruences and tilings of the plane. Tessellations have also been used in art and design throughout history, with examples ranging from Islamic geometric art to the cover designs of 18th-century English country houses.

Different types of tessellations include regular and irregular tessellations, canonical and non-canonical tessellations, and periodic and aperiodic tessellations. The most well-known example of a regular tessellation is the quincunx, consisting of atoms arranged on a surface, but other examples include the arrangement of pollen grains on an apple and the patterns created by soot burning in a stove.

Tessellations have captivated mathematicians and artists for centuries, with many examples still yet to be discovered and explored.