Tessellation

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Also referred to as tilling, tessellation is the covering of a two dimentional plane with regular polygons. A regular tessellation is a highly symmetric pattern made up of Congruent regular polygons. There are only three regular tessellations: those made up of Equilateral triangles, squares, or Hexagons. A semi-regular tessellation uses a variety of regular polygons, of which there are eight. The arrangement of polygons at every Vertex point is identical. An edge-to-edge tessellation is even less regular: the only requirement is that adjacent tiles only share full sides. Other types of tessellations exist, depending on types of figures and types of pattern. There are regular, irregular, periodic, nonperiodic, symmetric, asymmetric, and fractal tessellations.

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In 1619 Johannes Kepler made one of the first documented studies of tessellations when he wrote about regular and semiregular tessellation, which are coverings of a plane with regular polygons, in his Harmonices Mundi. Some two hundred years later in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the plane features one of seventeen different groups of isometries.