Tessellators

Tessellators are entities that create tessellations—the covering of a space by shapes with no gaps or overlaps. The term can refer to people, algorithms, or devices that generate tilings in two or three dimensions. In mathematics, tessellations describe plane or space-filling arrangements of polygons or polyhedra, and they are analyzed in terms of symmetry, vertex configurations, and the way shapes meet at borders.

In two dimensions, tessellations use polygons arranged so that every edge is shared by exactly two tiles and every region is a single tile. Regular tessellations use one type of regular polygon; in the plane, only equilateral triangles, squares, and regular hexagons tile periodically. Semi-regular (Archimedean) tilings use more than one polygon but share the same vertex arrangement at every vertex. Uniform tilings generalize these concepts. Aperiodic tessellations employ a finite set of prototiles that force non-periodicity, such as Penrose tilings, which exhibit long-range order without repeating patterns.

In three dimensions, space can be tessellated by space-filling polyhedra, though such tilings are more complex and diverse. Tessellators in art and design often focus on decorative, repeating patterns, as famously seen in the works of M. C. Escher, who used tessellations of animals and other motifs. In science and engineering, tessellators are essential in computer graphics for mesh generation and rendering, and in numerical methods for discretizing surfaces and volumes in finite element analysis.

Notable topics closely related to tessellators include tiling theory, crystallography, and geometric modeling.