tesselation

Tessellation, also known as tiling, is the arrangement of shapes that completely cover a surface without gaps or overlaps. In a tessellation, the tiles fit together edge to edge and meet at vertices in a way that leaves no space.

In the Euclidean plane, three regular tilings exist, using one type of polygon: by equilateral triangles, by squares, and by regular hexagons. Semi-regular (Archimedean) tilings use more than one type of polygon but maintain the same sequence of shapes around every vertex, such as 3.12.12 and 4.8.8.

Not all tilings are periodic. Irregular tilings vary; aperiodic or non-periodic tilings use a finite set of tiles that force a non-repeating pattern under certain matching rules. Tilings can be generated by substitution rules, inflation, or computational algorithms.

Tessellations also extend to space (three dimensions), where space can be filled by congruent or non-congruent polyhedra in a honeycomb pattern. The cubic honeycomb is the simplest example; more complex space-filling polyhedra exist.

Applications of tessellations are widespread. They appear in decorative art and architecture, computer graphics and texture mapping, crystallography and materials science, and various areas of mathematical theory. Tessellations help illustrate concepts of symmetry, periodicity, and aperiodicity, and they inform design and analysis across disciplines.

The term tessellation derives from Latin tessella, meaning a small tile. The concept has ancient roots in mosaics and paving, and it was developed by artists and mathematicians such as Johannes Kepler, who studied regular tilings in the early 17th century.