Tessellation is a process of using a repeating a shape to divide up a flat two-dimensional plane in such a way that there are no gaps or overlaps. The patterns that are produced in this way are seen all around us in everyday life in carpet patterns, tiles and in architecture, and be seen in the designs of ancient cultures as well as modern. We can even see the patterns in natural settings such as the way cells form or bees construct their honeycombs, so it is a fundamental feature of both nature and the things people create. Thus we see the maths working in science as well as art.
To use tessellation as part of an online jigsaw website, however, takes more than a designers eye to spot a recurring shape that will work. The designs must be expressed mathematically to allow them to be used as part of the computer coding that drives the puzzles on the website. We can trace this work back almost 400 years to the famous mathematician and astronomer Johannes Kepler. In 1619 Kepler made one of the first documented studies on the subject when he wrote about regular and semi-regular tessellation. In 1891 the Russian Yevgraf Fyodorov showed that all periodic tiling (another term for tessellation) of a flat surface features one of seventeen groups of specific shapes and this work began the formal mathematical study of tessellations.
Online jigsaw puzzles are designed using various tessellated shapes, each of which is a polygon, and the number of workable shapes that are available is huge. The classic shape of cardboard and wooden jigsaw puzzles became popular because its ideal to actually lock the pieces together, but for online jigsaw puzzles this is not a concern. So a far greater variety of shapes can be used and more options offered for the online players.