Tilings have been an object of study of scientists, mathematicians and artists since ancient times. Here we show four examples of periodic tilings, that is, tilings that can be fit upon themselves by translations in two (and hence, infinitely many) different directions. The first two pages present a special kind of tilings called semi-regular or Archimedean. These are edge-to-edge tilings made of copies of finitely many (but more than one) regular polygons in such a way that the arrangement of tiles around each vertex is the same. This arrangement uniquely defines the entire tiling and can be described by the numbers of sides of its tiles; thus, the symbol 3.4.6.4 denotes the tiling in which each vertex is surrounded by a triangle, a square, a hexagon, and another square in this order. It can be shown that there are 8 types of Archimedean tilings.

Tilings by non-regular polygons have also been studied. If only one shape of tile is allowed and, in addition, it must be convex, then it turns out that only n-gons with n=3, 4, 5, or 6 can tile the plane. There are no restrictions on the shape of triangular and quadrilateral tiles, and page 3 demonstrates how the plane can be periodically tiled with copies of any given quadrilateral, convex or not. It is known that there are only three different (parameterized) shapes of hexagonal tiles. For pentagons, 15 possible shapes were found, the last one in 2015, but whether this list is complete remains an open question. One of pentagonal tilings is found on page 4.