Tessellations are shapes that cover a plane without any gaps or overlaps. You can make tessellations with a single shape or a pattern with several shapes. A regular tiling makes use of one shape to cover the plane but a semi-regular tiling uses more than one shape.
Often tessellations are associated with tiles, such as a backsplash or kitchen floor. The Latin word, tessela, means “small square” which was associated with small pieces of clay, stone or glass used to make mosaics. Art and math connect with the use of tessellations.
The history of tessellations dates back as far as 4000 BC by the Sumarians creating wall decorations using tiles to make patterns. In Spain in the 1300s the Moors made tiles called Alhambra, original Spanish tiles. They are still the inspiration for many tiles produced today.
The first to document his study of tessellations is Johannes Kepler in 1619. He wrote about regular versus semi-regular tessellations. A regular tessellation is created with a regular polygon with the same angle between adjacent edges for every tile. Only 3 regular polygons can fit these constraints: the equilateral triangle, a square, and regular hexagon. In 1891, E.S. Fedorov, a Russian crystallographer, proved that every semi-regular tiling of the plane is constructed in accordance with one of seventeen different groups of isometries. This was the beginning of the study of tessellations as a mathematical topic.
The person most famous for his work with tessellations is the Dutch artist, M.C. Escher, who created surprising artwork often using tessellations to mystify the eye. Although Escher claimed he did not understand math, he created works of art that are both mathematically rigid and organic. Often he would create a pattern starting from a regular shape such as the triangle and its shape would transform into a bird as seen here.
This was a topic I loved teaching my seventh graders. Each student created their own tessellation and they were displayed around my classroom with all of my MC Escher artwork. See this video to learn how to make a simple regular tessellation.
To play with multiple shapes to make a tessellation you can make designs, like this one my son made, go to NCTM Tessellation Creator. Here he created a semi-regular pattern using triangles, squares and hexagons.
