Tessellations of Regular Polygons and Irregular Escher-like Shapes

A Tessellation is a repeating pattern of polygons that covers a plane with no gaps or overlaps. Typically, the shapes that make up a tessellation are polygons or similar regular shapes. The only regular polygons that tessellate on their own are equilateral triangles, squares, and regular hexagons. Regular polygons have equal sides and equal angles. 

Regular octagons tessellate with squares as space between the tessellations. So tessellations can be formed with more than one shape. Tessellations can be formed from irregular shapes, in architecture (tile floors, mosaics, and stone paths), and in nature (flowers and bees’ honeycombs). 

Currently at the Museum of Fine Arts in Boston, the art of Dutch mathematician, M.C. Escher (1896-1972) is on display. He created tessellations of translations, reflections, rotations, and glide reflections. He also explored impossible dimensions.

In class, we had the children create glide translations from a simple square. See the attached pdf. First draw a shape from one vertex of the square to an adjacent vertex without touching the edge in between the vertices. Cut out that shape and slide or glide it to the opposite side of the square and tape the two edges together, being careful not to cut away any part of the square. Then draw another shape from one of the two vertices used in the original shape to the other adjacent vertex and follow the above glide procedure. You now have a template that you can trace on any size piece of paper. Your design will tessellate on all four sides of the figure without having to rotate your template. You can then design inside of the shape and use color to make it come alive.

I also gave the children a regular hexagon from which they can create a rotational tessellation. This is much more difficult and should be attempted with a parent. Rotational tessellations can be done with the square as well.

The beauty of exploring this mathematical concept is that seemingly random shapes can allow their imagination to run wild as the resulting tessellation is usually unexpected.