Why do tessellations tessellate

If you look at a completed tessellation, you will see the original motif repeats in a pattern. One mathematical idea that can be emphasized through tessellations is symmetry. There are 17 possible ways that a pattern can be used to tile a flat surface or 'wallpaper'. Between and Escher produced 43 colored drawings with a wide variety of symmetry types while working on possible periodic tilings. He adopted a highly mathematical approach with a systematic study using a notation which he invented himself.

There are 4 ways of moving a motif to another position in the pattern. These were described by Escher. A translation is a shape that is simply translated, or slid, across the paper and drawn again in another place. The translation shows the geometric shape in the same alignment as the original; it does not turn or flip. A reflection is a shape that has been flipped. Most commonly flipped directly to the left or right over a "y" axis or flipped to the top or bottom over an "x" axis , reflections can also be done at an angle.

If a reflection has been done correctly, you can draw an imaginary line right through the middle, and the two parts will be symmetrical "mirror" images. Since a polygon is a closed figure, we can start. Let us first look at the three regular polygons that tessellate by themselves. These are equilateral triangle, square, and regular hexagon.

Kay, pg. Here are a couple of examples. Rannucci, pg. We can take a regular hexagon and translate the sides to form a different shape. What about circles? Circles are a type of oval —a convex, curved shape with no corners.

Circles can only tile the plane if the inward curves balance the outward curves, filling in all the gaps. While they can't tessellate on their own, they can be part of a tessellation There are three different types of tessellations source :. Tessellations figure prominently throughout art and architecture from various time periods throughout history, from the intricate mosaics of Ancient Rome, to the contemporary designs of M. Get creative and try making tessellated masterpieces of your own using this handy tessellation creator courtesy of the National Council of Teachers of Mathematics!