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Tessellations Tessellations are repeating patterns of distinct shapes. The word "repeating" means that the tessellation can be broken down into identical sections. These sections repeat throughout the design. Tessellations are different from patterns in that patterns usually do not have distinct closed shapes. A closed shape is a shape that has a definite interior (inside) and definite exterior (outside). On the left is a true tessellation; on the right is not a tessellation but a pattern. Patterns repeat but do not have clearly defined closed shapes. Tessellations repeat and do have clearly defined closed shapes. A pure tessellation consists of congruent copies. THERE are only three regular polygons that makeup a pure tessellation! What are they? Remember: Regular means that the sides and angles of the polygon are all the same length. Only three regular polygons tessellate in the Euclidean plane: triangles, squares or hexagons. We can't show the entire plane, but imagine that these are pieces taken from planes that have been tiled. Here are examples of a tessellation of triangles a tessellation of squares a tessellation of hexagons When you look at these three samples you can easily notice that the squares are lined up with each other while the triangles and hexagons are not. Also, if you look at 6 triangles at a time, they form a hexagon, so the tiling of triangles and the tiling of hexagons are similar and they cannot be formed by directly lining shapes up under each other - a slide (or a glide!) is involved. You can work out the interior measure of the angles for each of these polygons: One Interior Angle Shape triangle square pentagon hexagon more than six sides measure in degrees 60 90 108 120 more than 120 degrees Since the regular polygons in a tessellation must fill the plane at each vertex, the interior angle must be an exact divisor of 360 degrees. This works for the triangle, square, and hexagon, and you can show working tessellations for these figures. For all the others, the interior angles are not exact divisors of 360 degrees, and therefore those figures cannot tile the plane. These regular polygons can be transformed to create more complicated tessellations. Lets see how this can work: Let’s Tessellate Here are a few sample tessellations created by famous Dutch artist M. C. Escher