In this course, we want to take a broader perspective on the concept of a pattern, by allowing the basic units of a pattern to be similar but not necessarily identical, and to repeat but not necessarily with perfect regularity or symmetry. To understand and classify these patterns, we need to look at the various ways of tiling the plane . We will start with regular tilings , which are tilings with regular polygons . Such perfect and idealized tilings are not observed in nature but we will see that we can consider natural patterns as deformations of such perfect patterns.
Regular polygons are ones in which the sides have the same length and the angles are all equal.
One can show that the only regular polygonal tilings of the plane are by triangles, squares, and hexagons. Of course, you can find ways of tiling the plane with pentagons , but such polygons are not regular. You are all familiar with other forms of tilings, such as wallpaper tilings . These are periodic tilings of the plane which are built by translating or applying an orientation reversing symmetry to a tile that serves as a building block of the pattern. There are exactly 17 such independent patterns for the plane.