crystals, snowflakes, shells, geometric shapes, flowers, decorative arts such as the tilings of the alhambra palace in spain; they all share something besides what we might call "beauty," and that something that is imicit in the sort of regularity or self-similarity is symmetry, a symmetry we experience with satisfaction and pleasure in our everyday lives. often, that which eerience as symmetry is simply the final picture. but how did that beautiful, symmetric pattern come into being? with some close and careful inspection, anyone can see that symmetry starts with a basic motif which is then manipulated in time and space. mathematicians call such manipulations "transformations," or, though this might be confusing, we also call them "symmetries," thereby identifying the outcome with the transformation tt got us there. for example, start with a basic shape or pattern, say, the number 4. if we flip it over like it's on a mirror, we're using a kind of transformation called a reflection. and that reflection creates what we call bilateral symmetry. this bilateral symmetry seems fundamental to natu