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We as human beings see beauty. Theres a sense of balance, of order, to beautiful things. But where does that sense come from . And what does algebra or geometry have to do with it . Can we quantify the beauty of a butterfly . Welcome to the world of symmetry. 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 selfsimilarity 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 sy

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