Lecture 2 - 2 Examples of Symmetry groups Suppose that we...

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2. Examples of Symmetry groups Suppose that we take an equilateral triangle and look at its symmetry group. There are two obvious sets of symmetries. First one can rotate the triangle through 120 . Suppose that we choose clockwise as the posi- tive direction and denote rotation through 120 as R . It is natural to represent rotation through 240 as R 2 , where we think of R 2 as the effect of applying R twice. If we apply R three times, represented by R 3 , we would be back where we started. In other words we ought to include the trivial symmetry I , as a symmetry of the triangle (in just the same way that we think of zero as being a number). Note that the symmetry rotation through 120 anticlockwise, could be represented as R - 1 . Of course this is the same as rotation through 240 clockwise, so that R - 1 = R 2 . The other obvious sets of symmetries are flips. For example one can draw a vertical line through the top corner and flip about this line.
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