College Algebra Exam Review 98

College Algebra Exam Review 98 - n rotational symmetries in...

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108 2. BASIC THEORY OF GROUPS The group D n can appear as the symmetry group of a geometric figure, or of a physical object, in a slightly different form. Think, for example, of a five-petalled flower, or a star-fish, which look quite different from the top and from the bottom. Or think of a pentagonal card with its two faces of different colors. Such an object or figure does not admit rotational symmetries that exchange the top and bottom faces. However, a starfish or a flower does have reflection symmetries, as well as rotational symmetries that do not exchange top and bottom. Figure 2.3.3. Object with D 9 symmetry. Figure 2.3.4. Object with Z 5 symmetry. Consider a regular n -gon in the plane. The reflections in the lines passing through the centroid of the n-gon and a vertex, or through the centroid and the midpoint of an edge, are symmetries; there are n such reflection symmetries. We can show that the
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Unformatted text preview: n rotational symmetries in the plane together with the n reflection symmetries form a group that is isomorphic to D n . See Exercise 2.3.10 . Figure 2.3.3 (below) has D 9 symmetry, while Figure 2.3.4 possesses Z 5 symmetry, but not D 5 symmetry. Both of these figures were gener-ated by several million iterations of a discrete dynamical system exhibit-ing “chaotic” behavior; the figures are shaded according to the probability of the moving “particle” entering a region of the diagram — the darker regions are visited more frequently. A beautiful book by M. Field and M. Golubitsky, Symmetry in Chaos (Oxford University Press, 1992), discusses symmetric figures arising from chaotic dynamical systems and displays many extraordinary figures produced by such systems....
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