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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