College Algebra Exam Review 225

College Algebra Exam Review 225 - 4.5. THE FULL SYMMETRY...

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Unformatted text preview: 4.5. THE FULL SYMMETRY GROUP 235 group of a geometric figure is the group of g 2 SO.3; R/ that leave the figure invariant. The full symmetry group is the group of g 2 O.3; R/ that leave the figure invariant. The existence of reflection symmetries means that the full symmetry group is strictly larger than the rotation group. In the following discussion, I do not distinguish between linear isometries and their standard matrices. Theorem 4.5.1. Let S be a geometric figure with full symmetry group G and rotation group R D G \ SO.3; R/. Suppose S admits a reflection symmetry J . Then R is an index 2 subgroup of G and G D R [ RJ . Proof. Suppose A is an element of G n R. Then det.A/ D det.AJ / D 1, so AJ 2 R. Thus A D .AJ /J 2 RJ . 1 and I For example, the full symmetry group of the cube has 48 elements. What group is it? We could compute an injective homomorphism of the full symmetry group into S6 using the action on the faces of the cube, or into S8 using the action on the vertices. A more efficient method is given in Exercise 4.5.1; the result is that the full symmetry group is S4 Z2 . Are there geometric figures with a nontrivial rotation group but with no reflection symmetries? Such a figure must exhibit chirality or “handedness”; it must come in two versions that are mirror images of each other. Consider, for example, a belt that is given n half twists (n 2) and then fastened. There are two mirror image versions, with right–hand and left– hand twists. Either version has rotation group Dn , the rotation group of the n–gon, but no reflection symmetries. Reflections convert the right–handed version into the left–handed version. See Figure 4.5.1. Figure 4.5.1. Twisted band with symmetry D3 . There exist chiral convex polyhedra with two types of regular polygonal faces, for example, the “snubcube,” shown in Figure 4.5.2 on the next page. ...
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