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College Algebra Exam Review 217

# College Algebra Exam Review 217 - 2 are any two symmetries...

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4.3. WHAT ABOUT REFLECTIONS? 227 J a D 2 4 ± 1 0 0 0 1 0 0 0 1 3 5 J b D 2 4 1 0 0 0 ± 1 0 0 0 1 3 5 J r D 2 4 1 0 0 0 1 0 0 0 ± 1 3 5 J c D 2 4 0 ± 1 0 ± 1 0 0 0 0 1 3 5 J d D 2 4 0 1 0 1 0 0 0 0 1 3 5 : Figure 4.3.4. Reﬂections of the square tile. I claim that there are three additional symmetries that we must con- sider along with the ﬁve reﬂections and eight rotations; these symmetries are neither rotations nor reﬂections, but are products of a rotation and a re- ﬂection. One of these we can guess from our experience with the brick, the inversion, which is obtained, for example, as the product aj a , and which is implemented by the matrix ± E . If we can’t ﬁnd the other two by insight, we can ﬁnd them by computa- tion: If ± 1 and ±
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Unformatted text preview: 2 are any two symmetries, then their composition product ± 1 ± 2 is also a symmetry; and if the symmetries ± 1 and ± 2 are implemented by matrices F 1 and F 2 , then ± 1 ± 2 is implemented by the matrix product F 1 F 2 . So we can look for other symmetries by examining products of the matrices implementing the known symmetries. We have 14 matrices, so we can compute a lot of products before ﬁnd-ing something new. If you are lucky, after a bit of trial and error you will discover that the combinations to try are powers of R multiplied by the reﬂection matrix J r :...
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