College Algebra Exam Review 216

College Algebra - 226 4 SYMMETRIES OF POLYHEDRA and a reflection in several different ways This is the inversion which sends each corner of the

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Unformatted text preview: 226 4. SYMMETRIES OF POLYHEDRA and a reflection in several different ways. This is the inversion, which sends each corner of the rectangular solid to its opposite corner. This is implemented by the matrix E . Note that E D J1 R1 D J2 R2 D J3 R3 , so the inversion is equal to ji ri for each i . Having included the inversion as well as the three reflections, we again have a group. It is very easy to check closure under multiplication and inverse and to compute the multiplication table. The eight symmetries are represented by the eight 3—by—3 diagonal matrices with 1’s and 1’s on the diagonal; this set of matrices is clearly closed under matrix multiplication and inverse, and products of symmetries can be obtained immediately by multiplication of matrices. The product of symmetries (or of matrices) is a priori associative. Now consider a thickened version of the square card: a square tile, which we place with its centroid at the origin of coordinates, its square faces parallel with the .x; y/–plane, and its other faces parallel with the other coordinate planes. This figure has the same rotational symmetries as does the square card, and these are implemented by the matrices given in Section 1.5. See Figure 4.3.3. r c b a d Figure 4.3.3. Rotations of the square tile. In addition, we can readily detect five reflection symmetries: For each of the five axes of symmetry, the plane perpendicular to the axis and passing through the origin is a plane of symmetry. See Figure 4.3.4 on the next page Let us label the reflection through the plane perpendicular to the axis of the rotation a by ja , and similarly for the other four rotation axes. The reflections ja ; jb ; jc ; jd , and jr are implemented by the following matrices: ...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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