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College Algebra Exam Review 212

College Algebra Exam Review 212 - three pairs of opposite...

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222 4. SYMMETRIES OF POLYHEDRA Figure 4.2.4. Two–fold axis of the dodecahedron. task than picking out the four diagonals that are permuted by the rotations of the cube. But since each face has five edges, we might suspect that each object is some equivalence class of edges that includes one edge from each face. Since there are 30 edges, each such equivalence class of edges should contain six edges. Pursuing this idea, consider any edge of the dodecahedron and its op- posite edge. Notice that the plane containing these two edges bisects an- other pair of edges, and the plane containing that pair of edges bisects a third pair of edges. The resulting family of six edges contains one edge in each face of the dodecahedron. There are five such families that are permuted by the rotations of the dodecahedron. To understand these ideas, you are well advised to look closely at your physical model of the dodec- ahedron. There are several other ways to pick out five objects that are permuted by the rotation group. Consider one of our families of six edges. It contains
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Unformatted text preview: three pairs of opposite edges. Take the three lines joining the centers of the pairs of opposite edges. These three lines are mutually orthogonal; they are the axes of a cartesian coordinate system. There are five such coordinate systems that are permuted by the rotation group. Finally, given one such coordinate system, we can locate a cube whose faces are parallel to the coordinate planes and whose edges lie on the faces of the dodecahedron. Each edge of the cube is a diagonal of a face of the dodecahedron, and exactly one of the five diagonals of each face is an edge of the cube. There are five such cubes that are permuted by the rotation group. See Figure 4.2.5 on the next page . You are asked to show in Exercise 4.2.1 that the action of the rotation group on the set of five inscribed cubes is faithful; that is, the homomor-phism of the rotation group into S 5 is injective....
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