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College Algebra Exam Review 210

College Algebra Exam Review 210 - W n 2 Z g consists of one...

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220 4. SYMMETRIES OF POLYHEDRA 4.1.6. Let R be a convex polyhedron. (a) Show that if a rotational symmetry of R maps a certain face to itself, then it fixes the centroid of the face. Conclude that is a rotation about the line containing the centroid of R and the centroid of the face. (b) Similarly, if a rotational symmetry of R maps a certain edge onto itself, then is a rotation about the line containing the mid- point of the edge and the centroid of R . 4.1.7. Show that we have accounted for all of the rotational symmetries of the tetrahedron. Hint: Let be a symmetry and v a vertex. Show that the orbit of v under , namely the set f n .v/ W n
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Unformatted text preview: W n 2 Z g , consists of one, two, or three vertices. 4.1.8. Show that we have accounted for all of the rotational symmetries of the cube. 4.2. Rotations of the Dodecahedron and Icosahe-dron The dodecahedron and icosahedron are dual to each other and so have the same rotational symmetry group. We need only work out the rotation group of the dodecahedron. See Figure 4.2.1 . Figure 4.2.1. Dodecahedron and icosahedron. The dodecahedron has six 5–fold axes through the centroids of oppo-site faces, giving 24 rotations of order 5. See Figure 4.2.2 on the facing page ....
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