College Algebra Exam Review 211

# College Algebra Exam Review 211 - tries Now considering...

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4.2. ROTATIONS OF THE DODECAHEDRON AND ICOSAHEDRON 221 Figure 4.2.2. Five–fold axis of the dodecahedron. Figure 4.2.3. Three–fold axis of the dodecahedron. There are ten 3–fold axes through pairs of opposite vertices, giving 20 elements of order 3. See Figure 4.2.3 . Finally, the dodecahedron has 15 2–fold axis through centers of oppo- site edges, giving 15 elements of order 2. See Figure 4.2.4 on the following page . With the identity element, the dodecahedron has 60 rotational symme-
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Unformatted text preview: tries. Now considering that the rotation groups of the “smaller” regular polyhedra are A 4 and S 4 , and suspecting that there ought to be a lot of regularity in this subject, we might guess that the rotation group of the dodecahedron is isomorphic to the group A 5 of even permutations of ﬁve objects. So we are led to look for ﬁve geometric objects that are permuted by this rotation group. Finding the ﬁve objects is a perhaps a more subtle...
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