College Algebra Exam Review 205

College Algebra Exam Review 205 - tetrahedron....

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4.1. ROTATIONS OF REGULAR POLYHEDRA 215 the unit vector .1= p 3/ 2 4 1 1 1 3 5 is one symmetry of the tetrahedron. In Exer- cise 4.1.1 , you are asked to compute the matrix of this rotation. The result is the permutation matrix R D 2 4 0 0 1 1 0 0 0 1 0 3 5 : In Exercise 4.1.2 , you are asked to show that the matrices for rotations of order 2 are the diagonal matrices with two entries of ± 1 and one entry of 1 . These matrices generate the group of order 4 consisting of diagonal matrices with diagonal entries of ˙ 1 and determinant equal to 1. Finally, you can show (Exercise 4.1.3 ) that these diagonal matrices and the permutation matrix R generate the group of rotation matrices for the
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Unformatted text preview: tetrahedron. Consequently, we have the following result: Proposition 4.1.3. The group of rotational symmetries of the tetrahedron is isomorphic to the group of signed permutation matrices that can be writ-ten in the form DR k , where D is a diagonal signed permutation matrix with determinant 1, R D 2 4 0 0 1 1 0 0 0 1 0 3 5 , and ² k ² 2 . Now we consider the cube. The cube has four 3–fold axes through pairs of opposite vertices, giving eight rotations of order 3. See Fig-ure 4.1.5 . Figure 4.1.5. Three– fold axis of the cube. Figure 4.1.6. Four– fold axis of the cube....
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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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