College Algebra Exam Review 207

College Algebra Exam Review 207 - of the cube very easily...

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4.1. ROTATIONS OF REGULAR POLYHEDRA 217 to the cube. In fact, the choice of coordinates for the vertices of the tetra- hedron in the preceding discussion shows how to embed a tetrahedron in the cube: Take the vertices of the cube at the points 2 4 ˙ 1 ˙ 1 ˙ 1 3 5 . Then those four vertices that have the property that the product of their coordinates is 1 are the vertices of an embedded tetrahedron. The remaining vertices (those for which the product of the coordinates is ± 1 ) are the vertices of a complementary tetrahedron. The even permutations of the diagonals of the cube preserve the two tetrahedra; the odd permutations interchange the two tetrahedra See Figure 4.1.9 . Figure 4.1.9. Cube with inscribed tetrahedra. This observation also lets us compute the 24 matrices for the rotations
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Unformatted text preview: of the cube very easily. Note that the 3–fold axes for the tetrahedron are also 3–fold axes for the cube, and the 2–fold axes for the tetrahedron are 4–fold axes for the cube. In particular, the symmetries of the tetrahedron form a subgroup of the symmetries of the cube of index 2. Using these considerations, we obtain the following result; see Exercise 4.1.5 . Proposition 4.1.5. The group of rotation matrices of the cube is isomor-phic to the group of 3–by–3 signed permutation matrices with determinant 1. This group is the semidirect product of the group of order 4 consisting of diagonal signed permutation matrices with determinant 1, and the group of 3–by–3 permutation 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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