College Algebra Exam Review 218

College Algebra Exam Review 218 - However there is a more...

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228 4. SYMMETRIES OF POLYHEDRA J r R D RJ r D 2 4 0 1 0 1 0 0 0 0 1 3 5 J r R 3 D R 3 J r D 2 4 0 1 0 1 0 0 0 0 1 3 5 : These are new matrices. What symmetries do they implement? Both are reflection-rotations, reflections in a plane followed by a rotation about an axis perpendicular to the plane. (Here are all the combinations that, as we find by experimentation, will not give something new: We already know that the product of two rotations of the square tile is again a rotation. The product of two reflections appears always to be a rotation. The product of a reflection and a rotation by about the axis perpendicular to the plane of the reflection is always the inversion.) Now we have sixteen symmetries of the square tile, eight rotations (in- cluding the nonmotion), five reflections, one inversion, and two reflection- rotations. This set of sixteen symmetries is a group. It seems a bit daunting to work this out by computing 256 matrix products and recording the mul- tiplication table, but we could do it in an hour or two, or we could get a computer to work out the multiplication table in no time at all.
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Unformatted text preview: However, there is a more thoughtful method to work this out that se-riously reduces the necessary computation; this method is outlined in the Exercises. In the next section, we will develop a more general conceptual frame-work in which to place this exploration, which will allow us to understand the experimental observation that for both the brick and the square tile, the total number of symmetries is twice the number of rotations. We will also see, for example, that the product of two rotations matrices is again a rota-tion matrix, and the product of two matrices, each of which is a reflection or a rotation–reflection, is a rotation. Exercises 4.3 4.3.1. Verify the formula for the reflection J ˛ through the plane perpen-dicular to ˛ . 4.3.2. J ˛ is linear. Find its matrix with respect to the standard basis of R 3 . (Of course, the matrix involves the coordinates of ˛ .)...
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