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SOLUTION 7

# Algebra

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Math 150a: Modern Algebra Homework 7 Solutions 5.9.2 Identify the group of symmetries of a baseball, taking the stitching into account and allowing orienta- tion reversing symmetries. Solution: Carefully examine the stitching of a baseball. The stitches are angled from the seam...giving an orientation to the seam. So, in order to allow orientation reversing symmetries of the baseball, we will consider the seam as an undirected path to be preserved by each symmetry. Now notice that a baseball (with its seam) is essentially the same as a cube with an edge path representing the seam (see figure below). As we will see, the symmetry group of the baseball is the dihedral group D 4 ! Consider the square S in the middle of the cube with each of its vertices on the path representing the seam (as shown). The vertices represent the four inflection points of the seam of the (round) baseball. First, each symmetry of the baseball must preserve S . A face of the cube with two edges of the seam must be sent to a face with only two edges. So each symmetry must preserve the vertical fibers of the cube. In other words, the top of the cube can only be sent to the top or bottom—so the middle slice must be fixed. This gives a homomorphism from the symmetry group of the baseball to the symmetry group of thesquare, D 4 . This homomorphism is injective because each symmetry of the baseball is determined by its action on S .

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SOLUTION 7 - Math 150a Modern Algebra Homework 7 Solutions...

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