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Unformatted text preview: Exercise 1.4.4 ), and so takes edges to edges. Since a symmetry is an isometry, it must take each edge to an edge of the same length, and it must take the midpoints of edges to midpoints of edges. Let 2` and 2w denote the lengths of the edges; for the rectangle ` ¤ w , and for the square ` D w . The midpoints of the edges are at ˙ ` O e 1 and ˙ w O e 2 . A symmetry ± is determined by ±.` O e 1 / and ±.w O e 2 / , since the symmetry is linear and these two vectors are a basis of the plane, which contains the ﬁgure R . For the rectangle, ±.` O e 1 / must be ˙ ` O e 1 , since these are the only two midpoints of edges length 2w . Likewise, ±.w O e 2 / must be ˙ w O e 2 , since these are the only two midpoints of edges length 2` . Thus there are at most four possible symmetries of the rectangle. Since we have already found four distinct symmetries, there are exactly four....
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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.
 Fall '08
 EVERAGE
 Algebra, Matrices

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