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m348-assignment4sol

m348-assignment4sol - Math 348(2006 Assignment#4 Solutions...

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Math 348 (2006): Assignment #4: Solutions 1. We know that there are three types of orientation-reversing isometries of E 3 : reflection, glide, and rotatary reflection. We also know that a half-turn H about a line can be written as the composition r M 2 r M 1 of two reflections across any two perpendicular planes M 1 and M 2 intersecting in the line . Consider a reflection r M about a plane M . Let be any line contained in M and let M be the plane through perpendicular to M . Then H = r M r M and thus r M = r M r M r M = H r M . Consider a glide g . It can be written as g = r M 3 r M 2 r M 1 where M 1 M 3 and M 2 is perpendicular to both M 1 and M 3 . But then r M 3 r M 2 = H where is the line of intersection of M 2 and M 3 , and hence g = H r M 1 . Finally, consider a rotary reflection f . It can be written as f = r M 3 r M 2 r M 1 where M 1 and M 2 intersect in some line and are both perpendicular to M 3 . But then r M 3 r M 2 = H where is the line of intersection of M 2 and M 3 , and hence f = H r M 1 .
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