1.4. SYMMETRIES AND MATRICES 15 For the square (with sides of length 2w ), ±.w O e 1 / and ±.w O e 2 / must be contained in the set f˙ w O e 1 ; ˙ w O e 2 g . Furthermore if ±.w O e 1 / is ˙ w O e 1 , then ±.w O e 2 / is ˙ w O e 2 ; and if ±.w O e 1 / is ˙ w O e 2 , then ±.w O e 2 / is ˙ w O e 1 . Thus there are at most eight possible symmetries of the square. As we have already found eight distinct symmetries, there are exactly eight. Exercises 1.4 1.4.1. Work out the products of the matrices E , R , R 2 , R 3 , A , B , C , D , and verify that these products reproduce the multiplication table for the symmetries of the square, as expected. (Instead of computing all 64 prod-ucts, compute “sufﬁciently many” products, and show that your computa-tions sufﬁce to determine all other products.) 1.4.2. Find matrices implementing the six symmetries of the equilateral triangle. (Compare Exercise 1.3.1 .) In order to standardize our notation and our coordinates, let’s agree to put the vertices of the triangle at
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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.