466-HW-6 - 7 σ ‘ 1,ρ 7 O 2 π 7 σ ‘ 1(c Express the...

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Math 466 - Homework Set 6 (14.12.2011) 1. Consider the regular heptagon H in Figure 1. We know that the symmetry group of H is the dihedral group D 7 . Hence there are 7 reflection symmetries in lines 1 , ‘ 2 , ··· ,‘ 7 and 7 rotation symmetries around the origin through an angle of 2 kπ/ 7 radians for k = 1 , 2 , 3 ,..., 7 . (a) Find the images of 1,2,3,. ..,7 under each one of these 14 symme- tries using the example regarding the symmetries of an equilateral triangle. (b) Show that { ι,ρ O, 2 π/ 7 ,...,ρ O, 12 π/ 7 1 ,...,σ 7 } = { ρ O, 2 π/ 7 ,...,ρ 7 O, 2 π/ 7 1 O, 2 π/
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Unformatted text preview: 7 σ ‘ 1 ,...,ρ 7 O, 2 π/ 7 σ ‘ 1 } . (c) Express the elements of D 7 by using permutation notation in (i) two-line form, (ii) cyclic form. 2. Consider the symmetry group D 8 of the regular octagon in Figure 2. (a) Find the matrix forms of σ ‘ 6 and ρ 3 O,π/ 4 . (b) Find σ ‘ 6 ρ 3 O,π/ 4 by using the matrices you found in (a). (c) Use the matrix you found in (b) to write down the equations of the isometry. Which element of D 8 is this isometry? 1...
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This note was uploaded on 12/26/2011 for the course MATH 466 taught by Professor Ytalu during the Spring '11 term at Middle East Technical University.

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