Symmetries of an Equilateral Triangle

# Symmetries of an Equilateral Triangle - Set ρ = ρ O 2 π...

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Symmetries of an Equilateral Triangle It is easier to represent the symmetries of an equilateral triangle using permutation notation. Two line form Cyclic form ι = 1 2 3 1 2 3 ! (1)(2)(3) ρ O, 2 π/ 3 = 1 2 3 2 3 1 ! (1 2 3) ρ O, 4 π/ 3 = 1 2 3 3 1 2 ! (1 3 2) σ 1 = 1 2 3 1 3 2 ! (2 3) σ 2 = 1 2 3 3 2 1 ! (1 3) σ 3 = 1 2 3 2 1 3 ! (1 2) To ﬁll out the entries in the given table we perform the following calcu- lations : For example to compute σ 2 σ 3 , i.e. to compute row 3, column 4 entry we ﬁrst do σ 3 and then σ 2 . Then σ 2 σ 3 = 1 2 3 3 2 1 ! 1 2 3 2 1 3 ! = 1 2 3 2 3 1 ! = (1 2 3) = ρ O, 2 π/ 3
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Unformatted text preview: . Set ρ = ρ O, 2 π/ 3 and σ = σ ‘ 1 . Because ρσ is an involution ρσ = ( ρσ )-1 = σρ-1 and hence ρ k σ = σρ-k . If you use this equality and the permutations you can see that { ι,σ ‘ 1 ,σ ‘ 2 ,σ ‘ 3 ,ρ O, 2 π/ 3 ,ρ O, 4 π/ 3 } = { ι,ρ,ρ 2 ,σ,ρσ,ρ 2 σ } . Hence D 3 = { ι,ρ,ρ 2 ,σ,ρσ,ρ 2 σ } = < ρ, σ | ρ 3 = σ 2 = ι, σρσ-1 = ρ-1 > . 1...
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