By A. Baker
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Let the distinct elements of Γ be F1 , . . , Fn , where n = |Γ|. Let p ∈ R2 be (the position vector of) any point. Define 1 1 F1 (p) + · · · + Fn (p). n n For any k = 1, . . , n, by a result on Problem Sheet 4, we have 1 1 Fk (p0 ) = Fk F1 (p) + · · · + Fk Fn (p). n n p0 = Now if Fk Fi = Fk Fj , then Fk−1 Fk Fi = Fk−1 Fk Fj and so Fi = Fj . Also, every Fr can be written as Fr = Fk (Fk−1 Fr ) where Fk−1 Fr ∈ Γ has the form Fk−1 Fr = Fs for some s and therefore Fr = Fk Fs . So in the above expression for Fk (p0 ), the terms are the same as those in the formula for p0 apart from the order in which they appear.
It is always possible to arrange things so that k = 1. Notice that (A | 0) represents rotation about the origin through the angle 2π/d. 36. Let Γ O(2) be a finite subgroup. Then either Γ similar to a dihedral group D2n for some n 1. SO(2) or Γ is Proof. Again we will not give a proof. 7. Frieze patterns and their symmetry groups In this section we will see how frieze patterns can be understood by classifying their symmetry groups up to similarity. Recall that the group of translational symmetries of a frieze pattern is infinite and cyclic.
If we now write t = u + 2w where w is parallel to v1 and u is perpendicular to v1 , then (A | t) represents the composition of reflection in the plane v1 · x = v1 · w and rotation about the axis parallel to v1 and passing through the point with position vector c and satisfying the two conditions v1 · c = 0, (I − A)c = u. 3. The Euclidean group of R3 The set of all isometries R3 −→ R3 is denoted Euc(3) and forms the 3-dimensional Euclidean group (Euc(3), ◦) under composition of functions. Much of the general theory for Euc(2) discussed in Chapter 1 carries over to Euc(3) with obvious minor modifications.
Groups, Symmetry And Fractals by A. Baker
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