Groups, Symmetry And Fractals by A. Baker

By A. Baker

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A) If F and G are reflections in two distinct parallel lines, show that there is no point fixed by all the elements of Γ. Deduce that Γ is infinite. (b) If F is the reflection in a line L and G is a non-trivial rotation about a point p not on L, show that Γ is infinite. 15. Let Γ Euc(2) be a subgroup containing the isometries F, G : R2 −→ R2 and suppose that these generate Γ in the sense that every element of Γ is obtained by repeatedly composing powers of F and G. If a point p is fixed by both F and G, show that it is fixed by every element of Γ.

Let ⊆ R2 be an equilateral triangle with vertices A, B, C. B AI I   III  II  II   II    ·  O III  II  I  C A symmetry of is defined once we know where the vertices go, hence there are as many symmetries as permutations of the set {A, B, C}. Each symmetry can be described using permutation notation and we obtain the six distinct symmetries A B C A B C = ι, A B C B C A = (A B C), A B C C A B = (A C B), A B C A C B = (B C), A B C C B A = (A C), A B C B A C = (A B). Therefore we have | Euc(2) | = 6.

4. Find implicit and parametric equations for the plane P containing the points with position vectors p = (1, 0, 1), q = (1, 1, 1) and r = (0, 1, 0). Solution. Let us begin with a parametric equation. Notice that the vectors u = q − p = (0, 1, 0), v = r − p = (−1, 1, −1) are parallel to P and linearly independent since neither is a scalar multiple of the other. Thus a parametric equation is x = s(0, 1, 0) + t(−1, 1, −1) + (1, 0, 1) = (1 − t, s + t, 1 − t) (s, t ∈ R). To obtain an implicit equation we need a vector normal to P.

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