FractionalGeometry-Chap2

Because of the symmetry of the pieces these

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Unformatted text preview: upper left pieces are oriented differently from the lower left and upper right, which share the orientation of the whole shape. Because of the symmetry of the pieces, these orientations can be achieved through reflections or rotations. For example, the lower right piece can be obtained by reflecting the lower left piece across the y -axis and then translating by e = 1. Reflections across the y -axis can be obtained by making r negative; reflections across the x-axis by making s negative. Consequently, fractal (b) can be obtained by r 1/2 -1/2 1/2 1/4 s 1/2 1/2 -1/2 1/4 θ 0 0 0 0 ϕ 0 0 0 0 e 0 1 0 3/4 f 0 0 1 3/4 Rotations, too, can produce these orientations. With the given formulation of the transformations, rotations fix the origin, and positive angles are counterclockwise. Note also the order of the components of these transformations: scalings and reflections first, rotations second, and translations last. Using rotations, another set of transformations can produce fractal (...
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