FractionalGeometry-Chap2

260 as in example 292 each transformation can involve

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Unformatted text preview: l 224 fractals. This leaves 512 − 2 · 224 = 64. Because Peitgen, J¨ rgens, and Saupe have shown u so many that are not symmetrical across the diagonal, perhaps the reamining are symmetric across the diagonal. In Fig. 2.56 we see 8 fractals, all symmetric across the diagonal. Each can be generated by 8 diffrent IFS rules, so these account for the remaining 64 from the original 512 IFS rules. The gasket is easiest to visualize. With appropriate values of e and f , the transformations take (x, y ) to (x/2, y/2) + (e, f ), achieved by r = s = 1/2 and θ = ϕ = 0. With the same values e and f , but now with r = −1/2, s = 1/2, and θ = ϕ = −π/2, these transformations take (x, y ) to (y/2, x/2) + (e, f ). That is, these transformations reflect each piece across the y = x diagonal, consequently preserve the piece because of its symmetry. Three pieces, each generated by two distinct transformations, so 8 different IFS rules generate each image of Fig. 2.56. Figure 2.56: Fractals symmetric across the diagonal. 91 2.9. SOME TOPOLOGY OF IFS ATTRA...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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