This preview shows page 1. Sign up to view the full content.
Unformatted text preview: l 224 fractals. This
leaves 512 − 2 · 224 = 64. Because Peitgen, J¨ rgens, and Saupe have shown
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...
View Full Document