FractionalGeometry-Chap2

# 260 as in example 292 each transformation can involve

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

Ask a homework question - tutors are online