FractionalGeometry-Chap2

Iterated function systems figure 258 the rst four

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Unformatted text preview: tion of three transformation IFS. Example 2.9.1 A circle in parameter space Fig. 2.55 shows an example of attractors generated by points on a circular path in parameter space. Start with rules for the equilateral gasket rule 1 2 3 r s 1/2 1/2 1/2 1/2 1/2 1/2 IFS rules for θ ϕ e 0 0 0 0 0 1/2 0 0 1/4 the fractals of Fig. 2.55. f 0 0 √ 3/4 and step θ1 = ϕ1 from 0 to 2π . In Fig. 2.55 this is done in increments of π/4. Of √ course, all these fractals share the line between (1, 0) and (1/2, 3/2), but then placing them in their actual positions would result in overlaps obscuring the evolution of the attractors. Rather, we have placed each fractal at the location its θ1 on a large circle. Start at the equilateral gasket and follow the fractals counterclockwise arounf the circle. The continuity in Theorem 2.9.1 is revealed 2.9. SOME TOPOLOGY OF IFS ATTRACTORS 89 Figure 2.55: Left: Adding a rotation to the lower left corner of the equilateral gasket rules, in increments of π/4. Right: Adding a rotation of 0.05 to the lower left corner. through gradual changes in the...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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