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Unformatted text preview: rem, we develop some experience with
the inverse problem, ﬁnding an IFS to produce a given fractal.
The decomposition A = T1 (A) ∪ · · · ∪ Tn (A) suggests a method for solving
the inverse problem: given a selfsimilar or selfaﬃne set A, ﬁnd contractions Ti
producing A. In principle, the method is relatively simple:
1. Decompose A into scaled copies: A = A1 ∪ · · · ∪ An . 2. For each Ai , ﬁnd a transformation Ti with Ti (A) = Ai .
Example 2.1.1 Three IFS examples.
It is easy to see that fractal (a) consists of four copies of itself, three scaled
by a factor of 1/2 and one by 1/4. The translations are straightforward. The
parameters for the IFS generating this fractal can be presented in a table:
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
1/2
0
3/4 f
0
0
1/2
3/4 28 CHAPTER 2. ITERATED FUNCTION SYSTEMS (b) (a) (c) Figure 2.4: Fractals produced by three IFS. Each is contained in the unit square
with the origin at its lower left corner. Fractal (b) also consists of four copies of itself, but the lower right and...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.
 Fall '14
 AmandaFolsom
 Geometry, Fractal Geometry, Limits, The Land

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