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Unformatted text preview: tion of a portion of the middle. In our minds, this process can be
continued forever. Figure 2.1: Successive magniﬁcations of portions of the gasket.
The reappearance, under increasingly high magniﬁcation, of copies of the
whole shape, suggests two things. First, some sort of limiting process is involved here. Second, it is not the limiting process familiar from calculus, where
under magniﬁcation a smooth curve approaches its tangent line. Something different is happening here: the level of complexity remains about constant under
magniﬁcation. So we must determine what sort of limit produces fractals.
25 26 CHAPTER 2. ITERATED FUNCTION SYSTEMS 2.1 Iterated function system formalism Based on work of Mandelbrot  and Hutchinson , and popularized by
Barnsley , iterated function systems are a formalism for generating fractals
and for compressing images. First we recall some background on transformations
of the plane.
In the plane, a general linear transformation plus translation can be writ...
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