Unformatted text preview: re squares of
√
side length 2−k , hence of diameter 2 · 2−k . For the study of the random IFS
algorithm, the central fact is that the diameters decrease with increasing address
length.
Suppose we specify a resolution ǫ > 0 and want to render A to within
this resolution. That is, we want to approximate A by a ﬁnite set of points
V = {v1 , v2 , ..., vn } having the property that for every point w ∈ A, there is
√
some vj with d(w, vj ) < ǫ. Simply take k large enough that 2 · 2−k < ǫ. Then
construct V so it contains at least one point in each Ai1 i2 ...ik . For any point
w ∈ A, there is some Ai1 i2 ...ik with w ∈ Ai1 i2 ...ik . Then some vj ∈ Ai1 i2 ...ik , ad
√
so d(w, vj ) ≤ diam(Ai1 i2 ...ik ) = 2 · 2−k < ǫ.
Now of course these sets are not disjoint, points on some boundaries have
multiple addresses. This is another manifestation of the phenomenon whose
more familiar example is the nonuniqueness of decimal expansions: 1/10 can be
written as 0.1000... and as 0.09999.... (The geometric series 9/(102 ) + 9/(103 ) +
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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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