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Unformatted text preview: act sets A, B , and C , we show h(A, B )+
h(B, C ) ≥ h(A, C ). Write h(A, B ) = δ1 and h(B, C ) = δ2 .
A ⊆ Bδ1 (2.4) B ⊆ Cδ2 (2.5) and
From Eq (2.5), we see Bδ1 ⊆ (Cδ2 )δ1 ⊆ Cδ1 +δ2 , where the second containment
follows from Lemma 2.2.1. Together with Eq (2.4), this shows A ⊆ Cδ1 +δ2 . A
similar argument gives C ⊆ Aδ1 +δ2 , from which we see h(A, C ) ≤ δ1 + δ2 =
h(A, B ) + h(B, C ).
In the next section, we use the Hausdorﬀ metric to show that the deterministic IFS algorithm generates a sequence of compact sets converging to a unique
limit, the attractor of the IFS.
Practice problems
Prxs 2.2.1 (a) Find the Hausdorﬀ distance between Q, the set of rational numbers in [0, 1], and I , the set of irrational numbers in [0, 1].
(b) Interpret the answer to (a) in terms ofthe positivedefninteness of the Hausdorﬀ metric. 41 2.2. THE HAUSDORFF METRIC Prxs 2.2.2 In Fig. 2.20 we see the Sierpinski carpet O, constructed from the
ﬁlledin unit square by removing the...
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 Fall '14
 AmandaFolsom
 Geometry, Fractal Geometry, Limits, The Land

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