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Ck+1 : Figure 2.19: The ﬁrst six stages, C1 through C6 , of the Hilbert curve.
So certainly no point of the ﬁlled-in unit square S is farther than ǫk+1 =
3−k 2/2 from a point of Ck+1 , so
S ⊆ (Ck+1 )ǫk+1 2.2. THE HAUSDORFF METRIC 39 Of course, for each k ,
Ck+1 ⊆ S = S0
h(S, Ck+1 ) ≤ ǫk+1
Once we know that C∞ is a curve (Prob. 2.2.7), and that h is a metric (Prop.
2.2.1), hence positive-deﬁnite, then the observation
lim h(S, Ck+1 ) = 0 k→∞ shows that the curve C∞ is the ﬁlled-in unit square. That is, C∞ : [0, 1] →
[0, 1] × [0, 1] is onto, and so the name space-ﬁlling curve is sensible.
Space-ﬁlling curves provided an example of an unexpected complication in
mathematics. They are continuous functions from a 1-dimensional set (the
unit interval) onto a 2-dimensional set (the unit square). That is, continuous
functions need not preserve dimensions. In Sect. 3.13, we see that (some notions
of) dimensions are preserved under a more restrictive class of functions.
To establish the convergence result for all IFS, we must take these steps:
1. Show the Hausdorﬀ dist...
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