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Unformatted text preview: 38 CHAPTER 2. ITERATED FUNCTION SYSTEMS
√
Similar calculations give h(K, Kn ) = ( 3/6) · 3−n , and so
lim h(K, Kn ) = 0. n→∞ This is the sense in which the sequence K0 , K1 , K2 , ... converges to the Koch
curve K.
We end our examples with one that has a surprising consequence, the source
of considerable worry in the foundations of mathematics in the late 19th and
early 20th centuries.
Example 2.2.3 Convergence in the Hausdorﬀ distance: spaceﬁlling curves.
For now we consider just the sequence of images. Details of the construction,
given in Example 4.7.3, require an elaboration of IFS to hierarchical IFS, presented in Sect. 4.7. Starting with C0 , the line segment from (0, 0 to (1, 0),
successive generations C1 through C6 are shown in Fig. 2.19. Certainly, the
limit of this sequence of curves appears to ﬁll the unit square. Can we make
this precise? First, note the corner points of the Ci :
C1 : {(0, 0), (1, 0), (0, 1), (1, 1)} =
C2 : ij
,
30 30 ij
,
31 31 : 0 ≤ i ≤ 31 , 0 ≤ j ≤ 31 ij
,
3k 3k : 0 ≤ i ≤ 30 , 0 ≤ j ≤ 30 : 0 ≤ i ≤ 3k , 0 ≤ j...
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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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