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Unformatted text preview: middle ninth, leaving eight 1/9 × 1/9
squares, then removing the middle ninth of these, and continuing.
(a) Find IFS rules to generate the Sierpinsk carpet.
(b) Find the minimum ǫ for which Oǫ is simply-connected, that is, has no holes.
(c) Find the minimum ǫ for which Oǫ has exactly 1 hole.
(d) Find the minimum ǫ for which Oǫ has exactly 1 + 8 holes.
(e) For every nonnegative integer k , ﬁnd the minimum ǫ for which Oǫ has exactly
1 + 8 + 82 + · · · + 8k holes. Figure 2.20: The Sierpinski carpet, subject of Prxs 2.2.2.
Practice problem solutions
PrxsSol 2.2.1 (a) For every ǫ > 0, and for each rational q , there is an irrational
within ǫ of q , and for each irrational i, there is a rational within ǫ of i. (This is
a simple consequence of the fact that both the rationals and the irrationals are
densein [0, 1].) That is, for every ǫ > 0
Q ⊆ Iǫ and I ⊆ Qǫ and so h(Q, I ) ≤ ǫ for all ǫ > 0. That is, h(Q, I ) = 0.
(b) The result of (a) certainly appears to contradict positive-deﬁniteness. We
have two sets, Q and I , which...
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