FractionalGeometry-Chap2

# The next example shows that we can get adequate

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 0 ) ∪ T3 (A0 ) ∪ T4 (A0 ) = T (A0 ), and continue with An+1 = T (An ). Pictured in Fig. 2.17 are A0 , A1 , A2 , and A3 . Figure 2.17: Iterates converging to the product of two Cantor sets. The containments A0 ⊃ A1 ⊃ A2 ⊃ · · · ⊃ A 37 2.2. THE HAUSDORFF METRIC are clear. Moreover, it is not diﬃcult to see A ⊂ (A0 )0 and A0 ⊂ A√2/4 , A ⊂ (A1 )0 √ 2/4 √ so h(A1 , A) = 2/42 and A1 ⊂ A√2/42 , so h(A0 , A) = ... A ⊂ (An )0 and An ⊂ A√2/4n+1 , so h(An , A) = √ 2/4n+1 Consequently, limn→∞ h(A, An ) = 0. This is the sense in which the sequence A0 , A1 , · · · converges to the fractal A. This calculation is easy. The next example shows that we can get adequate estimates in situations with geometry suﬃciently complex to make soberingly diﬃcult (and unnecessary) the exact calculation of Hausdorﬀ distance. Example 2.2.2 Convergence in the Hausdorﬀ distance: the Koch curve The Koch curve K is generated by this IFS r 1/3 1/3 1/3 1/3 s 1/3 1/3 1/3 1/3 θ 0 π/3 −π/3 0 ϕ 0 π/3 −π/3 0 e 0 1/3 1/2 2/3 f 0 0 √ 3...
View Full Document

Ask a homework question - tutors are online