Estimate the fractal dimension of the set {1/n}_{n=1,...infinity} = 1, 1/2, 1/3, . . . . Note that the answer is NOT 0. (An exact answer may be too difficult to find.)

Referring to the covering of [0,1]² by grids with squares of size s_n = (1/2)² (cf. Question #10 on Homework #2), find a consistent covering of a subset A of [0,1]² that has fractal dimension log3/log2 >1 and has no area (i.e., is just dust and/or lines). Explain why A has no area (you should be able to justify this mathematically).

Find the IFS's for the square von Koch curve and for the Sierpinski carpet (Fig. 2.20, page 81). That is, find the matrices and shifts that make up the affine transformations of the IFS's. Show that W(F) = F where F is the fractal.

Find the fixed points of the transformations w_i that make up the von Koch and square von Koch IFS's. Sketch the locations of the fixed points on the fractals. (As a check on your answer, the fixed points should lie on the fractal.)

Calculate the Hausdorff distance h(A, B) between the following two sets;

(5a*) A = the (solid) unit square, B is the (solid) disc of diameter 1 enscribed within the square.

(5b*) Same as above but now let B be the (solid) disc without the centre point.

(5c*) Same as in (5a) but with B = (solid) disc enscribed within the square union the point (1,0) (here we are placing the centre of the square and circle at (0,0)).

Let h(A, B) denote the Hausdorff distance between the sets A and B (see Page 268). Verify the Contraction Mapping Principle for the Sierpinski IFS:

Let T_o be the solid equilateral triangle (sides of length 1) and T be the Sierpinski triangle. Show that h(Wⁿ(T_o), T) -> 0 as n -> infinity. Here W denotes the IFS of the Sierpinski triangle. It will be easier to find a (simple) function f(n) such that h(Wⁿ(T_o), T) <= f(n) and such that f(n) -> 0 (because it may be difficult to determine exactly what h(Wⁿ(T_o), T) is). See also page 4 of the W2000 final exam for a similar question (and with diagram).

(7a) Let W be the IFS for the Cantor set (which produces the Cantor set along the x-axis between 0 and 1).

• (i) Let S = [0,1]. What is the set Wⁿ(S)? What is the limit of Wⁿ(S) as n tends to infinity?

• (ii) Let S = { (0,0), (1, 0) }. What is the set Wⁿ(S)? What is the set Wⁿ(S) as n tends to infinity? How do you reconcile this with the fact that W(C) = C (that is, that the Cantor set C is the fixed point of W) and the Contraction Mapping Principle?

(7b) Let W be the IFS for the Sierpinski triangle. Let S={(0,0), (1,0), (0.5, 1)}. What is Wⁿ(S)? Let TT be the set lim_{n-> infinity}Wⁿ(S). What is the set TT? Is this the same set we obtained by the "removing the middle 1/2 triangle" construction? (the latter is what we've been calling the Sierpinski triangle T) Verify that W(TT) = TT and W(T) = T. Does this contradict the Contraction Mapping Principle (which states that any contraction has only one fixed point)?

Starting with a square of size 500 by 500 pixels, and assuming your computer can draw 10,000 squares per second, estimate how long your computer will take to draw the von Koch and the square von Koch curves (see page 260 in the text).

What is the fractal dimension of the set of numbers in [0,1] that have no occurence of the string '73' in their decimal expansion? (This a hard one - I think, so even just and estimate (like "the fractal dimension is greater than y but less than x") would be something.)

End of Homework #3

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