Show that the fractal dimension D for the set of rational numbers in the interval [0,1] is 1. (Hint: Use the fact that between any two numbers is a rational and an irrational number.)

What is the fractal dimension of the set {1, 1/2, 1/3, .... }? (This is hard!)

(3a) Show that the fractal dimension D = [log(4)]/[log(5)] = .861 for the Cantor middle 1/5th set (see Question #5 of Homework #1 for a description of the Cantor middle 1/5th set). Since D = .631 for the Cantor middle 1/3rd set, we see that the Cantor middle 1/5th set is more 'complicated' that the Cantor middle 1/3rd set; the points of the Cantor middle 1/5th set are more spread out over [0,1] than the points of the Cantor middle 1/3rd set - or in other words, the gaps (i.e., the intervals removed) in the Cantor middle 1/5th set are smaller than the gaps in the Cantor middle 1/3rd set (even though a total length of 1 is removed from the interval [0,1] in both cases).

(3b*) Let m=2k+1 be an odd positive integer and let C_m be the Cantor middle 1/mth set, i.e., C_m is the set obtained by removing from [0,1] the middle 1/mth interval at each stage (similar to the procedure for obtaining the Cantor middle 1/3rds set). Show that the fractal dimension dimension D = [log(2k)]/[log(2k+1)] for C_m. Thus, as m (and so k) tends to infinity, D tends to one. Comment on this fact (it is helpful to compare this to the set of rational numbers computed in question 3).

Compute the fractal dimension of the Menger Sponge (Figure 2.43). (Answer: 2.727....)

Show that for curves, D = d+1 where D is the fractal dimension of the curve and d is the exponent obtained from the length vs scale measurements.

Explain why if A is a subset of B, then D_A < = D_B. Here, D_A and D_B are the fractal dimensions of A and B respectively.

(Thus, if A is a subset of the interval [0,1] then D_A < = 1, and if A is a subset of the square [0,1]² then D_A < = 2, and if A is a subset of the cube [0,1]³ then D_A < = 3. This also holds if A is a subset of a regular curve, a regular area, or a regular volume.)

Calculate the fractal dimension of the Cantor set using (a) (2-dimensional) squares to cover it, and (b) (3-dimensional) cubes to cover it. Calculate the fractal dimension of the Sierpinski triangle using a 3-dimensional covering.

This question refers to the square [0,1]² and the covering of it with squares of size s_n = (1/3)ⁿ (similar to what we did in class and as described in the weekly summary for January 20).

• (8a*) Sketch the first two steps in the covering of a set with dimension D=log 4/log 3.
  
  
• (8b*) Suppose some subset A of the square requires a(n) squares of size (1/2)ⁿ to cover itself for each n and that A has fractal dimension D . If a set B requires 2n*a(n) squares to cover itself for each n, does B necessarily have a larger fractal dimension than A? What if kn*a(n), k any positive integer, squares were needed at each n to cover B?
  
  
• (8c) Can you find a covering of a subset A of the square [0,1]² in such a way that A has fractal dimension greater than 1 and is totally disconnected (i.e., is dust)?

Show that the fractal dimension of any piece of the Cantor set C is the same as the fractal dimension of the whole Cantor set. In particular, calculate the fractal dimension of the sets C_{a1 a2 ... an} as described in the summary for 13 Jan (self_cantor).
(Fractal dimension is independent of the size of the set.)

Explain why the box counting method (Section 4.4) gives (an approximation of) the fractal dimension of a set. (Does the box counting method at a particular scale give the best, i.e., minimum, covering at that scale?)

Write out the matrices A for the following transformations (see pages 234-237 in the text);

• (11a) Rotation clockwise by 30 degrees followed by reflection through the x-axis.
• (11b) Reflection through the x-axis followed by clockwise rotation of 30 degrees.
• (11c) Rotation counterclockwise by pi/4 radians, followed by reflection through the line y = -x, followed by clockwise rotation by pi/6 radians.

Sketch the action of each of these transformation on the square with vertices at (3,1), (4,1), (3,2), (4,2).

Show that the set A produced by iterating an IFS that has k lenses, each of which shrinks by a factor of r (0 < r < 1 ) and whose images do not overlap, has fractal dimension log k / log (1/r). What happens if the images produced by the lenses do overlap? Can you estimate the fractal dimension of A if the lenses have different reduction factors r and the images produced by each lens do not overlap? (These last two questions are difficult; see page 271-272 for some discussion.)

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;

(15a) A is a (solid) square with diagonal of length d that is enscribed within the (solid) disc B of diameter d. (The centres of the square and disc coincide.)

(15b) Same as above but now remove a line from B that cuts it in half (so the new set B is a solid disc minus a line cutting through it).

(15c) A is the same square as in (15a) but now B is the (same) disc union the point (d,0). (We are placing the centres of the square and disc 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 (so that Wⁿ( T_o ) converges to T ). 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).

(17a) 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?

(17b) 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)?

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