Show that the exponent d for the square von Koch curve is larger than the exponent for the (triangle) von Koch curve (so the square von Koch curve is more "complicated" than the triangle von Koch curve).

Let K₁₀ be the curve obtained in the 10^th stage of the von Koch curve construction. Show that the plot of log(u) vs log(1/s) for K₁₀ is composed of two straight lines, one having positive slope and one having slope zero (use s_n = (1/3)ⁿ ). The line of slope zero extends arbitrarily far to the right ( i.e., towards smaller scales). What is the slope of the straight line segment of positive slope? At what value of s does the slope change?

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.)

(4a) 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).

(4b*) 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.

Show that the fractal dimension of the Peano space-filling curve is 2. (The Peano curve is described in section 2.5.)

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 = (1/2)ⁿ, n = 0, 1, 2, 3, .... (as we did in class and as described in the weekly summary for January 21).

• (10a) 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?
  
  
• (10b) Find a covering of a subset A of the square [0,1]² in such a way that A has fractal dimension 1 and is totally disconnected (i.e., is dust). (Hint: Start with the covering of the diagonal line.)

Explain why the box counting method (Section 4.4) gives (an approximation of) the fractal dimension of a set.

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

• (12a) Rotation clockwise by 30 degrees followed by reflection through the x-axis.
• (12b) Reflection through the x-axis followed by clockwise rotation of 30 degrees.
• (12c) 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.)

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