Homework #1: MAT335 - Winter, 2001

• (1) Show that the binary representation (i.e., base 2) of 1/6 is .001(01) (here, (..) means the string in the brackets repeats forever).(Hint: first determine what nunmber has the binary representation .01(01). )
  Show that the ternary representation (i.e., base 3) of 5/8 is .12(12).
  Answer these in two ways; one way using a geometric series, and one way using the trick I showed in class (i.e., multiply the number by an appropriate number, etc).
• (2) Find the ternary representation of 23/27 and 7/5 to 4 places.
• (3) Indicate on a sketch of the interval [0,1] the subinterval that contains the numbers whose ternary expansion begins with .201
  Also identify the point whose ternary expansion is .11(11) on your sketch.
• (4) Use the identification between the Cantor set and [0,1] discussed in class (and in the text on page 75) to find the numbers in the Cantor set that are matched with the following numbers from [0,1]; 0, 1, 3/4, 1/8, 3/8, 1/6.
• (5) Show that every point in the Cantor set C is an accumulation point of points from the set E of end points of intervals removed during the construction of C (E = {0,1,1/3, 2/3, 1/9, 2/9, 7/9, 8/9, ...}). That is, show that for any p in C and any d>0 (small number), there is a q in E such that |p-q|< d.
  (Hints: Note that if q in E, then the ternary expansion of q ends in infinitely many zeros or infinitely many 2's (correction), and also contains no 1's. And that if x and y are two numbers in [0,1] whose ternary expansions agree to the nth place, then |x-y|<1/(3^n). )
• (6) Show that the 'Cantor middle-fifth' set has the same properties as the Cantor middle-thirds set (the one we discussed in class). So show that the Cantor middle-fifth set 1) has length 0, 2) has at least as many points as the interval [0,1], 3) is totally disconnected (i.e., contains no intervals), and 4) is self-similar. The proofs of these assertions should be similar to the proofs in class, and at the same level of mathematical rigour.
  (The construction of the Cantor middle-fifth set removes the middle fifth open interval of each subinterval that occurs at that stage. Note that in the second step 4 intervals are removed; in contrast, only 2 intervals were removed in the second step of the construction of the Cantor middle-thirds set.)
• (7) Show that the length of the Sierpinski triangle curve T is infinite.
• (8) Let l be the bottom edge of the Sierpinski triangle T. Show that arbitrarily close to any point p on l there is a branching point of T. That is, for any p in l and any d>0 (small number), there is a q in T such that |p-q|< d. Branching points are described on page 118 of the text.
  (Hint: Note that the branching points on l occur at the points 1/(2^n) for some n>=0, including the point 0. Now consider the binary expansion of such points and of the point p; this is similar to question (5) above.)
• (9) Suppose the graph of log(u) vs log(1/s) for a curve is a straight line with slope 3/2. If u(1) = 100, what is the length of this curve at the scale s = 1/4? What is the length of this curve?
• (10) (a) Can a curve of finite length have exponent d > 0? Explain.
  (b) Can a curve of infinite length have exponent d = 0? Explain.
• (11) 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).
• (12) Let K_m be the curve obtained in the mth stage of the von Koch curve construction. Show that the plot of log(u) vs log(1/s) for K_m 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 (smaller scales). What is the slope of the straight line segment of positive slope? At what value of s does the slope change?
• (13) 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.)
• (14) Sketch and describe the image that is produced by the IFS with the following blueprints;
  a) Two squares of size 1/2 placed along side of each other on the bottom edge of the intial square (this would be the blueprint labeled 'Stage 1' in Figure 5.8 except the small square on the top would be absent).
  b) Four squares of size 1/2 placed adjacent to each other in the four quadrants of the intial square (so like a) above with two more smaller squares just on top of those bottom two).
  c) Two smaller squares like in part a) but the right one rotated counter clockwise by 90 degrees. (Here just make some general comments about the structure of the final image like; How many parts does it have? Are there any 'gaps'?, etc.)
• (15) Write down the IFS for the square von Koch curve. That is, find the parameters a,b,c,d,e,f for each transformation in the IFS.
• (16) Use the program Fractal Pattern to iterate the IFS with transformations
  w_1 = (a,b,c,d) + (e,f) = (0.5, 0, 0, 0.5) + (0, 0)
  w_2 = (a,b,c,d) + (e,f) = (0.5, 0, 0, 0.5) + (0.5, 0)
  w_3 = (a,b,c,d) + (e,f) = (1, 0, 0, 1) + (0, 0)
  Note that this is a variation of the Sierpinski IFS, so you can just modify the Sirepinski data. Also, because the Sierpinski IFS actually uses an initial image of a triangle instead of a square (like the other IFS), you will see triangles instead of squares in the image.
  Iterate, say 6 or 8 times and explain the pattern you see (be sure to explain, not just describe what you see).
  If you are unable to access the program, I printed out copies of the pattern and handed them out in class on Feb. 9 (extra copies are on my office door).
• (17) Calculate the fractal dimension of the fractals produced by the IFS with blueprints;
  • (a) 5 small squares of size 1/3, 4 of the squares are located in the corners of the original (1 by 1) square and the fifth one sits in the middle (so here (a,b,c,d) = (1/3, 0, 0, 1/3) for each w_1, ..., w_5, with shifts (e, f) of (0, 0), (2/3, 0), (2/3, 2/3), (0, 2/3), and (1/3, 1/3)). (Hint: Cover the fractal with squares of size s_n = (1/3)^n )
  • (b)
    • (bi) Two squares of size 1/2 lying on the bottom edge of the original square, with no rotations (so here (a,b,c,d) = (1/2, 0, 0, 1/2) for w_1 and w_2, with shift (0,0) for w_1 and (1/2, 0) for w_2).
    • (bii) Same as part (bi) except the second 'lens' is rotated counterclockwise by 90 degrees; w_2 = (0, -1/2, 1/2, 0) and with corresponding shift to place it in the bottom right corner (this is the blueprint of question (14c) above).
    
    What general conclusion can you draw from your results from (bi) and (bii)?
    
    
    • (biii) Four small squares of size 1/2 - a/2 placed in the corners of the original square (so the distance between sides of adjacent squares is a). Note that your answer will depend on a. What do you expect the fractal dimension will be in the limit as a tends to zero? (first think about the fractal that is produced for a = 0, and then verify with your formula).
• (18) Find the fixed points for the transformations w_i that make up the IFS for the von Koch curve and the square von Koch curve. Sketch the location of the fixed points on the curves.
• (19) Calculate the Hausdorff distance h(A,B) where A is the square of size one centred at the origin, and B is the circle of radius 1/2 centred at the origin plus the point p with coordinates (1,0).
• (20) Let T_o be the solid equilateral triangle with sides of length 1, T the Sierpinski triangle, and W the IFS used to draw the Sierpinski triangle (so W^n(T_o) --> T as n --> infinity). Show that h(W^n(T_o), T) --> 0 as n--> infinity, where h(A,B) is the Hausdorff distance between the sets A and B.
• (21) 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).

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