Exercises for Chaos Under Control

Chapter 2: Collage Theorem Problems

In this set of exercises "DET" and "RAND" refer to Options in the TreenessEmerging Program.

In these problems we get practice using the Collage Theorem to find the IFS parameters producing given fractal pictures.

28. For both (a) and (b) in the picture, find a three function IFS to generate the picture. (That is, for each picture specify the R, S, Theta, Phi, E and F values for each of the three functions.) Answer

29. Describe the Collage Theorem and illustrate how it is used by referring to the picture. Answer

30. Below is a picture of a windswept shrub. Explain how one might extract IFS rules from this photograph to construct an approximate computer image of the shrub. Make a sketch illustrating the procedure.

31. Determine the IFS rules to generate this fractal. Answer

32. For each of (a), (b), (c) and (d), find a five function IFS to generate the picture. (That is, for each picture specify the R, S, Theta, Phi, E and F values for each of the five functions.) Answer

33. (a) Find a three-function IFS to produce this fractal. Answer

34. We investigate fractals resulting by performing Cantor set constructions in two directions -- we call this the product of the Cantor sets. For example, the product of two Cantor MTS is obtained by starting with the unit square, removing the vertical strip above 1/3 < x < 2/3 and removing the horizontal strip to the right of 1/3 < y < 2/3 (the first stage of the construction), then repeating this process on each of the four resulting squares, ... .

A Cantor Middle 1/2 Set is formed from the unit interval by removing the middle 1/2, leaving two intervals, each of length 1/4, then removing the middle 1/2 of both of these intervals, and continuing.

(a) Find the IFS parameters for (a)the product of two Cantor sets removing the middle 1/2. Answer

(b) Find the IFS parameters for (b)the product of two Cantor sets removing the middle 2/3.

(c) Find the IFS parameters for (c)the product of two Cantor sets removing the middle 1/5.

(e) Find the IFS parameters for the product of two Cantor sets removing the middle s, for any s, 0 < s < 1. Answer

(f) How can we use this construction to produce a solid square? Answer

A given set of rules constructs a unique picture. The converse is not true, however; a picture does not define a unique set of rules. This point is explored in problems 35 - 39.

35. In the first rule of the Eq. Gasket table, change Theta and Phi both to 120ยก and change E to 0.500. You will get the same Gasket as before when you run the program with these parameters. Can you see why? (Hint: what do the new transformations do to an equilateral triangle?) Can you find four rules, different from those in the original parameter table, that produce the same gasket?

36. In RAND the Eq. Gasket rule is an IFS code for the equilateral gasket of side length 1 and vertices at the points (0,0), (1,0), and (1/2,sqrt(3)/2).

(a) Find the IFS code for the equilateral gasket if the lower right corner (labeled B in the schematic picture) includes a 120 degree counterclockwise rotation. Answer

(b) Find the IFS code for the equilateral gasket if the upper corner (labeled C in the schematic picture) includes a 120 degree counterclockwise rotation. Answer

(c) Find the IFS code for the equilateral gasket if the lower left corner (labeled A in the schematic picture) includes a horizontal reflection. Answer

(d) Find the IFS code for the equilateral gasket if the lower right corner (labeled B in the schematic picture) includes a horizontal reflection. Answer

(e) Find the IFS code for the equilateral gasket if the upper corner (labeled C in the schematic picture) includes a horizontal reflection. Answer

37.[C] Use RAND to check your answers to parts (a) - (e) of problem 36.

38. (a) Find the IFS code for a product of two Cantor sets removing the middle 1/2, with 90 degree (counterclockwise) rotation for the lower left corner (labeled A in the schematic picture). Answer

(b) Find the IFS code for a product of two Cantor sets removing the middle 1/2, with 180 degree (counterclockwise) rotation for the lower left corner A. Answer

(c) Find the IFS code for a product of two Cantor sets removing the middle 1/2, with 270 degree (counterclockwise) rotation for the lower left corner A. Answer

(d) Find the IFS code for a product of two Cantor sets removing the middle 1/2, with vertical reflection for the lower left corner A. Answer

(e) Find the IFS code for a product of two Cantor sets removing the middle 1/2, with horizontal reflection for the lower left corner A. Answer

39.[C] Use RAND to check your answers to parts (a) - (e) of problem 38.

40.[E] Try making an IFS image of a picture of a natural object of your own choosing. Leaves (especially maple), flowers, vegetables such as broccoli and cauliflower, all make good subjects. So, sometimes, do trees. To find the IFS parameters you will have to make a covering of the original picture. You can do this approximately by hand (make rough sketches), a little more accurately using a reducing copying machine, or even more accurately using a computer "paint" program which allows variable scaling. Keep track of the scalings, translations, and rotations, enter the values in the parameter table in RAND, and run to see how well you did. (One of our students made a nine-function IFS of his face -- a reasonable likeness, but the result of a lot of trial and error. Moral: don't start with really complicated pictures.)