# Exercises for Chaos Under Control

## Chapter 3: Some Meaning

10. (a) Using Equation (3), compute the dimension of the fractal in part (a) of the Picture: a right gasket of side length 1. Answer

(b) Compute the dimension of the fractal in part (b) of the Picture: a right gasket of side length 1.5. Answer

(c) Compute the dimension of the fractal in part (c) of the Picture: a right gasket of side length 2. Answer

(d) Compute the dimension of the right gasket of side length s for any positive number s. Answer

(e) Find IFS parameters to generate each of the fractals in parts (a) - (d).

(f) Do the E and F parameters in the IFS program have any effect on the dimension of the resulting fractal? Answer

11. How disconnected is the Cantor MTS? Here are two simple ways to get some indication of just how badly the Cantor MTS is chopped up.

(a) First, measure the length at each stage in constructing the Cantor MTS. Since the MTS is made by removing intervals, each of our measurements exceeds the length of the MTS, but we can look for a pattern and deduce the actual length. To begin, we cover the MTS with one interval of length 1. Next, we cover it with two intervals of length 1/3, so its length is less than 2/3. Continue this line of reasoning and make some conclusion about the length of the Cantor MTS. Answer

(b) Now measure the lengths of the intervals removed in forming the Cantor MTS and see what remains. For example, first we remove one interval of length 1/3, then two intervals of length 1/9, and so on. Using a calculator, add up the lengths removed (1/3 + 2/9 + ... ). What do you get for the length of the MTS? Answer

(c) Do your answers for (a) and (b) agree?

12. (a) Using Equation (3), compute the dimension of the Cantor set formed from the unit interval by removing the middle half of the interval. (Part (a) of the picture for problem 7, Chapter 2, shows this Cantor set.) Answer

(b) Compute the dimension of the Cantor set formed from the unit interval by removing the middle 2/3 of the interval. (Part (b) of the picture shows this Cantor set.) Answer

(c) Compute the dimension of the Cantor set formed from the unit interval by removing the middle 1/5 of the interval. (Part (c) of the picture shows this Cantor set.) Answer

(d) Compute the dimension of the Cantor set formed from the unit interval by removing the middle segment of length t, where t is any number 0 < t < 1. Answer

13. (a) Using Equation (3), compute the dimension of the fractal shown in part (a) of the picture for problem 34, Chapter 2 - the product of two copies of the Cantor set of 12(a). Answer

(b) Compute the dimension of the fractal shown in part (b) of this picture - the product of two copies of the Cantor set of 12(b). Answer

(c) Compute the dimension of the fractal shown in part (c) of this picture - the product of two copies of the Cantor set of 12(c). Answer

(d) Compute the dimension of the product of two copies of the Cantor set of 12(d). Answer

(e) Comparing these results with those of the corresponding parts of exercise 12, what general rule suggests itself to you? Answer

14. (a) Using Equation (3), compute the dimension of the fractal formed from the unit interval by dividing it into fifths and removing the second and fourth fifths. (See Figure (a).) Answer

(b) Compute the dimension of the fractal formed from the unit interval by dividing it into sevenths and removing the second, fourth, and sixth sevenths. (See Figure (b).) Answer

(c) Compute the dimension of the fractal formed from the unit interval by dividing it into sevenths and removing the second, third, and fifth sevenths. (See Figure (c).) Answer

(d) Considering parts (b) and (c), can you make some conclusion about existence of different fractals having the same dimension. Answer

15. Build a self-similar fractal in this way: start with the unit interval along the x-axis in the plane. Replace this line segment with two smaller segments, one with ends (0,0) and (1/2,1/2), the other with ends (1/2,1/2) and (1,0). (See Figure (b).) Now repeat this process, replacing each of these segments with two still smaller (Figure (c)), and so on.

(a) What are N and s for the resulting fractal? (Hint: to find s use the Pythagorean Theorem.) Answer

(b) Using Equation (3), compute the dimension. Answer

(c) Does the answer make sense? (Draw stages 3, 4, and 5 of the construction.)

16. What is the similarity dimension of the fractal shown in part (b) of the Figure (b)for problem 28, Chapter 2?. Explain. Answer

17. Determine the dimension of the fractal in the box. Explain. Answer

18. Determine the dimension of this fractal. Explain. Answer