# Exercises for Chaos Under Control

## Chapter 3: The Importance of Being Fractal in the real World

26. The following data are for two identically constructed objects. What is the mass dimension of all objects constructed in exactly the same way? Explain. Answer
Average length Mass
(inches) (ounces)
1 2
2 16

27.[E] Make a copy of fine graph paper on a overhead foil. Use this as one of the sheets in the "blob of goo between two sheets of plastic" experiment described in this section. Put a small blob of goo on this sheet, cover with a plain overhead transparency foil, and press firmly on the top foil until the small blob is squeezed out to a large, thin blob. Pull the foils apart, producing a fingery pattern. Do box counting to estimate the average dimension of the boundary of pattern. Try the experiment with different pulling speeds. Does the dimension depend on the speed with which the sheets are separated? Also, try different "goos." Does the dimension depend on the material used?

28.[E] Make a notch in the middle of one side of a piece of typing paper. Hold the sides of the paper firmly and pull them apart (in the plane of the paper) until the paper tears. The tear edge will be jagged and fractal-like. Overlay the torn edge on a piece of graph paper and estimate the dimension of the edge by box counting. Does the dimension depend on the speed with which the paper is torn? Try tissue paper instead of typing paper. Does the character of the edge depend on the structure of the paper?

29. In a sense, "the bigger is a DLA, the less there is of it." Explain.

30. A natural fractal has a dimension of 1.2. It is most likely a (a) cloud, (b) mountain range, (c) diffusion limited aggregate, (d) river bed. Explain. Answer

31. A natural fractal has a dimension of 3.3. It is most likely a (a) cloud, (b) mountain range, (c) diffusion limited aggregate, (d) river bed. Explain. Answer

32. A sack of dried peas has a dimension of 3. That is because the peas (a) pack so tightly that there are no holes, (b) pack so uniformly that the holes between the peas are roughly all the same size, (c) pack so irregularly that the holes between the peas are of many different sizes, (d) individually each have dimension 3. Explain. Answer

33. Wads of paper have a smallest possible dimension and a largest possible dimension. What are they? Describe the internal structures of paper wads with dimension close to the smallest value and, alternatively, close to the largest value. Answer

34. The graph shows actual data taken from an attempt to measure the dimension of some real fractal objects. Determine, approximately, the dimension of these objects. Are they most likely tear tracks (like coastlines), DLAs, or wadded paper balls? Explain. Answer