Exercises for Chaos Under Control

Chapter 3: Dimensions of Physical Objects

19. The graph shows the plot of log(N) and log(1/s ) for two natural fractals, one represented by squares, the other by circles. Which has the higher box-counting dimension? Answer

20.[E] Here's a paper-crumpling experiment. Make a series of tightly wadded balls of tissue paper (very un-stiff) and of brown paper bag paper (very stiff). Before making any measurements, guess which collection has the higher dimension. What reasons can you give to support your answer? Now do the experiment. Were you right or wrong?

21.[E] Instead of crumpling paper, suppose we packed dried grains into tight clusters. Get some plastic food wrap. Weigh out different amounts of some dried grain, make tight balls using the wrap, and, in each case, measure the diameter. (Since the "balls" may be irregular you may want to find an average diameter for each ball.) Guess what the dimension of the collection of such balls will be. Do your measurements agree with your guess? (Does rice give a different dimension from lentils, peas, ... ?) Are the dimensions of paper balls and balls of grains similar? Explain why or why not.

22. (a) Consider the fractal pictured here: the product of the Cantor middle half set and the unit interval. This is not a self-similar set, so compute its box-counting dimension. (Hint: use squares of side length 1/4, then 1/16, ..., 1/4n.) Answer

(b) Do you want to reconsider your answer to part (e) of exercise 13? Answer

23. Suppose a rock is dropped onto a concrete sidewalk from a great height, causing cracks in the concrete spreading out from around the point of impact. Describe how you might measure the dimension of this pattern of cracks.

24.[E] Measure the box-counting dimension of the unit circle (by this we mean just the curve x2 + y2 = 1, not the interior). Do it in this way: draw a circle of radius about three inches on a piece of graph paper. Count the number of 1 inch x 1 inch squares touched by the circle. Then count the number of 1/2 x 1/2, the number of 1/4 x 1/4, the number of 1/8 x 1/8, and (if your patience and eyesight still hold out), the number of 1/16 x 1/16. Make a log-log plot to determine the dimension of the circle. Does the measurement agree with your expectation?

25. Describe the difference between "self-similar" and "self-affine." Is one a subclass of the other? Give examples.

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