| CHAPTER RESOURCES: PROJECTS
CHAPTER 4 |
| The Ant Walks Around Lake Tahoe In the Project for Chapter 4 you used data gathered by "walking" the shore of Lake Tahoe on a map to find an expression for the number of steps y as a function of the step size x. This expression has the form y = k/xn, where x is the step size in inches and y is the number of steps needed to walk all the way around the shore on the map. For the data given on page 393 you found that k is about 23.5 and the exponent n is about 1.153. For these activities we'll assume that the same model describes the actual shore of Lake Tahoe. We don't know the scale of the map used to generate the data for the Chapter 4 project, but since Lake Tahoe is about 20 miles long and 10 miles wide, we can guess that the map scale was about 1 inch to the mile. So using real distances the formula says  where y is the number of steps and x is now the step size in miles. ACTIVITIES 1. In the Chapter 4 project we imagined an ant walking the coastline, seeing every stone as a mountain to be scaled. Our ant starts out on a trip around Lake Tahoe, taking steps 0.1 inch in length. Use the scaling formula to determine how far the ant travels in walking all the way around. Remember that you'll have to convert the step size of 0.1 inch to miles, since x is the step size in miles. 2. The length of your stride might be around 2 feet. How far will you have to walk to go all the way around the lake? 3. The scaling formula implies that you can make the shore length as long as you want by taking small enough steps. What step size would you have to use to make the length come out to 1000 miles? 4. Suppose that instead of having the irregular shape shown in the picture on page 393 Lake Tahoe was a perfect circle. Imagine carrying out the same "compass-walking" procedure with different step sizes and building a table like the one on page 393. When you find the constants k and n, what value do you think you'll get for the exponent n? FURTHER READING ON THE WEB The British mathematician Lewis Fry Richardson was the first to investigate the relation between the scale used to measure a coastline and the resulting estimate of its length. He published a graph showing the relation between scale and length for a variety of coasts, using logarithmic scales so that the exponent in the length formula can be read from the slope of a line fitted to the data points. You'll learn more about logarithms in Chapter 5, but to take a look at Richardson's graph and to see a nice illustration of the measurement process, visit http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html For an account of Richardson and his many interests, which included the statistical analysis of wars, see T. W. Körner's wonderful book The Pleasures of Counting (Cambridge University Press). Körner quotes Richardson's description of exactly how he measured coastline lengths and provides many interesting references. Richardson's work was the starting point for the Chapter 5 of Benoit Mandelbrot's famous book The Fractal Geometry of Nature (WH Freeman, 1983), which uses an exploration of the length of the coastline of Britain measured at various scales to introduce the notion of fractal dimension. The Web site referred to above gives a few details about Mandlebrot's definition of fractal dimension and includes this quotation from Mandelbrot: "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." (Activity 4 is about an ideal circular coastline, but as Mandelbrot says, "coastlines are not circles.") There is a brief profile of Mandelbrot at http://www.research.ibm.com/research/mandelbrot2.html A good place to see a variety of fractal shapes, including fractal plants and landscapes, is http://www.mhri.edu.au/~pdb/fractals/fracintro/ |
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