The term fractal (derived from the Latin adjective fracus meaning “to break”) was introd...

The term fractal (derived from the Latin adjective fracus meaning “to break”) was introduced by Benoit Mandelbrot (b. 1924) to refer to objects with a broken and irregular appearance. One of the most complex and beautiful fractals is the Mandelbrot set shown next.

Mandelbrot’s definition of a fractal is too technical for this book, but a hallmark of fractals is a property called self-similarity as illustrated next. Consider the following fractal, which is called the Sierpinski gasket.

If we magnify one portion of the Sierpinski gasket, as shown next, it looks just the same as the original. Magnifying again yields the same results. No matter how the figure is magnified, we will see the original figure. Because of this property, the Sierpinski gasket is said to be self-similar.

We now describe how the Sierpinski gasket is constructed. (Other common names for the Sierpinski gasket are the Sierpinski triangle and the Sierpinski sieve.) The Sierpinski gasket is constructed using an infinite sequence of operations. Begin with one equilateral triangle, labeled S₀. Next divide S₀ into four identical equilateral triangles and remove the middle triangle. The second stage of this infinite construction process is labeled S₁.

At each subsequent stage of the construction, every black triangle in the collection is divided into four identical equilateral triangles, and the middle triangle in each is removed. The next two stages of the construction are shown as here.

The Sierpinski gasket is the figure formed by repeating this process indefinitely. The following four problems show different beginning figures, which lead to other fractals.

The Sierpinski gasket is a mathematical object as opposed to an object found in nature. The current popular interest in fractals is motivated by the surprising fact that fractals often occur in nature. A good example of a naturally occurring fractal is the coastline of Britain. You would probably expect that the coastline of Britain has some welldefined length. The length is usually computed by taking a map of Britain and measuring a path along the coast using a straightedge marked to a certain length. To get a more precise estimate of the length of the coastline you might use a larger map and a straightedge marked to a smaller length.

In 1967, Mandelbrot seriously investigated the question “How long is the coast of Britain?”

Mandelbrot discovered that the answer depends on the size of the map and the length used to do the measurement. If a straightedge marked to a length of 200 miles (based on the map scale) is used, some inlets and details will be overlooked, as shown in the following figure. The approximate coastline length in that case is 1600 miles. However, if a straightedge marked to a length of 10 miles is used, more of the jagged coast line is measured. Thus, the approximate length of the coastline increases to approximately 5184 miles.

If you look at a map, you will see that the coastline of Britain is jagged, and it is jagged at all scales. Such jaggedness at all scales is one indication that you have found a natural fractal.

a. Using a map and a marked straightedge, which border will probably be estimated more accurately, the border of Illinois or the border of Iowa? Explain your reasoning.

b. Using a road map for the state in which you live and a marked straightedge, find the approximate length of the border. Use the map scale to mark off 200 miles on the straightedge. Use that length to approximate the length of the border. Then, use the map scale to mark off 100 miles on the straightedge. Use that length to approximate the length of the border.