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 first two stages of the Cantor set are shown, in which a line segment is divided into three equal segments and the middle third is removed.

a. Construct the next three stages of the Cantor set.

b. If the length of the line segment in stage 0 is 81 units, find the length of each segment in each stage and the combined lengths of the segments in each of the five stages.