8.7 Non-Euclidean Geometry and Fractal Geometry 531 Learning Catalytics Keyword: Angel-SOM-8.7 (See Preface for additional details.) Step 1 Step 2 Step 3 Step 4 Figure 8.104 Figure 8.105 The word fractal (from the Latin word fractus, “broken up, fragmented”) was first used in the mid-1970s by mathematician Benoit Mandelbrot to describe shapes that had several common characteristics, including some form of “self-similarity,” as will be seen shortly in the Koch snowflake. Fractals are developed by applying the same rule over and over again, with the end point of each simple step becoming the starting point for the next step, in a process called recursion. Using the recursive process, we will develop a famous fractal called the Koch snowflake named after Niels Fabian Helge von Koch, a Swedish mathematician who first discovered its remarkable characteristics. The Koch snowflake illustrates a property of all fractals called self-similarity; that is, each smaller piece of the curve resembles the whole curve. To develop the Koch snowflake: 1. Start with an equilateral triangle (Step 1, Fig. 8.104). 2. Whenever you see an edge replace it with (Steps 2– 4). What is the perimeter of the snowflake in Fig. 8.104, and what is its area? A portion of the boundary of the Koch snowflake known as the Koch curve, or the snowflake curve, is represented in Fig. 8.105. The Koch curve consists of infinitely many pieces of the form . It can be shown that after each step, the perimeter is 4 3 times the perimeter of the previous step. Therefore, the Koch snowflake has an infinite perimeter. It can be shown that the area of the snowflake is 1.6 times the area of the starting equilateral triangle. Thus, the area of the snowflake is finite. The Koch snowflake has a finite area enclosed by an infinite boundary! This fact may seem difficult to accept, but it is true. However, the Koch snowflake, like other fractals, is not an everyday run-ofthe-mill geometric shape. Let us look at a few more fractals made using the recursive process. We will now construct what is known as a fractal tree. Start with a tree trunk (Fig. 8.106a). Draw two branches, each one a bit smaller than the trunk (Fig. 8.106b). Draw two branches from each of those branches, and continue; see Fig. 8.106(c) and (d). Ideally, we continue the process forever.
RkJQdWJsaXNoZXIy NjM5ODQ=