Topics in Mathematics in a Fractal Class

Coordinate Systems

Instead of a teacher telling students to learn about a given set of axes and its subsequent coordinate system, fractals actually encourage students to ask the teacher about these systems. This stems from the students' preliminary searches through the set to discover attractive or bizarre pictures. Many students want to tell their friends where to look in the set to find certain patterns, or they want to record their discovery so they can return to them later. These desires lead them to ask how they can accomplish these goals. Once they learn that they can pinpoint specific areas with two numbers and a magnification, they begin to eagerly exchange information with their classmates using a coordinate system. Students are also fascinated to learn that they can plot their point using an angle and a distance as opposed to using two points. Thus, a polar coordinate system does not have to be abstract and baseless when students can immediately use it as a tool to accomplish a desired task. There is no need to complicate this concept with the fact that they are on the complex plane, as opposed to the real plane, at least on the elementary level. The important outcome of this process is that the students begin to learn how to isolate positions on an axis system.

 

Negative Numbers, Scientific Notation

Two other concepts also seem to fall naturally into place at this point. First, students immediately notice that some numbers are negative while others are not. They want to learn why, which leads to the notion of positive and negative positions on a graph. Second, students quickly learn decimal placement because they have the need and desire to do so in order to continue their explorations. This is because the computer uses scientific notation to report numbers greater or equal to ten and less than one (due to the way it was programmed). For example, 0.071583 may be shown as 7.158364e-2. If the students ignore the exponent they miscommunicate their information and want to know why.

 

Simple Arithmetic

Teachers can use children's enjoyment of zooming in and around the Mandelbrot set to make games utilizing counting methods such as simple addition and multiplication. If we look at the original set as being composed of a body (the main cardioid), a head (the next largest lobe to the left of the body), two arms (the third largest lobes above and below the body), and numerous other lobes around the edges of the set, we can find consistent relationships between the positions of the lobes and their periods. The period just refers to the number of positions a point in the set will ultimately touch or orbit after an infinite number of iterations.

Figure 1a. Selected points in the interior of the "body" yield different paths and periods.

For example, any starting point within the main body will eventually settle down with a period of one (figure 1a).

Figure 1b. A point chosen from the "head".

Figure 1c. The "arm" is a period 3 lobe.

Any initial starting point inside the head orbits between two final points and the two arms each have a period of three, as seen in figures 1b and 1c. Now if we look at the largest lobe laying along the edge of the body between the head and an arm, we find that it has a period of five (figure 1d).

Figure 1d. A 5 period lobe.

Likewise, the largest lobe between the three period lobe and the five period lobe has a period of eight (figure 1e). This addition pattern continues infinitely around the edge of the main body, as long as we select the largest lobe between the two previously largest lobes.

Figure 1e. An 8 period lobe, the largest lobe between 3 and 5.

 

Multiplication

Multiplication properties come into play once we begin to travel away from the main body and explore the other lobes as main bodies themselves. If we zoom in on a lobe of period five from the previous process, we can re-identify the lobe as a body, head, two arms, and subsequent smaller lobes similar (actually quasi self-similar) to the main Mandelbrot set picture. It turns out that the arm of this lobe has a period of 15, which is just the period of the parent lobe (five) times the relative position of the lobe containing the initial point (three) selected. We have to remember that bodies have a relative period of one, heads have a relative period of two, arms have a relative period of three, the next largest lobe between the head and an arm has a relative period of five (using our addition properties), and so forth. For example, if we go back and examine the main arm of the Mandelbrot set, which we have already determined to have a period of three, we should notice that its arm (or the arm of the arm of the main set) has a period of nine. This is easy to calculate because the period of all arms is just a multiple of three, and we are looking at the arm of an arm. Likewise, if we look at the head of this arm we should find that its period is six, since heads are all multiples of two, arms are multiples of three, and we are only one level deep.

 

Distributive Law

Now that we know we can use addition to figure the periods of in-between lobes and multiplication to figure the period of lobes-on-lobes, it should be pointed out that the distributive law holds as well. For example, let us look at the largest lobe between the head and arm from the original set. Because the head has a period of two, and the arm has a period of three, we know that our chosen lobe has a period of five. Now if we look at this lobe's head we know its period is 10 because heads are figured by multiplying two by the body lobe (in this case, five). In other words, 5 x 2 = 10. Likewise, this lobe's arm has a period of 5 x 3 = 15. So what is the period of the largest lobe between the head and arm of this period five lobe (figure 2)?

Figure 2. A Period 5 lobe and some of its adjoining lobes.

Just like the main Mandelbrot set, we can think of this picture as containing a body, head (the largest lobe stemming from the body), arm (the next largest lobe), and smaller lobes. The head is always twice the period of the body. The arm's period is always three times that of the body. With this pattern, we can find the period of the mystery lobe in two different ways, as seen below.

There are two ways to figure it out. We could add 10 and 15, since we are now looking at the largest lobe between the period 10 and period 15 lobes. This can also be written as (5 x 2) + (5 x 3), since this is the full equation starting from the period five lobe. Or we could multiply five and five, since it is a period five lobe off of another period five lobe. Using the main period five lobe as a base, this can also be written as 5 x (2 + 3), since we are looking at the largest lobe between a head (with a relative period of 2) and an arm (with a relative period of 3). The point of this is that by using the Mandelbrot set as a reference frame, students can easily learn that (5 x 2) + (5 x 3) = 5 x (2 + 3) because they can actually follow pictorial paths and envision relationships instead of struggling with abstract numerical manipulations devoid of concrete meaning.

 


Copyright 1993, University of Illinois Board of Trustees
National Center for Supercomputing Applications, Education Group
rpanoff@ncsa.uiuc.edu and jgasaway@ncsa.uiuc.edu