Fractal Geometry

Cheryl Fallis, Boyd Herforth, Dale Nowlin,

Greg Somers, and Grace Williams

Abstract

Geometry has a very important role in the study of change. In algebra students have been introduced to the notion of change through simple algebraic functions. In future courses students will deepen their understanding of change through investigating logarithmic, exponential, polynomial, and other functions. Among the goals of mathematics today is the ability to understand this change both visually and numerically, and geometry is an important component in achieving that goal.

The purpose of this unit is to provide an initial experience with fractals that is rich in creativity and inquiry for geometry students and to provide an introduction into fractal geometry for teachers. This material can be taught as a four or five day unit that can be inserted into an existing curriculum. The only prerequisite skills for the unit are conceptual understandings of similarity and whole number exponents. The unit introduces fractals with activity based lessons on the following topics.

• How long is the coastline of England?

• Iteration on a segment

• Iteration with closed figures

• Iteration as a random process

• Fractal dimension

Our hope is that an early introduction to fractals from a geometric perspective will serve as a conceptual foundation for a later, more complex encounter with these topics.

How Long Is the Coast of England?

Historically, one of the early questions leading to the geometry of fractals concerned the length of the coastline of Britain. In different atlases, the value given for the length of the coastline varies widely, much more than could be attributed to the errors in measurement. Working in groups of three or four, select the map of an island and measure its perimeter several times, using a different "ruler length" for each trial. As a check, be sure that each measurement is verified independently by another person in the group, and record the measurements in a table like the one suggested in step 7 below.

Possible islands include Great Britain, Australia, the Celebes, Prince George's Island or Vancouver Island in Canada, Luzon in the Philippines, Honshu or Hokkaido in Japan.

Use the compass method of measuring your coastline:

1) Set the compass to a distance L, called the ruler unit. Choose this unit so that you can get around the island in four or five "steps."

2) Pick any point on the island as a starting position. Label it P_{0. With the} compass tip at P_0, rotate the compass counterclockwise until it intersects the coast at a point P₁.

3) Use P₁ as the new center and rotate the compass to intersect the coastline at another point. Call this point P₂. Do not change the opening of the compass.

4) Repeat step 3 as many times as needed to go almost completely around the island, so that the last distance back to P₀ is less than the compass opening.

5) To find the perimeter (the approximation to the length of the coastline) for this compass opening, multiply the length of your compass opening by the number of steps.

6) Repeat steps 1 through 5 for a sequence of decreasing "ruler lengths."

7) Make a table with column headings for compass opening (cm), number of steps around the island, and perimeter.

An excellent homework assignment that can be given as a follow up to this activity is for the students to graph the results with compass opening on the horizontal axis and perimeter on the vertical axis. They can then write a paragraph discussing their observations.

Iteration on a Segment

In this activity students will generate fractal stages that are initiated from a segment. Students can use square or isometric grid paper to sketch the first two iterations on a segment. The first iteration is performed by replacing the original line segment, called step 0 or the initiator, with four segments, each one third the length of the original segment (see Figure 1). The figure used to replace the line segment is called step 1 or the generator.

Figure 1

To sketch step 2, the next iteration is performed by replacing each of the four segments with a reduced copy of the generator. Students should conjecture as to the number of segments that would be used to draw step 3, the third iteration. A continuation of this process leads to the Koch curve. Some students may also be interested in considering changes in length of the figure and area under the figure from one stage to the next although this will be covered in greater depth in the next lesson.

At this point students should work in small groups to complete the worksheet (see the accompanying worksheet). On the worksheet students are given an initial segment length (step 0) and a generator, the result after one iteration (step 1). The students should then sketch the next two iterations, steps 2 and 3, for each problem.

Two problems from the worksheet are not self-explanatory (see Figure 2). In problem 3, the direction in which the generator is drawn is important. The direction, as seen in Figure 2, is A to B, B to C, C to B, and B to D. In problem 6 the segments added by the generator are half the length of the previous segment, i.e. CD = 1/2(AB).

Figure 2

If computers are available, students can now be shown how to use Fract-O-Graph software. Example 1 from the worksheet can be used for demonstration purposes. On Fract-O-Graph the generator is drawn on a square grid (see Figure 3) by clicking the mouse at each vertex and double clicking on the last vertex. When the mouse is clicked on the Fractify icon the computer replaces each segment with a copy of the generator. Each subsequent click on the Fractify icon results in the next step in the iteration process. The pictures can be rescaled for easier viewing.

Figure 3

Students would benefit greatly by working on the computer with a partner while using Fract-O-Graph to generate fractals. They can draw the generators from the worksheet problems and verify their drawings by fractifying on the computer. They should verbally predict the next step in the iteration process to their partner before generating it on the computer. Clicking the mouse on the Defractify icon returns to the previous step in the iteration process. The students should continue to fractify enabling them to see the result of many iterations. Problems 4 and 6 cannot be done on Fract-O-Graph because they are generated by discontinuous sketching.

Students should then be given more time to make their own fractal designs. One challenge that could be placed before them is to find a generator that produces a reasonable coastline when a segment is iterated.

Iteration on a Segment Activity

For each problem the initiator is a segment with the given length. Use the generator to determine the sketch of the next two steps.

1. Initiator length = 27 units

2. Initiator length = 27 units

3. Initiator length = 8 units

4. Initiator length = 9 units

5. Initiator length = 4 units

6. Initiator length = 8 units

7. Initiator length = 8 units

Iteration on Closed Figures

Figure 4

Consider now a closed figure along with its perimeter and area. Using the same procedures learned on the Koch curve, students will begin to construct a Koch snowflake on triangular grid paper. Let the unit of length be one side of the triangle in the grid and let the area be measured by the number of such triangles contained in the closed figure. The work will be easier if students begin with a triangle with side length twenty-seven. The iterative procedure will be to remove the middle third of each side and draw a Koch curve generator. A figure drawn in this manner with three iterations appears as Figure 5. After each iteration, record the perimeter and area of the closed figure produced. For the perimeter, encourage students to actually count units along one side of the figure until they begin to see patterns or shortcuts for counting. Table1 contains the results for three iterations.

Figure 5

Table 1

When students record the differences between successive perimeters they should see both that the perimeter is increasing and that the amount of increase is larger each time. This suggests the notion of unlimited growth in the perimeter. The ratio between perimeters with each iteration is greater than one, a fact which again suggests unlimited growth. (In advanced algebra the set of perimeter values would be referred to as a divergent geometric sequence.)

Table 2

Remember that area is measured in units of the smallest triangle on the grid paper. For the first triangle with side length twenty-seven units, finding the area without actually counting all the triangles is a mathematical exercise in itself! After that first step, help the students to realize that since each iteration changes the linear measure by a factor of 1/3, it produces a triangle with area 1/9 that of the previous triangle. This concept is normally taught in the study of similar figures. If students are not familiar with this idea, a drawing such as Figure 6 should be helpful.

Figure 6

Beginning with step 2, the students can see that each iteration adds four new triangles to each Koch curve generator. Because these new triangles have 1/9 the area of the previous step, each iteration adds 4/9 of the area added the previous time. Although the area is increasing, just as the perimeter did, the amount of increase in this case is smaller each time. The ratios between successively larger areas are also decreasing. Because the rate of change in area is decreasing, there must be some limit to the total area. If students have difficulty in grasping that concept, perhaps it would help to show them the circle which circumscribes the triangle. That circle definitely is an upper bound for the area of the snowflake; in fact, the area is much less. An advanced algebra class can look at the area as the following sum:

729 + 1*243 + 4/9*243 + 4/9*4/9*243 + 4/9*4/9*4/9*243 + . . . = 1166.4.

Table 3 shows the result of counting areas through three iterations, with an extension of the pattern to the fourth iteration.

Table 3

Many students will be amazed that the perimeter has no limit, yet the area does have a limit. An extension for this lesson would be to have them repeat the exercise for another generator which produces a closed curve. Not all generators produce the same results. For instance, using the generator in Figure 7 on each side of a square will produce a closed figure whose area never changes. The closed figure itself appears as Figure 8.

Figure 7

Figure 8

Geometric Iterations as a Random Process

An element of randomization can be incorporated into the iterative procedure that produces the Koch snowflake. Consider step 1 of the snowflake with a new triangle added to the middle third of each side producing the shape in figure 9 below. A variation on this procedure would be to subtract a triangle from each side as in figure 10. Now, to introduce some randomization to the process, suppose three coins are tossed before step 1 to determine whether each side has a triangle added to it or subtracted from it. Figure 11 below shows step 1 with triangles added to two sides and subtracted from one side. Before constructing step 2 a coin would be tossed twelve times, once for each copy of the generator being used to form that step. The toss of the coin will determine whether a triangle is added or subtracted to that portion of the snowflake, thus incorporating an element of randomness into the curve. Figure 12 shows an example of a step 2 generated in this manner. Considering the continuation of this process students might predict changes in area. If the randomness is obtained from dice or random number tables the students could vary the probabilities to say 1/3 versus 2/3. What effect might this have on the change in areas?

This same randomization process can be applied to other closed figures or segment generators. The process can be enhanced by using dice or random number generators to introduce other elements of randomness into the iterative process. For example, a tree branching iteration could be produced by using randomness to determine the distance to the next branching, the side on which the next branching occurs, and the length of the branch.

This activity also offers an excellent opportunity for students to investigate their own fractal creation. They could draw an initiator, a generator, and two or three more steps with some characteristic of the generating unit determined by probabilistic methods.

Random Iteration and Chaos

Suppose you were to determine which of three points on a triangular island you were to visit by the outcome of the roll of one die. If a one or two came up, you would spend the day on the beach near point A, if you rolled a three or four, your surfing would take place off point B, and set off toward C if a 5 or 6 came up. Now, if you were to keep track of the number of days you had spent at each beach, draw your idea of what a graph showing these numbers for A, B, and C would look like.

Now suppose the rules are changed. Starting at A, when the die is rolled the player moves half the distance to the designated vertex. (For a first roll of 1 or 2 remain on A.) Mark the position at that time. Again roll the die and move half the distance to the vertex determined by the die. Mark a new point and continue the process. Use the island below to go through 15 or 20 rolls of your die, continuing the process given above. Remember to mark each point halfway to the corner determined by the roll of the die.

Most people are surprised that a process involving so much chance would turn out to have such a clear pattern. Can you think of other examples of similar situations, where the individual events are determined by chance, but when the process is repeated many times, a strong pattern or trend becomes clear ? The shape that is appearing is called the Sierpinski triangle.

Fractal Dimension

In geometry we usually deal with objects having one, two, or three dimensions. We recognize that one-dimensional objects have length, two-dimensional objects have length and width, and three dimensional objects have length, width, and depth. (Points have zero dimension. Time is sometimes considered a fourth dimension.) Does the Koch curve have dimension 1 or dimension 2? Does the Sierpinski triangle have dimension 2 or dimension 0 (as it appears to break up into scattered points)? To decide, we will first investigate the dimensions of segments, squares, and cubes by considering reduction factors.

Suppose a segment is reduced by a factor of two (see Figure 14). It takes two copies of the reduced segment to cover the original segment. This can be expressed as an equation by saying the number of copies is equal to the reduction factor to some power which in this case is 1, i.e. 2 = 2¹. Similarly if the segment is reduced by a factor of three, three copies will be required to cover the original segment. The number of copies is equal to the reduction factor raised to the first power, i.e. 3 = 3^1.

Figure 14

If a square is reduced by a factor of two, four copies of the reduced square will cover the original (see figure 15). The number of copies is again equal to the reduction factor to some power, which in this case is two, i.e. 4 = 2². If the reduction factor for the square is changed to three, it would take nine copies of the reduced square to cover the original. The number of copies is still equal to the reduction factor to the second power (9 = 3²).

Figure 15

An excellent way to concretely model the above situations is to make a 50% reduction of a figure on a photocopy machine and see how many of the copies are required to physically cover the original figure. This should help students visualize the difference between reducing in one or two dimensions.

Now consider a cube with a reduction factor of 2. In this case eight copies of the reduced cube will fill the original cube. We get the equation 8 = 2³. If a reduction factor of three is applied to a cube, 27 copies of the reduced cube will fill the original cube. The resulting equation is 27 = 3³. The general pattern that we have seen in these one, two, and three dimensional examples is that the number of copies used is equal to the reduction factor raised to some power (N = R^D). That power is the dimension.

Figure 16

Now consider the Koch curve with the initiator and generator shown above in Figure 16. The reduction factor is 3 and the original segment is replaced by four copies of that reduced segment. In the equation N = R^D , N = 4 and R = 3 which produces the equation 4 = 3^D. Students should see that D > 1 because 3¹ = 3 and D < 2 because 3² = 9. They could then use their calculators to estimate the value of D. Successive approximations should result in an exponent of 1.26 to the nearest hundredth. Logarithms could be introduced as a means for finding this exponent , but the guess and test method on the calculator makes it possible to complete this task without the use of logarithms. Thus, before second year algebra, students can gain exposure to exponents other than integers, while expanding their concept of dimension. After calculating values of D for various fractals, students can be guided to compare them, and to observe that larger values of D belong to "rougher" fractals.

Resource Materials

Printed Material:

Jorgensen and Brown. Geometry, Houghton Mifflin Company.

Peitgen, H., Jurgens, H., and Saupe, D. Fractals for the Classroom, Part One, Introduction to Fractals and Chaos, Springer-Verlag, New York, 1992.

Peitgen, H., et al. Fractals for the Classroom: Strategic Activities Volume One, Springer-Verlag, New York, 1991.

Software:

The Chaos Game, freeware.

Caswell, Dennis. Fract-O-Graph, 1988. freeware.

Woodrow Wilson Leadership Program in Mathematics lpt@www.woodrow.org
The Woodrow Wilson National Fellowship Foundation webmaster@woodrow.org
CN 5281, Princeton NJ 08543-5281 Tel:(609)452-7007 Fax:(609)452-0066