Mathematics 350 Section 6.3 Introduction to Fractals A fractal is generally "a rough or fragmented geometric shape that is self-similar, which means it can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole." The term fractal was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." Mathematical fractals are usually based on one or more simple processes or equations that are applied recursively. This means that there is a starting figure or situation, the process is applied to it, then applied to the out put of the first application, then applied to the output of the second application, and so on infinitely many times. Classical fractals began to be studied by mathematicians over 100 years ago. But during the 1970s methods were developed to display them using a computer, and striking images appeared. Also, Mandelbrot began to champion them as a way of describing the geometry of many objects in the real world that were not smooth like the lines and circles of Euclidean geometry. First we will describe the process for creating a classical fractal called the Sierpinski Triangle. Start with the region enclosed by an equilateral triangle. This triangle is called the Stage 0 figure. Perform two steps: 1. Locate the midpoints of the sides of the Stage 0 triangle, and connect them with line segments. This divides the region inside the Stage 0 triangle into four congruent subregions bounded by equilateral triangles. 2. Remove the middle triangular region (this is usually done by coloring it dark), leaving three triangular regions touching only at vertices. You have now completed Stage 1 of the process and arrived at the Stage 1 figure, as shown on the next page.
Stage 1 Thereafter, at each stage steps 1 and 2 on the first page are applied to all the triangular regions remaining after the previous stage. Classwork 1 Create Stage 2 by applying the Process to the result of Stage 1 shown below. (The shaded triangle has been removed.)
Classwork 2 Create Stage 3 by applying the process to the result of Stage 2 shown below. (The shaded triangles have been removed.) The Sierpinski Triangle is what remains after completing an infinite number of stages. After a few stages the changes caused by the next stage cannot be seen by the naked eye, so an excellent approximation to the Sierpinski Triangle can be created by stopping after the first few (usually about 5-8) stages of the construction procedure. Because the instructions at each stage depend on the result of the previous stage, this is called a recursive procedure. (Compound interest is also a recursive procedure. Assuming you don t deposit or withdraw money from your account, your balance during the next period will be (1 + interest rate) x balance at end of current period. So the next balance is always computed from the current one.) It is important to note that the triangles that appear at a given stage are similar to, but at 1/2 scale of the triangles that appeared at the previous stage. Classwork 3 Complete the entries in the table on the next page. To make the patterns in the table more apparent, it is assumed that the sides of the Stage 0 triangle are each 1 unit long. Because the computation of the area of such a triangle in square units would involve 2 and obscure the patterns in the area columns, we have used the area of the Stage 1 triangle as a (nonstandard) unit of area and calculated the other entries as fractions of that area. The entries at a given stage may be calculated using the idea of scaling factor: a triangle that appears at a particular stage is 1 2 2 the side length and 1 2 the area of a triangle that appeared at the previous stage.
Stage Number of Triangles Perimeter of a Triangle in units Sum of perimeters of all triangles (in units ) Area of each triangle as a fraction of Stage 0 area Total area as a fraction of Stage 0 area 0 1 3 3 1 1 1 2 3 If you need it, here is the Stage 3 Triangle: Stage 3
Classwork 4 Answer the following questions about your table: 1. What pattern do you see in the numbers in the Number of triangles column? What should the entry be for Stage 4? For Stage n? 2. What pattern do you see in the numbers in the Perimeter of a triangle column? What should the entry be for Stage 4? For Stage n? 3. What pattern do you see in the numbers in the Sum of perimeters column? What should the entry be for Stage 4? For Stage n? 4. What pattern do you see in the numbers in the Area of each triangle column? What should the entry be for Stage 4? For Stage n? 5. What pattern do you see in the numbers in the Total area column? What should the entry be for Stage 4? For Stage n? 6. The picture at any Stage is just an approximation to the Sierpinski Triangle. The actual Sierpinski Triangle is what results if you do this process of middle triangle removing forever. More precisely, the Sierpinski Triangle is the limit of the process. What do you think should be the perimeter of the Sierpinski Triangle (i. e. of the limit of the perimeters of the Stages)? Why? 7. What should be the area of the Sierpinski Triangle (i.e. the limit of the areas of the Stages)?
The Sierpinski triangle is self-similar. That means that if we look at a small piece of the object under a microscope, it will look like the original object. For example, at left below is a Sierpinski triangle with 5 stages completed. At right is what we get by blowing up one of the stage 3 triangles in it. The two figures are similar. Many things in nature are structured like fractals, such as coastlines, clouds, plants, mountain ranges. Much of the animated versions of these that appear in the movies such as Star Wars have been generated from more advanced fractal techniques.
Fractal Dimension A point has no dimensions - no length, no width, no height. A That dot is obviously way too big to really represent a point. But we'll live with it, if we all just agree what a point really is. A line has one dimension - length. It has no width and no height, but infinite length. L Again, this model of a line is really not very good, but until we learn how to draw a line with 0 width and infinite length, it'll have to do. A plane has two dimensions - length and width, no depth. It's an absolutely flat tabletop extending out both ways to infinity. Space, a huge empty box, has three dimensions, length, width, and depth, extending to infinity in all three directions. Obviously this isn't a good representation of 3-D. Besides its size, it's just a hexagon drawn to fool you into thinking it's a box. Fractals can have fractional (or fractal) dimension. A fractal might have dimension of 1.6 or 2.4. How could that be? Let's investigate. Just as the images above weren't very good pictures of a point, line, plane, or space, the drawing meant to be the Sierpinski Triangle has limitations. Remember that fractals are really formed by infinitely many steps. So there are infinitely many smaller and smaller triangles inside the figure, and infinitely many holes (the black triangles that were removed). Let's look further at what we mean by dimension. Take a self-similar figure like a line segment, and double its length. Doubling the length gives two copies of the original segment. 1 1 1
Take another self-similar figure, this time a square 1 unit by 1 unit. Now multiply the length and width by 2. How many copies of the original size square do you get? Doubling the sides gives four copies. 1 1 1 Take a 1 by 1 by 1 cube and double its length, width, and height. How many copies of the original size cube do you get? Doubling the side gives eight copies. Let's organize our information into a table. Figure Line segment Square Cube Dimension 1 2 3 Number of Copies When scale doubled 2 = 2 1 4 = 2 2 8 = 2 3 Do you see a pattern? It appears that the dimension is the exponent - and it is! So when we double the scale and get a similar figure, we write the number of copies as a power of 2 and the exponent will be the dimension. Let's add that as a row to the table.
Figure Line segment Square Cube Self-similar using n copies Dimension 1 2 3 d Number of Copies When scale doubled 2 = 2 1 4 = 2 2 8 = 2 3 n = 2 d We can use this to figure out the dimension of the Sierpinski Triangle because when you double the length of the sides, you get another Sierpinski Triangle similar to the first. Start with a Sierpinski triangle of 1-inch sides. Double the length of the sides. Now how many copies of the original triangle do you have? (Remember that the black triangles are holes, so we can't count them.) Doubling the scale of the Sierpinski Triangle gives us 3 copies, so 3 = 2 d, where d = the dimension. But wait, 2 = 2 1, and 4 = 2 2, so what number could this be? It has to be somewhere between 1 and 2, right? Let's add this to our table. Figure Dimension Number of Copies When scale doubled Line segment Sierpinski Triangle Square Cube Self-similar using n copies So the dimension of Sierpinski's Triangle is between 1 and 2. 1? 2 3 d 2 = 2 1 3 = 2? 4 = 2 2 8 = 2 3 n = 2 d
Classwork 5 Use a calculator with an exponent key (usually ^) to find a value of d so that 2 d = 3. (For example, try 1.1. Type 2^1.1 and you get 2.143547.) Determine d to 2 decimal places. Now solve 2 d = 3 algebraically by taking the logarithm of both sides and using a property of logarithms. Note: The base 2 in the number of copies column is of the table is the scale factor used to scale up to the larger similar copy. In other fractals the scale factor may be 3, or 4, or whatever. But in general, if scaling up by a scale factor s yields a similar copy made up of n copies of the original, then n = s d The Koch Fractal Use the triangular grids on the next page. In the first draw in pencil a horizontal line segment that is 9 units long. This is stage 0. Then: 1. Divide each straight line segment (at first there is only one) into three equal segments, and remove (erase) the middle part. 2. Replace each removed part with two segments each as long as the segment removed, making a v whose ends attach where the removed part was, so the v points outward. Here is a diagram of what happens at the first stage: each becomes The result of applying the steps to the original is called Stage 1. Create it in the top grid. It has 4 segments. To get Stage 2 apply the two steps above to each segment in Stage 1. You may do this in the first grid on top of Stage 1, or (if you don t like erasing) you may create it in the middle grid. To get stage 3 apply all the steps to the segments in Stage 2. You may create these in your Stage 2 or (if you don t like erasing) create it in the bottom grid. The Koch Fractal Curve is the result of iterating the process infinitely many times.
Once you have created the first three stages of the Koch Fractal, complete the chart below. Stage Number of Segments Length of each segment Total length at this stage 0 1 2 3 Classwork 6 Answer the following questions about your table: 1. What pattern do you see in the numbers in the Number of segments column? What should the entry be for Stage 4? For Stage n? 2. What pattern do you see in the numbers in the Length of each segment column? What should the entry be for Stage 4? For Stage n? 3. What pattern do you see in the numbers in the Total length at this stage column? What should the entry be for Stage 4? For Stage n? 4. The picture at any Stage is just an approximation to the Koch Fractal. The actual Koch fractal is what results if you do this process forever. More precisely, the Koch fractal is the limit of the process. What do you think should be the total length of the Koch fractal (i. e. of the limit of the total lengths of the Stages)? Why? 5. What should be the fractal dimension of the Koch fractal? (Hint: How many copies do we need to put together to get a copy that is twice as big as the original?)
Homework for Section 6.3 Your homework is to repeat the work done in class, but with the Sierpinski Gasket. In this process we start with a square, think of it as 3 by 3 array of smaller squares whose sides are 1/3 as long as the original square, and then remove the middle square. Here is stage one, where the darkened square has been removed: 1. To create stage two, apply the process to each of the remaining 8 squares. (Use the figure above to show this.) 2. Suppose the outer square has sides of length 1 unit, and so has area 1 square unit. How much total area remains at stage 1? Justify your answer. 3. Continuing as in 2., how much total area remains at stage 2? Justify your answer.
4. Continuing as in 2. and 3., how much total area remains at stage 3? Justify your answer. 5. Continuing as in 2. - 4., how much total area remains at stage n? 6. The Sierpinski Gasket is the limit of this process of removing the inner square in each of the squares at the previous stage. Based on your answers to 2. - 5., what should be the area of the Sierpinski Gasket? Justify your answer. 7. What is the fractal dimension of the Sierpinski Gasket? Answer this exactly, and give a decimal approximation to 2 decimal places. Justify your answers. (Hint: The scale factor is not 2 here!)
References The beauty of fractal images is best appreciated on a computer. There are many web sites that provide java or other programs that are appropriate for teachers and students. Here are a few: 1. http://math.rice.edu/~lanius/frac/ This site has a sequence of lesson on fractals that are appropriate for middle grades students. Some of the material in this handout was adapted from this web site. 2. http://www.pbs.org/wgbh/nova/fractals/ This reference is to a NOVA program on fractals. Excellent images. 3. http://math.bu.edu/dysys/applets/index.html This site has a number of applets that generate fractals. 4. http://illuminations.nctm.org/lessondetail.aspx?id=l236 This lesson relates to construction of the Sierpinski Triangle. A good reference book is: Fractals for the Classroom, by Peitgen et. al., Springer Verlag, 1992