“Fractal geometry will make you see everything differently. There is a danger in reading further. You risk the loss of your childhood vision of clouds, forests, flowers, galaxies, leaves, feathers, rocks, mountains, torrents of water, carpet, bricks, and much else besides. Never again will your interpretation of these things be quite the same.” – Michael Barnsley
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As promised in my last blog, we will continue our discussion of dimensions in fractal geometry (i.e. fractional dimensions). Unlike last time, however, we will focus more so on the actual mathematics of fractals as opposed to the intuition behind them. (If you need a refresher on the definition of a fractal, please go back and reread from last week; it will be to your advantage to do so.) Have paper and a pencil handy: You will definitely need them for this blog!
Last week I ended with the notion of fractional dimensions. It was a tad remiss of me, however, to claim something so profound yet never explain the true definition of a dimension. (Now, when I say “true definition of a dimension,” I mean no insults to intelligence. You are all very bright. I am only here for clarification purposes.) So, what is a dimension? Stop reading and think as broadly as you can about the term “dimension.”
Are you still having difficulty conjuring up a reasonable definition? (I know I did when I first thought about dimensions.) To start, let us simplify the question slightly. With what dimensions can we associate? Well, over our years of growing mathematical observations, we have all come to associate one-dimension with a line, two-dimensions with a plane, and three-dimensions with a space. To put more rigorously, a line has only one independent path of travel (east-west), a plane has two independent paths of travel (east-west and north-south), and a space has three independent paths of travel (east-west, north-south, and up-down). Therefore, by inductive reasoning, we assume that dimensions can indicate the number of independent paths able to be taken in some region.
This definition makes a lot of sense, and it certainly seems to unify our understanding of dimensions in the first, second, and third degrees. However, if fractals have fractional dimension (as I mentioned last week), how is it possible to have a region with a fractional number of independent paths able to be taken? It simply is not possible! Therefore, our definition of dimensions breaks down. To replace this broken definition requires a new, more inventive interpretation of dimensions. (However, please keep the idea of independent paths of travel in your head. It is very important for the fundamental definition of a dimension.)
Consider a line segment. If we double the length of a line segment, we get two copies of the original line segment. If we triple the length of the line segment, we get three copies of the original line segment. If we quadruple the length of the line segment, we get four copies of the original line segment. Get the picture? For any multiple n of a line segment, we get n copies of the original line segment.
Consider a square. If we double both sides of a square, we get a larger square which can fit four of the original squares inside of it. Similarly, if we triple both sides of a square, we get a larger square which can fit nine of the original squares inside of it. If we quadruple both sides of a square, we get a larger square which can fit sixteen of the original squares inside of it. Therefore, in general, for any multiple n applied to both sides of a square, n^{2} number of copies of the original square can be produced in the resulting square.
Consider a cube. If we double all sides of a cube, we get a larger cube which can fit eight original cubes inside of it. (If you are having trouble visualizing the reasons behind these observations, consider the unit cube below.)
Likewise, if we triple all sides of a cube, we get a larger cube which can fit twenty-seven of the original cubes inside of it. If we quadruple all sides of a cube, we get a larger cube which can fit sixty-four original cubes inside it. Therefore, in general, for any multiple n applied to all three sides of a cube, n^{3} copies of the original cube will be produced in the resulting cube.
Let us now organize all this information in a table so that we may be able to understand dimensions more explicitly.
Figure |
Dimension | Linear Multiple |
Number of Copies |
Line | 1 | 2 | (2)^{(1)} = 2 |
Square | 2 | 2 | (2)^{(2)} = 4 |
Cube | 3 | 2 | (2)^{(3)} = 8 |
General Case | D | L | L^{D} = S |
Note: The term linear multiple is used by mathematicians to describe the n values that I have described in the thought experiments above. Linear multiples tell us how large we wish to scale each independent path of travel in the figure. For instance, if we wanted to double the distance along a line, the linear multiple would be equal to two because we wish to double the east-west direction of travel on the line. If we wanted to double the distances along a square, the linear multiple would still be two because we wish to double the east-west direction of travel on the square as well as the north-south direction of travel on the square. (If you double the distances along a cube, you would have a linear multiple of two as well: Each independent path of travel—east-west, north-south, and up-down—will be scaled by two.)
Clarifications aside, notice the pattern presented in the table. If we wish to find the number of copies of some generic figure, we simply take the linear multiple we want (how large we want to scale each independent path of travel) and raise that number to the dimension of the figure. Herein, we establish a rigorous definition of dimension: It is a number which quantifies the effect of changing sizes of linear independent paths of travel on the number of original figure copies produced in the resulting figure. (That definition is quite a mouthful to explain, and if you do not quite understand, I suggest rereading the last three paragraphs or so to let the definition sink into your mind.)
Let us use this definition now to calculate the dimension of the second-most famous fractal—the Sierpinski’s Triangle. Get your paper and pencil!
Creating this fractal is actually fairly easy. (I find it easier than creating the von Koch snowflake.) Begin with an equilateral triangle (a triangle of equally measured sides). Emphasize the midpoints (i.e. halfway points) of the three line segments creating this equilateral triangle with modest dots. Connect these dots to form another equilateral triangle. Shade in this triangle. You should now have a figure with three equilateral triangles (pointed right side up) and a fourth darkened equilateral triangle (pointed upside down). For each of the three equilateral triangles that are not shaded, emphasize the midpoints of the three segments which create the triangle. Draw an equilateral triangle which connects these three points; shade in this equilateral triangle. (Remember to do this process for each of the three equilateral triangles that were created in the first iteration.) You should now have a figure with one large shaded triangle (pointed upside down), three small shaded triangles (pointed upside down), and nine small triangles that are not shaded (pointed right side up). If you are able to see the pattern, I encourage you to continue. If not, this picture you have created is sufficient enough to continue.
For clarification purposes, I have drawn above the next few iterations of Sierpinski’s Triangle.
Now, let us consider a linear multiple of two. (In other words, let us multiply the independent paths of travel in the triangle by two; notice in this figure that there are two independent paths of travel—east-west and north-south.) The result of using a linear multiple of two on Sierpinski’s Triangle is depicted below.
After we multiplied both independent paths of travel by two, how many copies of Sierpinksi’s Triangle did we obtain? Well, it appears we got three original Sierpinski’s Triangles. Recall that we can use this information to establish the dimension of Sierpinski’s Triangle. In other words, recall S = L^{D}. In this case, (3) = (2)^{D}. Notice that if we choose D=1, we undervalue the number of copies of the original Sierpinski’s Triangle we obtained by using a linear multiple of two. Similarly, if we choose D=2, we overvalue the number of copies of the original Sierpinski’s Triangle we obtained by using a linear multiple of two. Therefore, by logical deduction, the dimension of Sierpinski’s Triangle must lie between one and two. It must be fractional or irrational!
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Mathematician’s Side Note
For those of you who are dying to find the exact value for the dimension of Sierpinski’s Triangle, I will explain briefly, but please note that I am assuming you are familiar with certain mathematical operations. As we discovered, S = L^{D}. We would really like to express this equation as D in terms of S and L. In order to isolate D, I will take the natural logarithm of both sides of the equation: ln(S) = ln(L^{D}). After doing so, I will apply the power property of logarithms to rewrite the equation as ln(S) = Dln(L). Lastly, I will divide both sides of the equation by ln(L): ln(S)/ln(L) = D. This equation now expresses dimension as a function of a linear multiple and the number of produced original figures as a result of the chosen linear multiple. In the case of Sierpinski’s Triangle, we have ln(3)/ln(2) = D or, using a calculator to evaluate the logarithms, D ≈1.5850.
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I also promised in this blog that I would show you the most famous fractal of all time. I will not go into the mathematics behind it, which requires an extensive knowledge of complex numbers and complex analysis. Instead, I leave it below for you to ponder. The name of the fractal is the Mandelbrot Set. Based on the information I have provided to you over the last two blogs, you should be able to know how the fractal received that name.
I also wanted to shed light on a question asked by Lewis last week. Fractals are everywhere in nature, Lewis. You just have to go looking for them. This would be a great place to start.
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The way in which you explained, step by step, the concepts needed to understand this fractal triangle’s mathematics was very down to earth and quite engaging. I never find myself losing focus as I read one of your blogs.
I don’t know if it fits into any other ideas that you had for blogs, but I liked how you added the link to see the principle in nature, and I think it would be awesome if there was a portion to each blog that tied the concept to a practical idea, like looking into nature.
I agree with Lewis! You have put so much into this blog and its as fascinating as ever! The way you break the broad concept of fractal geometry down into small segments for better understanding enhances what I said last week about you making an excellent teacher. Your shout-out to Lewis also reinforced the consideration you put into this each week because you were willing to make your blog even more extensive if it meant granting your viewers even better understanding. Great job.
I must commend you for the effort you put forth in this blog; it is definitely evident. I also want to admit I found this post particularly engaging. Who knew I would enjoy the mathematics of fractals! For the record, I never find math fascinating.
I also thank you for the additional post. I love the connection between math and natural structures.