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Fractal Geometry
- 1. FRACTAL GEOMETRY: THE MATHEMATICAL LANGUAGE OF NATURE Marisa Hahn December 10, 2015 Concordia University Wisconsin
- 2. What is a Fractal? No uniform definition Displays certain properties Geometric figure that consists of an identical motif repeating itself on an ever-reducing scale
- 3. Overview Brief history Properties of fractals Famous Fractals
- 4. History: Discovery of Fractals Theoretical mathematics Ancient Greeks Proofs, facts, rules, axioms, etc. Euclidian Geometry
- 5. History: Discover of Fractals Experimental and observational science Other fields of science Computers Both theoretical and experimental are needed Geometric figure that consists of an identical motif repeating itself on an ever-reducing scale
- 6. Benoit Mandelbrot Curious about geometry at young age Work at IBM The Fractal Geometry of Nature (1977)
- 8. What Makes Fractals Different? Geometric figure Fine structure Unique No concrete definition Example: Triangle
- 9. Properties of Fractals Iteration Recursion Self-similarity Fractal dimension
- 10. Iteration Feedback process Next input is previous output
- 11. Recursion Repeating operation Replacement Rule Example: Sierpinski Triangle
- 12. Self-Similarity Example: tree Smaller replicas of itself
- 13. How long is the coast of Britain? Fractal Dimension introduction Start with by using rulers to outline 13 rulers * 200 𝑘𝑚 1 𝑟𝑢𝑙𝑒𝑟 2,600 km 38 rulers * 100 𝑘𝑚 1 𝑟𝑢𝑙𝑒𝑟 3,800 km
- 14. How long is the coast of Britain? Smaller measurements increase precision 320 rulers * 27 𝑘𝑚 1 𝑟𝑢𝑙𝑒𝑟 8,640 km 107 rulers * 54 𝑘𝑚 1 𝑟𝑢𝑙𝑒𝑟 5,778 km
- 15. Defining Dimension Number of coordinate axes needed to determine location of point in space Example: line, plane, object, point Reduction factor Dimension =Replacement number
- 16. Defining Dimension Reduction factorDimension =Replacement number r D =n 3Dimension = 27 3Dimension = 33 Dimension = 3
- 17. Defining Dimension Reduction factorDimension =Replacement number r D =n log (r D) = log (n) D log (r) = log (n) D = 𝐥𝐨𝐠(𝒏) 𝐥𝐨𝐠(𝒓) = log(𝑟𝑒𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑛𝑢𝑚𝑏𝑒𝑟) log(𝑟𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟)
- 18. Fractal Dimension Not whole numbers Example: British Coast Relationship Jaggedness acts like fractals dimension 1.58
- 19. Fractal Dimension Jaggedness similar to fractal dimension British coastline vs. Norwegian coastline Between dimensions? 1-D object in a 2-D plane 2-D object in a 3-D plane
- 20. Famous Fractals Koch Curve Julia Set
- 21. Koch Curve Fine Structure Self-similar Recursion Iteration Fractal dimension n=0 𝟒 𝟑 𝟏𝟔 𝟗 1
- 22. Gaston Julia (1893-1978) Early success Iteration and rational functions Forgotten work
- 23. Complex Numbers Combination of real and imaginary numbers z = x+iy Example: z = 2-3i
- 24. Julia Set – Iteration & Recursion Set of points on the complex plane that is defined through a process of function iteration Given sequence: x2+c x → x2+c → (x2+c)2+c → ((x2+c)2+c)2+c → … Iteration Each output is the new input Recursive rule Continuous substitution
- 25. Attractors Attractors of iteration Figure that comes by iterating linear transformations Example: xnew = x2 old Let xold=0.9 0.9 x 0.9 = 0.81 0.81 x 0.81 = 0.6561 0.6561 x 0.6561 = 0.43046… …and after ten iterations = 1.39 x10-47 Attractor point is zero
- 26. Attractors Attractors of iteration Figure that comes by iterating linear transformations Example: xnew = x2 old Let xold=1.1 1.1 x 1.1 = 1.21 1.21 x 1.21 = 1.4641 1.4641x 1.4641= 2.14358… …and after ten iterations = 2.43 x1042 Attractor point is infinity
- 27. Julia Points z = x + iy Prisoner Points - Approach the attractor as a limit - Set of all Prisoner Points is “Prisoner Set” Escaping Points - Tend towards infinity - Set of all Escaping Points is “Escaping Set” Julia Points - Do not approach an attracting fixed point - Do not tend towards infinity - Set of all Julia Points is “Julia Set”
- 28. Julia Points Attractors of iteration xnew = x2 old Figure that comes by iterating linear transformations What happens when xold=1.0 ? Unstable Points jump around
- 29. Complex Numbers Use same formula in complex plane znew=z2 old and starting point |z|=1 If inside circle, attractor is zero (Prisoner Set) If outside circle, attractor is infinity (Escaping Set) If on the circle, no attractor and unstable and jumps around on circle (Julia Set)
- 30. Julia Sets Now we add a complex constant c to the equation: znew= z2 old + c Result is not a circle Determined by c Fall within |z|=2 Symmetric about origin
- 31. Julia Sets Black points are not attracted to infinity (outline is Julia Set) White points are attracted to infinity (Escaping points) znew= z2 old + c
- 32. Julia Set Iteration of: z → z2+c z=x+iy c=a+ib -4<x<4 -3<y<3 -16<b<16
- 33. Benoit Mandelbrot “Fractal geometry is not just a chapter of mathematics, but one that helps Everyman to see the same old world differently.”
- 34. Sources Bedford, C.W. (1998). Introduction to Fractals and Chaos: Mathematics and Meaning. Andover, MA: Venture Publishing. Gleick, James. (1987). Chaos: Making a New Science. Harrisonburg, VA: R.R. Donnelley & Sons Co. Lauwerier, H. A. (1991). Fractals: Endlessly Repeated Geometrical Figures. Princeton, NJ: Princeton University Press. Mandelbrot, Benoit B. (1977). The Fractal Geometry of Nature. New York: W.H. Freeman and Company. McGuire, Michael. (1991). An Eye For Fractals. United States of America: Addison-Wesley Publishing Co., Inc. Peitgen, H.O., Jurgens, H., Saupe, D., Maletsky, E., Perciante, T., & Yunker, L. (1992). Fractals for the Classroom: Part One Introduction to Fractals and Chaos. Rensselaer, NY: Springer- Verlag New York, Inc. Peitgen, H.O., Jurgens, H., Saupe, D., Maletsky, E., Perciante, T., & Yunker, L. (1991). Fractals for the Classroom: Strategic Activities Volume One. Baltimore: Springer- Verlag New York, Inc. Peitgen, H.O., Jurgens, H., Saupe, D., & Zahlten C. (Producers). (1990). Fractals: An Animated Discussion [VHS]. New York: W.H. Freeman and Company. Image Fractal Geometry: https://en.wikipedia.org/wiki/The_Fractal_Geometry_of_Nature Video: https://www.youtube.com/watch?v=RAgR_KVWtcg
Editor's Notes
- Well, fractals are odd because there is no one concrete definition that works for all fractals However, each different fractal displays certain properties and characteristics which we will talk about later But for now, this is a good working way to describe a fractal is a ….. Geometric figure that consists of an identical motif repeating itself on an ever-reducing scale
- To begin, we will go through a brief history of fractals, Followed by the main properties of fractals And end by looking we will look at two famous fractals in depth.
- In the world of mathematics, the study of fractals has taken off in the last 50 years. The word “fractal” had not even been coined until the 1970s. So Why had it taken mathematicians so long to begin studying these geometric figures? Part of the answer goes back to the Ancient Greeks. Dating back to the 300s BC, mathematics has been as a general rule conducted theoretically, that is Logical rules, principles, theorems, axioms, corollaries, etc. are the main means of reasoning. And using these rules and theorems, gives birth to proofs which become tools by which other facts can be made into theorems or corollaries, and the same thing happens over again. CLICK This has been how most mathematics, particularly Euclidian geometry, has generally been done for thousands of years. Like if you think back to when you were in high school, this has been especially true for the field of Euclidian geometry…it’s was all proofs and theorems and axioms and corollaries. Some famous fractals had been discovered way before the 1970s , CLICK but there was uniform name for them. It was something mathematicians looked at as like “Wow isn’t that neat,” and they could notice all the cool characteristics the fractals displayed but they had no way to explain them in depth! All they could do was use…..
- Use Experimental and observational science. Which is how other science fields have been conduct their research for hundred of year, but it was not that common in mathematics. Benoit Mandelbrot One of the main contributors to fractal geometry said that In order to study fractal geometry, both the theoretical and observational science methods are needed Another reason that the study of fractals has exploded in the past few decades is because of the development of technology. Going back to the description of a fractal as being a geometric figure repeating itself on an ever reducing scale The computer allows mathematicians analyze fractals at magnitudes that would never have been possible 70 years ago.
- The fusion of observational and theoretical mathematics broadened the horizons for mathematicians which leads us to talk about Benoit Mandelbrot, who is called the founder of fractal geometry. In his early life, Mandelbrot recalls being fascinated by geometry and always looking for structure in the world around him It wasn’t until he worked at IBM in the 1970s that he began to notice consistent patterns and geometric relationships between bursts of errors in the communications, which the engineers there had overlooked or disregarded In the 1970s he went on to study at these patterns and geometrical relationships in nature and wrote…. The fractal geometry of Nature
- Mandelbrot realized that there must be some structure and mathematics behind what we see in the natural world like mountains, trees, lightning, and clouds. He argued that mountains do not look like not cones, and clouds are not spheres and so on
- Why are fractals different from traditional geometric figures? Well, to begin, each fractal has fine structure – meaning that no matter how much you zoom in or out, there is always infinitely more to see. Gets more and more complex Each fractal is unique in that there is no one concrete definition for all fractals For ex…By definition a triangle is a one dimensional figure that has three sides connected by three vertices, and this holds true for any triangle ever. As we said before, fractals can display certain properties but there is no one solid definition for a fractal
- Now we are going to look at some of those properties that makes a fractal a fractal Four main properties that we will focus on are iteration Recursion Self-similartiy, And fractal dimension
- Iteration is very important to the study of fractal geometry Iteration can be defined as a repeating operation where the next input is the previous output It can be also though of as the process of feedback as seen in the diagram.
- So when we talked about iteration as a repeating operation where the next input is the previous output Recursion is that repeating operation or a repeating rule that informs how the next stage of a figure will be constructed One graphical object is replaced with another which is usually more complex but it still fits into the place of the original figure. By just looking at the Sierpinski triangles, we could say that the rule is that for every you create a new figure by connecting the midpoints of each side of the black triangle and removing the resulting triangle.
- Another property fractals display is that of self-simliartiy To introduce the concept of self-similarity, it will help us to look at a tree (or a picture of a tree). Well start by looking at where the trunk meets the ground. Slowly bring your eyes up to where the trunk begins to branch off. These two branches continues for a while and then the larger branch breaks off into more smaller branches. And this process continues until we are left with the “ends” of the tries with the tiniest branches. This is the concept of self-similarity. A figure that is self-similar if part of the figure contains a smaller replica of the whole. Looking again at the sierpinski triangle, you can see that each part of the new triangle in the set is an exact replica of the one before it and therefore strictly self-similar
- To introduce the concept of fractal dimension, we will look at the question “How long is the coastline of Britain?” This question and what I’m about to explain was put forth by Mandelbrot So to begin, we start by outlining it with a ruler that is 200 km long. The result is that it takes 13 rulers to outline the coast, showing that Britain’s coast is 2,600 long. Next, we repeat the process with a shorter ruler of 100 km long. This results in using 38 smaller rulers. Doing the math, our coast is 3,800 km long
- Again, we use smaller rulers representing 54 km and the coast is 5,778 and using ones that represent 27 km show the coast to be 8,640 km. So that’s great, but what is this telling us? Well as we use smaller and smaller measurements, we are able to include more bays and coastlines and capes, Illustrating the mathematical concept that the length of a smooth curve can be as precise as you want it to be by using smaller and smaller measurements.
- Now we are going to step away from the British coastline problem and look closer at how we define the word dimension A loose definition is Number of coordinate axes needed to determine location of point in space For example, a line in space requires only one dimension to be defined, a plane requires two dimension, and figures such as squares or spheres require 3 dimensions . In order to examine fractal dimension in a little bit better we are going to use the formula Reduction factor Dimension =Replacement number This equation says that You start with a d-dimensional shape and enlarge it by a reduction factor r. Then its d-dimensional volume is multiplied by r to the D.
- That might not mean a lot to you right now, So we’re going to use this formula to find the dimension of this cube. The picture tells us that the reduction factor is 3 because there are 3 linear lengths of the smaller cube in the larger cube, or the linear lengths of the larger original cube are reduced by 1/3. Next our replacement number is 27 because it is the amount of smaller cubes needed to replace the larger original cube. So using the formula we find that the cube has a dimension of 3
- If we rearrange this formula using logarithms we end up with Dimension equaling the log of the replacement number over the log of the reduction factor So what does this have to do with fractals?
- Well, when we looked at the cube, the dimension was three, a positive, whole integer This is a key difference between the dimensions of traditional Euclidian geometric figures and the dimensions of fractals. Fractal dimension is rarely a whole number, and doesn’t even have to be positive. Now remember the British coastline example, we see by this chart that there is a relationship between the number of rulers used and the reciprocal length of the ruler itself When we use the equation for dimension stated previously, we can find the “jaggedness” or fractal dimension of the British coastline…. It comes out to be 1.58 When I first came across this, I thought “How can you have something with a 1.58- dimensional figure?” And a good way to think about this fratal dimension, is that it takes up more space than a one-dimensional figure of a line line but less space than a two-dimensional figure, like a circle. When we look at the jaggedness of coastlines, it acts like a fractal dimension.
- If we look at the Norwegian coast, we can see that it is much more jagged, and therefore has a higher “fractal dimension” of 1.7 These coastlines are at type of fractal curve So this whole coastline question put forth by Benoit Mandelbrot enables us to look at the dimensions of fractal curves. They appear to be a one-dimensional (line) in a two dimensional plane, therefore it’s fractal dimension lies between 1 and 2. It could also be that a fractal surface has a dimension like 2.3 meaning that it acts like a 2 dimensional object (plane) but it is defined in 3 dimensional space.
- Okay, now that we have those properties of fractals down, we will see how they are displayed in two famous fractals……. The koch curve And the Julia Set
- 21
- Before we look at the Julia Set, here is a little bit about the man who discovered this fractal, Gaston Julia. He became a famous mathematician at the age of 25 when he published his first article that focused on the iteration of rational functions. He later became a mathematics professor, and most of his work remained forgotten until Mandelbrot used it decades later. Mandelbrot was able to use Julia’s work to display beautiful fractals like the one seen on the slide.
- So like we did with the Koch Curve we are going to look at the basic properties of fractals that the Julia Set displays To do this, some of the most beautiful fractals are defined using complex numbers. And Julia sets live in the complex plane. So we’re going to do a quick review of complex numbers because it has been a while since most us have used them in our mathematics classes Complex numbers are a combination of real and imaginary numbers and It’s important to realize that real and imaginary numbers are just as real and imaginary as any other type of number and complex numbers can be manipulated under the mathematical operations like addition, subtraction, multiplication, and so on. the common notation for representing complex number is written as ….. With x being a variable on the real axis and y being a variable along the imaginary axis and i being square root of -1
- The Julia Set is the set of points on the complex plane that is defined through a process of function iteration So given the sequence x square plus c Where c is some arbitrary fixed position in the complex plane We can keep iterating it continuously because each output of the next stage is the input for the new stage Also, recursion is displayed because our rule is given to us by the formula x squared + c
- Now, we’re going to take a step away from complex number and examine only the real numbers to define attractors “the figure that arise from iterating linear transformations are said to be the attractors of the iteration,” and “a really simple example of attraction is what happens when a number on the real line is iteratively squared; that is xnew = x2old If we begin by letting xold=0.9, then the following results from the formula above (x2old = xnew): 0.9 x 0.9 = 0.81 0.81 x 0.81 = 0.6561 0.6561 x 0.6561 = 0.43046… …and after ten iterations = 1.39 x10-47
- This time we will let x equal 1.1 If we begin by letting xold=1.1, then the following results from the formula above (x2old = xnew): The values keep getting larger and larger So the attractor point of this set is infinity This brings us to....
- …us being able to define Julia Points First, Prisoner points approach the attractor as a limit….. so like our first example when the attractor was zero Second, are escaping points…. In which all the points tend towards infinity Then there are Julia Points which do not approach a fixed point at all and do not tend towards infinity The set of all these points is a Julia Set
- Looking again at our function in the reals, we saw that when the original x was 0.9 the attractor was 0 When the original x was 1.1 the points tended towards infinity But what happens when 1 is the original x? It would appear to be fixed at 1, but the function is actually unstable This because if the value is changed to even the lightest bit below 1 it is attracted to zero And if it even the tiniest bit above 1, then it will go to infinity.
- Now using the same function under the complex number plane, we use z where z is some complex number. And we start by letting z = 1 ………… If inside circle, attractor is zero ( or the Prisoner Set) CLICK CLICK If outside circle, attractor is infinity ( or Escaping Set) CLICK CLICK If on the circle, no attractor and is unstable and jumps around on circle (or the Julia Set) CLICK CLICK
- So now, we will add a complex constant to our equation on the complex plane. Unlike the previous slide, the set of boundary points is not a circle (except for the trivial case when c is equal to 0) But the boundary is a fractal which depends on the value of c These give us different Julia sets Some characteristics of Julia sets is that they are determined by c Fall within the absolute value of z being 2 And they are symmetric about the origin
- The white part are the points attracted to infinity (the escaping points) The black part are those not attracted to infinity (this is the prisoner points and the Julia points) But the Julia Points are just the boundary of the black part. And you can imagine these on a complex plane if you add in the coordinate axes
- Now we’ll will look at julia set illustrated by the iteration of z squred +c CLICK So if the absolute value of z becomes greater than 2 during iteration, then the initial value of z is attracted to infinity and the iteration stops. The different colors are used to keep count of the number of iterations of z for each point up to some maximum number (as long as it is within the absolute value of 2) This creates a contour map of colors where the numer of iterations are represented by number of iterations. https://www.youtube.com/watch?v=RAgR_KVWtcg
- So in conclusion, fractals are extremely complex geometric figures and they are all around us in the natural world. And I will leave you with a quote by Benoit Mandelbrot “Fractal geometry is not just a chapter of mathematics, But one that helps Everyman to see the same old world differently.” Thanks.