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Random Fractals Random fractals Logan Axon Notre Dame March 29, 2010 Introduction What is a fractal? There is no universally agreed upon definition of \fractal". Mandelbrot took the from the Latin \fractus" meaning broken. He used it to describe sets that were too badly behaved for traditional geometry. Outline 1 Bad behavior by example 2 Fractal dimension 3 Random fractals Logan Axon (Notre Dame) Random fractals March 29, 2010 2 / 36 The Cantor middle thirds set Start with the unit interval [0; 1]. Remove the middle third to get 1 2 0; 3 [ 3 ; 1 . Continue removing middle thirds. The Cantor set C is what is left. 0 1 . C Logan Axon (Notre Dame) Random fractals March 29, 2010 3 / 36 The Cantor middle thirds set In symbols ::: E0 = [0; 1] 1 2 E1 = 0; [ ; 1 3 3 0 1 1 2 1 2 7 8 E = 0; [ ; [ ; [ ; 1 2 9 9 3 3 9 9 . 1 \ C C = Ei i=0 Exercise What is the total length of all the intervals removed in the construction of C? Logan Axon (Notre Dame) Random fractals March 29, 2010 4 / 36 Properties of the Cantor set C 1 1 It is self-similar: The part of C in 0; 3 is a scaled copy of all of C. 2 It has detail at all scales. 3 It is simple to define but hard to describe geometrically. 4 Its \local geometry" is strange: every open interval containing one point of C must contain other points of C as well as points not in C. All fractals should have these properties. Logan Axon (Notre Dame) Random fractals March 29, 2010 5 / 36 The von Koch curve 0) Begin with a unit line. 1) Replace the middle third with the top of an equilateral triangle. 2) Replace the middle third of each of the 4 line segments. The result is a continuous, nowhere differentiable curve with infinite length called the von Koch curve. Logan Axon (Notre Dame) Random fractals March 29, 2010 6 / 36 The von Koch curve Logan Axon (Notre Dame) Random fractals March 29, 2010 7 / 36 Generalizations of the von Koch curve The construction of the von Koch curve can be modified in a number of ways. The quadratic von Koch curve or Minkowski sausage: Logan Axon (Notre Dame) Random fractals March 29, 2010 8 / 36 The dragon curve Image source: Wikipedia Logan Axon (Notre Dame) Random fractals March 29, 2010 9 / 36 The Mandelbrot set Image source: http://www.math.utah.edu/~pa/math/mandelbrot/mandelbrot.html Constructed by looking at iterations of a function on C. We won't deal with this kind of fractal. Logan Axon (Notre Dame) Random fractals March 29, 2010 10 / 36 Dimension of a fractal One of the main tools in fractal geometry is fractal dimension. Warning: There are a lot of different ways to define fractal dimension (Hausdorff dimension is the most widely used). Our definition is almost Hausdorff dimension but only works for certain kinds of fractals. Idea Look at covering the fractal with smaller and smaller sets. As the sets get smaller we need more of them to cover the fractal. Measure how fast the number of sets increases as the size of the sets decreases. Logan Axon (Notre Dame) Random fractals March 29, 2010 11 / 36 3−n. Covering the Cantor set The construction of the Cantor set shows how to cover it with smaller and smaller sets. 0 1 We can cover C with: 0) 1 set of size 1; 1 1) 2 sets of size 3 ; 1 2) 4 sets of size 9 ; . n . n)2 sets of size ::: Logan Axon (Notre Dame) Random fractals March 29, 2010 12 / 36 Covering the Cantor set The construction of the Cantor set shows how to cover it with smaller and smaller sets. 0 1 We can cover C with: 0) 1 set of size 1; 1 1) 2 sets of size 3 ; 1 2) 4 sets of size 9 ; . n −n . n)2 sets of size ::: 3 . Logan Axon (Notre Dame) Random fractals March 29, 2010 12 / 36 8 −x >1 if 2 · 3 > 1 n < lim 3−nx (2n) = lim 2 · 3−x = 1 if 2 · 3−x = 1 n!1 n!1 :>0 if 2 · 3−x < 1 Fractal dimension of the Cantor set We want to measure how fast 2n grows relative to how fast 3−n shrinks. Do this by looking at lim 3−nx 2n n!1 for different values of x. Logan Axon (Notre Dame) Random fractals March 29, 2010 13 / 36 Fractal dimension of the Cantor set We want to measure how fast 2n grows relative to how fast 3−n shrinks. Do this by looking at lim 3−nx 2n n!1 for different values of x. 8 −x >1 if 2 · 3 > 1 n < lim 3−nx (2n) = lim 2 · 3−x = 1 if 2 · 3−x = 1 n!1 n!1 :>0 if 2 · 3−x < 1 Logan Axon (Notre Dame) Random fractals March 29, 2010 13 / 36 Fractal dimension of the Cantor set 8 1 0 d x y = lim 3−nx 2n n!1 The fractal dimension of the Cantor set is the value d where −nx n −x limn!1 3 (2 ) switches from 1 to 0. This happens when 2 · 3 = 1: Logan Axon (Notre Dame) Random fractals March 29, 2010 14 / 36 We wish to do this more generally. We won't always be covering with straight line segments so we need a good definition for the size of a more general set. Fractal dimension of the Cantor set 2 · 3−x = 1 () 2 = 3x () log 2 = x log 3 log 2 () x = log 3 log 2 Therefore the fractal dimension of the Cantor set is log 3 ≈ 0:6309: Logan Axon (Notre Dame) Random fractals March 29, 2010 15 / 36 Fractal dimension of the Cantor set 2 · 3−x = 1 () 2 = 3x () log 2 = x log 3 log 2 () x = log 3 log 2 Therefore the fractal dimension of the Cantor set is log 3 ≈ 0:6309: We wish to do this more generally. We won't always be covering with straight line segments so we need a good definition for the size of a more general set. Logan Axon (Notre Dame) Random fractals March 29, 2010 15 / 36 Diameter Definition The diameter of a set is the largest distance between two points in the set. To be technically correct, the diameter of a set is the supremum of the distances between two points in the set. For example: The diameter of the line segment [0; 1] is 1; The diameter of a triangle is the length of its longest side; The diameter of a rectangle is the length of its diagonal; The diameter of a disk is twice the radius. Logan Axon (Notre Dame) Random fractals March 29, 2010 16 / 36 Fractal dimension Suppose Nδ is the minimum number of sets of diameter δ needed to cover the set F . We compare how fast Nδ grows relative to the how fast δ shrinks by looking at x lim δ Nδ: δ!0+ 8 m 0 d x x y = lim δ Nδ δ!0+ Logan Axon (Notre Dame) Random fractals March 29, 2010 17 / 36 Fractal dimension x The fractal dimension of F is the value d where limδ!0+ δ Nδ switches from 1 to 0. 8 In symbols: x d = inf x : lim δ Nδ = 0 m δ!0+ 0 d The actual value limδ!0+ δ Nδd mayx be 0, 1, or anything in between. This value is called the d-dimensional measure of the set. Logan Axon (Notre Dame) Random fractals March 29, 2010 18 / 36 Why \dimension"? Why is this called fractal dimension? It agrees with our ideas of what dimension should be: The fractal dimension of a point is 0; The fractal dimension of a line is 1; . And the fractal dimension of the Cantor set is ≈ 0:6309. The Cantor set had properties \between" those of a point and a line. It makes sense that the fractal dimension of the Cantor set is between 0 and 1. Logan Axon (Notre Dame) Random fractals March 29, 2010 19 / 36 Fractal dimension of the von Koch curve Generalize the technique from the calculation for C. Again the construction tells us how to get covers of the curve. We can cover the curve with: (0) 1 full-sized copy of the curve; (1) 4 copies of the curve scaled by 1 3 ; (2) 16 copies of the curve scaled by 1 9 ; . (n)4 n copies of the curve scaled by 3−n. The diameter of the curve itself is 1. So ::: Logan Axon (Notre Dame) Random fractals March 29, 2010 20 / 36 Bestiary The dimension of a point is 0. The dimension of the Cantor middle thirds set is log 2 log 3 ≈ 0:6309: The dimension of a line is 1. log 4 The dimension of the von Koch curve is log 3 ≈ 1:262: The dimension of the Minkowski sausage is 1:5. The dimension of the dragon curve is 2. Many more on Wikipedia. Logan Axon (Notre Dame) Random fractals March 29, 2010 21 / 36 Romanesco broccoli (image source: Wikipedia) Logan Axon (Notre Dame) Random fractals March 29, 2010 22 / 36 The coast of Britain In one of his early papers on fractals Mandelbrot addressed the question, \How long is the coast of Britain?" This is a hard question in part because you get different answers depending on how much detail you include.
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