Transcription

1 FRACTALS

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11 The term fractal was coined by mathematician Benoit Mandelbrot

12 A fractal object, unlike a circle or any regular object, has complexity at all scales

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14 Natural Fractal Objects

15 Natural fractals occupy space in a complex way. They make the most out of a limited space.

16 Iteration as a Feedback Machine A repeat procedure where the output of one step becomes the input of the next step. The feedback machine with IU = input unit, OU=output unit, CU = control unit

17 The construction of the Koch curve by iteration

18 The Koch curve can also be produced by means of an L- System Axiom : F++F++F Production rules: F F F++F F

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20 Cantor set

21 Constructing the Menger Sponge

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23 Multiple Reduction Copy Machine (MRCM)

24 The Attractor of the MRCM

25 MRCM Applied to MRCM

26 Limit Object Koch Curve

27 Blueprint of Barnsley s Fern

28 The first Iterates

29 Fern

30 Koch Curve Transformed into the fern

31 Self-Similarity

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34 No Self-Similarity at Finite Stage

35 A structure is self-similar if it is composed of arbitrarily small pieces, each being a small copy of the entire structure. More precisely: A structure is self-similar if any arbitrarily small pieces can be can be obtained from the whole structure by a similarity transformation.

36 A fractal is produced by a system of transformations. These transformations may include all kinds of similarity (or sometimes affine) transformations: Rotations, translations, reflections Scaling Shear and stretch.

37 IFS for a Twig

38 A Tree

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41 Crystal with Four Similarity transformation

42 Crystal with Five Transformations

43 The self-similarity of a fractal is due to the fact that: the same, simple rules are applied across scales.

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45 A rough definition A fractal is a geometric object whose fractal dimension is greater than its topological dimension We need to understand these expressions: Dimension Topology Fractal dimension

46 Many of the topics involved in gluing pertain to a field of mathematics known as: TOPOLOGY The word has a Greek origin: TOPOS = place LOGY = analysis, study It was once also known as analysis situs (Latin for study of place )

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48 Which of the following questions is most important for a traveler? (a) The exact distance between two stations? (b) Is there is a link between two stations? (c) Is the railway line between two stations straight or curved?

49 Topology is not concerned with metric characteristics (it takes no account of distances or length). Topology is also not concerned with the question of whether lines are straight or curved.

50 So what is topology about?

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54 TOPOLOGICAL DIMENSION Topological ideas can be used to describe the dimensions of objects. There are several ways to define a topological dimension, using: A. Boundaries B. Cuts C. Covering

55 TOPOLOGICAL DIMENSION 1. BOUNDARIES We describe our physical space as 3- dimensional because the walls of a prison are 2- dimensional. (H. Weyl) A volume (dimension 3) can be surrounded with planes (dimension two). A plane (dimension two) is bound by lines (dimension one). A line (dimension one) is bound by points (dimension zero).

56 2. CUTS A space of n-dimensions is cut by a space of n-1 dimensions. Every point (dimension 1) cuts the real line (dimension 2). A point cannot cut the plane. Only a line (dimension 1) cuts the plane (dimension 2).

57 3. COVERING DIMENSION

58 We can arrange a covering of a line or curve in such a way that every intersection is the intersection of no more than two discs.

59 Any covering with intersections of more than two discs can be refined so that every intersection is the intersection of two (and no more than two) discs.

60 A curve has dimension one because it is possible to cover it with open disks of small radius so that only pairs of disks have a nonempty intersection.

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62 A figure has dimension N if: it can be covered with open disks so that: only N + 1 disks have a nonempty intersection.

63 A space is zero dimensional if it can be covered with small open discs that are disjoint.

64 A two-dimensional surface can be covered with balls but every intersection is the intersection of three rather than two balls.

65 The boundary dimension, cut dimension, and covering dimension normally give us the same results. A figure will always have the same dimension, no matter what method is used.

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67 The topological dimension tells us HOW A FIGURE IS CONNECTED. This is the information that topological dimension tells us. Topological dimension has nothing to do with measurement!

68 Whatever definition we choose, the topological dimension of a figure is topologically invariant. The dimension will never change for objects which are topologically equivalent. A straight line can, for instance, be bent into a curve, but it will remain a one-dimensional object. Its topological dimension is preserved when it is bent.

69 We cannot change the dimension of a square by distorting it. To make it into a three-dimensional object, we must glue it to form (for example) a cube.

70 There are, however, other (non-topological) definitions of dimension. These do not make reference to topological properties but to metric properties (concerning distance). These other definitions are the basis for the concept of FRACTAL DIMENSION.

71 FRACTAL DIMENSION There are various kinds of fractal dimension. The easiest one to understand is the SELF-SIMILARITY DIMENSION.

72 SELF-SIMILARITY The idea is to use the self-similarity of an object to compute its dimension.

73 Self-Similarity in the Decimal System

74 Self-similarity of the line, square and cube

75 Begin with simple examples: a line, a square, and a cube. For each object we will: Reduce its scale by a factor of r. Combine N reduced objects so that we fill the same space as the original object.

76 Here is a one-dimensional line segment. First we reduce its size by a factor of two. Then fill up the available space. We need to use two lines. Next we reduce the size of the original line by a factor of three, and again fill the available space. Now we need three lines.

77 We discover the law: N = r D D is the self-similarity dimension of the object.

78 Another example: the square. The formula N = r D gives dimension 2.

79 For a cube, the formula N = r D gives dimension 3.

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81 KEY CONCEPT There is a relation between: a. the reduction (or scaling) factor and b. the number of reduced pieces that occupy the same space as the original.

82 The self-similarity dimension is only one type of fractal dimension. Another type is the Hausdorff dimension. We will not study it here, since it is a more advanced topic.

83 Many of these ideas were introduced by a Jewish mathematician named Felix HAUSDORFF As a result of the Nazi Nuremberg Laws of 1935, Hausdorff was forced to leave his academic position at Bonn.. His research could only be published outside Germany. When the Nazis scheduled him to go to a concentration camp, he committed suicide together with his wife and his wife's sister (1942)

84 Hausdorff made many contributions to modern topology and set theory. He introduced the notion of Hausdorff dimension, the Hausdorff metric, and the term "metric space. He also developed the concept of a partially ordered set.

85 Let us now compute the fractal dimension of the Koch Curve

86 Each copy is reduced by 1/3. 4 copies of this piece are produced. 4 = 3 D Each iteration increases the total length by a ratio of 4/3. The self-similarity dimension is. D = log 4 / log 3 =

87 The self-similarity dimension indicates how the object occupies space.

88 Topologically, this is a curve with topological dimension 1. The curve s self-similarity dimension is greater than its topological dimension. Self-similarity dimension = Topological dimension = 1

89 Topological dimension is always an integer. Self-similarity dimension can be an integer, but it does not have to be.

90 The Peano curve. Each generation is composed of 9 n segments. The reduction factor is 1/3. This gives us D = 2. Topological dimension = 1 Self-similarity dimension = 2

91 The Sierpinksi Triangle Fractal dimension = log 3 / log 4 =

92 For some objects the topological dimension = the self-similarity dimension. These include ordinary straight lines, squares, cubes, etc. For other objects the self-similarity dimension is greater than the topological dimension. These objects are FRACTALS.

93 THE DEFINITION AGAIN FRACTAL A geometric object whose fractal dimension is greater than its topological dimension

94 Not every self-similar object is a fractal. A line segment, a square, and a cube are all self-similar, but they are not fractals. The topological dimension is the same as the self-similarity dimension.

95 Measuring the length of the coast of Britain with straight line segments of 200 miles

96 Measuring the length of the coast of Britain with straight line segments of 100 miles

97 Measuring the length of the coast of Britain with straight line segments of 50 miles

98 Measuring the length of the coast of Britain with straight line segments of 25 miles

99 Measuring the length of the coast of Britain

100 As the length of each interval becomes smaller and smaller, the total length of the coast becomes larger and larger. We are able to penetrate ever finer twists and turns of the curve Measurement cannot settle down to a stable length. As the length of the device approaches 0, the length of the coast approaches infinity. As the measurements become more precise, the lengths increase without bound.

101 "The result is most peculiar: coastline length turns out to be an elusive notion that slips between the fingers of one who wants to grasp it. All measurement methods ultimately lead to the conclusion that the typical coastline's length is very large and so ill determined that it is best considered infinite." Mandelbrot

102 Outside the world of regular mathematical objects, length is relative. It depends on our choice of scale and on the precision of our technical instruments of measurement. It is not entirely objective. The observer inevitably intervenes in its definition." (Mandelbrot)

103 "Nature does exist apart from Man, and anyone who gives too much weight to any specific [measuring device and total length] lets the study of Nature be dominated by Man, either through his typical yardstick size or his highly variable technical reach. If coastlines are ever to become an object of scientific inquiry, the uncertainty concerning their lengths cannot be legislated away." Mandelbrot

104 Contrast the coast of England with the measurement of a circumference. The true measure of a circumference can be determined.

105 The procedure for determining the length of the side of a polygon inscribed inside a circle

106 A curve is rectifiable if, as the number of intervals tends to infinity, and the length of each interval tends to zero, the total length of the polygonal line approaches a finite limit. The circle is rectifiable. The coast of England is NOT rectifiable.

107 Graph of the length of the coast of Britain and the circumference

108 For a smooth, rectifiable curve, as the length r of each interval is reduced (goes to zero), the product approaches a finite limit, which gives the length L of the curve.

109 What is the main difference between rectifiable and nonrectifiable curves? It has something to do with the spacefilling or space-occupying properties of the object: the spatial intricacy of the object.

110 A fractal object, unlike a circle, has complexity at all scales

111 The measurements are relative. Is there anything that is not relative in the act of measurement? To approximate the length L(e) of the coast we need roughly F(ε) 1-D intervals of length ε.

112 This relationship was found by observation. It was not a priori. It was not discovered through pure mathematics.

113 The value of D depends on the coastline, so that different coastlines, even different pieces of the same coastline, may produce different values of D. But its value is independent of the yardstick or method used to measure the coast. It is not relative to our measurement choice.

114 Use the expression scaling fractals The term fractal indicates the aspect of irregularity. The term scaling indicates the aspect of regularity or order.

115 The fractal dimension gives an index of the rate at which your estimated lengths for the coastline keep on growing as your resolution becomes finer. This indicates how the object occupies space. It gives a measure of its spatial complexity.

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118 When a curve makes a sharp turn, its tangent is not uniquely defined.

119 Mathematicians like G. Peano and D. Hilbert invented curves that are everywhere continuous but every point is a sharp turn!

120 David Hilbert invented a space-filling curve

121 Hilbert s Paper Space-filling curve

122 Peano Curve Construction

123 There is a sharp 90 degree turn after every segment. After a certain number of iterations, a graphical representation of the curve is not visually distinguishable from a region of the plane. Infinite iterations will fill a whole region of the plane.

124 Non-Linear Curve as 2D space: It is a space-filling curve.

125 About the construction of a fractal: mr. bean

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