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The Pennsylvania State University the Graduate School Department of Philosophy The Pennsylvania State University The Graduate School Department of Philosophy PICTURES OF THOUGHT: THE REPRESENTATIONAL FUNCTION OF VISUAL MODELS A Thesis in Philosophy by Karim Joost Benammar Copyright 1993 Karim Joost Benammar Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 1993 Table of Contents Chapter 4. Fractal images as visual models . 83 4.1. Self-similarity and the projection of mathematical monsters . .. 85 4.2. Fractal geometry and fractal dimension . 92 4.3. Fractal images as models . 99 4.4. The limits of modelling . 105 Chapter 5. Strange attractors . 110 5.1. Strange attractors in the study of dynamical systems . 110 5.2. Modelling and approximation . 115 5.3. Mathematics and the world . 120 5.4. Generating interpretations . 122 Chapter 6. Generative models and representation . 125 6.1. Necessity and analogy . 126 6.2. Projection and representation . 133 6.3. Generative models . 137 Chapter 7. Visual models and knowledge of the world . 144 7.1. Visual models and language . 145 7.2. Visualization and the visible . 151 7.3. The construction of reality . 154 Bibliography . 158 List of figures 4.1. The Koch curve . 87 4.2. The Koch Snowflake . 87 4.3. Construction of the Cantor set . 87 4.4. The Sierpinski Gasket (1) . 89 4.5. The Sierpinski Gasket (2) . 89 4.6. The Sierpinski pyramid . 89 4.7. The fractal dimension of the coast of Norway . 94 4.8. The Mandelbrot set and corresponding Julia sets . 96 4.9. The Mandelbrot set . 97 4.10. Fractal planetrise . 102 5.1. A Poincaré section . 113 5.2. A strange attractor . 113 5.3. The Lorenz butterfly with equations . 116 5.4. A more detailed Lorenz butterfly . 117 6.1. the necessary function of visual models . 132 6.2. Analogies between relations . 139 6.3. Generative representation between theory and empirical system . 140 6.4. Properties of theory and analogical model . 140 6.5. Properties of the relation between generative model and theory . 141 6.6. Properties of the relation between theory and model . 141 6.7. Characteristics of generative models . 142 7.1. Correspondences between theory and model . 150 7.2. Projection and representation . 150 83 4. Fractal images as visual models Fractals are self-similar images; parts of the image are exactly the same as the whole. They are produced by a variety of different mathematical techniques. Self-similar transformations of images, geometrical projection of numerical sets, or plotting the results of iterative equations on a complex plane are three methods which produce fractal images. Fractals form a new, "fractal" geometry, since the dimension of a fractal image need not be an integer, but can instead be a fraction. This dimension has proved to be the most important mathematical property of fractals thus far. Benoit Mandelbrot is widely credited for his dogged perseverance in trying to interest the scientific community in fractals. He coined the word "fractal," provided a definition of fractal dimension, and studied the impact of fractals as models for a wide variety of natural and artificial phenomena (Mandelbrot 1982). He resurrected the obscure equations and images that had already been studied at the turn of the century by mathematicians who lacked the technology to further visualize their creations. Fractal images are unusual and perplexing; infinitely complicated images spring from simple equations, 84 mathematical constructs turn out to model empirical systems, and complex phenomena can be visualized, in a restricted sense of the term. What kind of visual model is a fractal? What empirical system does it represent? What is the representational relation between the theory of fractal images and images and objects in the world? Fractals images are strangely beautiful images that are produced by mathematical equations or from measurements of empirical processes; this makes them visual models par excellence. Fractals visually model a wide variety of physical and non-physical phenomena, ranging from clouds and lightning strikes to population dynamics. Sometimes, this modelling is connected to an image of an empirical phenomenon (the way a thing "looks"), while at other times it is a visualization of abstract variables, comparable to the graphical imaging of turbulence. These two senses of visual modelling will allow us to sharpen the distinction between projection and resemblance which was raised in the second chapter. In the last chapter we discussed the increased complexity in projection methods and graphical manipulation. The increase in complexity of projection method used for images will come from the radical concept of fractal dimension, and from novel projection methods based on plotting iterated results. These projections are not simply topological transformations. The use of fractal dimension 85 for measuring a wide variety of graphical images modelling phenomena has been a tremendous, though controversial, boost in the effort to link fundamental properties of systems to the topological properties of the visual models that represent them. The furor over fractals has now abated somewhat, and the new perception is that they have been overrated as a revolutionary scientific tool. They are not a panacea for all visualization quandaries, and cannot yet be fruitfully applied in all fields. But this only seems a predictable backlash after their meteoric rise in the scientific and lay consciousness, which has not always led to revolutionary results. 4.1. Self-similarity and the projection of mathematical monsters Although the widespread fascination with fractal images and their applications is barely more than a decade old, visionary mathematicians at the turn of the century thought about endlessly self-similar geometrical objects to illustrate paradoxical conclusions about infinity and dimension. These "mathematical monsters" (Mandelbrot 1982, 35-6; Voss 1985, 25), provide a good introduction to fractals produced by geometric projection methods and to the 86 Hausdorff dimension DH, the most common and straightforward fractal dimension. These fractal images are constructed by starting with a geometrical shape, the initiator, and endlessly repeating the same geometrical transformation, called the generator. After several generations, the image has become so detailed that it can no longer be reproduced exactly on paper. However, since the image is exactly self-similar, zooming in on any part of the image gives us another representation of the whole: exactly self-similar fractal images stretch on into infinity. Infinite fractals are virtual images; they can never be completely represented. Figure 4.1 details the construction of the Koch curve, named after the Swedish mathematician Helge von Koch who proposed it in 1904 as an example of a continuous curve that is nowhere differentiable (Schroeder 1990, 7). Figure 4.2 shows the Koch "snowflake," another geometrical construction of the same curve. Figure 4.3 explains the construction of the Cantor set obtained by repeatedly erasing the middle third of a line segment. After infinitely many repetitions of this operation, we are left with an infinite number of points which have no total length; this has been aptly named "Cantor dust." Georg Cantor proposed this set to demonstrate the possibility of sets with uncountably many numbers and measure (length) zero (Schroeder 1990, 16). 87 Figure 4.1. The Koch curve. Figure 4.2. The Koch Snowflake. Figure 4.3. Construction of the Cantor set. 88 Figures 4.4 and 4.5 show the construction of a two- dimensional Cantor set called the Sierpinski gasket. Figure 4.6 shows a three-dimensional Sierpinski "pyramid". The fractal (Hausdorff) dimension can be calculated by a simple formula1; it is a way of measuring the space- filling capacity of our fractal images. This is an intuitive concept: a very squiggly line will fill more space than a straight line. We can distinguish three different dimensions here: the topological dimension of a line, which is always equal to 1; its Euclidean dimension, which is 1 for a straight line and 2 for a curved line (since it is represented in a plane); and its fractal dimension, which depends on the squiggliness and complexity of the line, and is superior to 1 in all cases other than the straight line (Fan, Neogi, and Yashima 1991, 15). The Koch curve and snowflake in figure 4.1 and 4.2 have a Hausdorff dimension DH = 1.26..., indicating their moderate space-filling capacity. The Cantor set has fractal dimension DH = 0.63..., since it is less than a line, but this dimension is much more than 0, which could be suggested by infinite dust. The Sierpinski gasket has dimension DH = 1.58..., and the pyramid has dimension DH = 2, although it is embedded in three dimensions. The computation of the 1 The Hausdorff dimension DH is given by the formula: DH = lim log N , where N is the number of straight-length r- >0 log (1/r) segments of length r needed to step along the curve from one end to the other (Schroeder 1990, 9-10). 89 Figure 4.4. The Sierpinski Gasket (1). From Schroeder 1990, 17. Figure 4.5. The Sierpinski Gasket (2). From Schroeder 1990, 29. Figure 4.6. The Sierpinski pyramid. From Mandelbrot 1982, 143. 90 fractal dimension is a powerful mathematical tool to typify fractal images, and so to differentiate between the space- occupying capacity of different dusts, curves, and solids. These early mathematical monsters function as visual representations of mathematical properties, demonstrating the existence of peculiar sets or curves. But they need not be visual images, and can be conceived as sets of numbers; there are arithmetic transformations which correspond to the geometric operations to generate self-similar images. The Cantor set can be represented by a series of numbers, and numerical transformations correspond to the geometrical operation of cutting out the middle third (Schroeder 1990, 162-64). These mathematical monsters are fractal images produced by geometrical transformations, and they illustrate theories about numbers.
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