Fractal Geometry
Contents Introduction The Fractal Geometry of Nature Self-Similarity Some Pioneering Fractals Dimension and Fractal Dimension Cellular Automata Particle Systems Scope of Fractal Geometry 2
Introduction Recent advances in computer graphics have made it possible to visualize mathematical models of natural structures and processes with unprecedented realism. The resulting images, animations, and interactive systems are useful as scientific, research and education tools in computer science, engineering, biosciences, and many other domains. Applications include computer-assisted landscape architecture, design of new varieties of plants, crop yield prediction, the study of developmental and growth processes, and the modeling and synthesis of an innumerable amount of natural patterns and phenomena. 3
Why is geometry often described as cold and dry? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. The existence of these patterns challenges us to study those forms that Euclid leaves aside as being formless, to investigate the morphology of the amorphous. (Mandelbrot, 1983; p. 1) 4
B. Mandelbrot (1983) coined the term fractal to identify a family of shapes that describe irregular and fragmented patterns in nature, thus differentiating them from the pure geometric forms from the Euclidean geometry. Fractal geometry is the geometry of the irregular shapes found in nature, and, in general, fractals are characterized by infinite details, infinite length, self-similarity, fractal dimensions, and the absence of smoothness or derivative. 5
Examples of fractals in nature 6
Self-Similarity 15cm 5.5cm 7
Self-Similarity Small copies are variations of the whole So-called statistical self-similarity or selfsimilarity in the statistical sense. Furthermore, the reduced copies may be distorted in other ways, for instance squeezed. In such cases there is the concept of selfaffinity. 8
Some Pioneering Fractals The first fractal was discovered in 1861 by K. Weierstrass. He discovered a nowhere differentiable continuous function; that is, a curve consisting solely of corners. Other early-discovered fractals were introduced by G. Cantor, H. von Koch, and W. Sierpinski, among others. These fractals were considered mathematical monsters because of some nonintuitive properties they possessed. 9
Some Pioneering Fractals: The Cantor Set Take a line; remove its middle third, leaving two equal lines. Likewise, remove the middle third from each of the remaining lines. Repeat this process an infinite number of times, and you have (in the limit) the Cantor set. 10
Some Pioneering Fractals: Koch Curve Start with a line segment and iteratively apply the following transformation: 1) take each line segment of the Koch curve and remove the middle third; and 2) replace the middle third with two new line segments, each with length equal to the removed part, forming an equilateral triangle with no base. 11
Some Pioneering Fractals: Sierpinski Gasket Start with an equilateral triangle (that can be filled or not), and divide it into four smaller equilateral triangles. 12
Some Pioneering Fractals: Peano Curve The Peano curve starts with an initial line segment. For each step, all line segments are replaced by a curve consisting of nine smaller line segments. 13
Dimension and Fractal Dimension The discovery of space-filling curves, such as the Peano curve, had a major impact in the development of the concept of dimension: questioned the intuitive perception of curves as one-dimensional objects. Points are zero-dimensional, lines and curves are onedimensional, planes and surfaces are two-dimensional, and solids, such as cubes and spheres, are threedimensional. Roughly speaking, a set is d-dimensional if d independent variables (coordinates) are needed to describe the neighborhood of any point. This notion of dimension is called the topological dimension of a set. 14
Dimension and Fractal Dimension: By the early 1900s, describing what dimension means and what are its properties was one of the major problems in mathematics. Notions of dimension: topological dimension, Hausdorff dimension, fractal dimension, self-similarity dimension, box-counting dimension, capacity dimension, Euclidean dimension, and so forth. In some cases all of them are the same and in others not. 15
Dimension and Fractal Dimension If the dimension of the object is known, and this is the case for the Euclidean shapes, powers or exponents allow us to work out how many smaller copies of itself the object contains, of any particular size. In the case of regular shapes, such as lines, squares, and cubes, a d-dimensional shape is composed of N (1/m) copies of itself, where each copy has a size m in relation to the original shape; m is called the reduction factor. The relationship is thus given by N = (1/m) d. 16
Example: A square that can be divided into: four (N = (1/m) d = 2 2 = 4) one-half sized (m = 1/2) squares nine (N = (1/m) d = 3 2 = 9) one-third sized (m = 1/3) squares. 17
Self-Similarity Dimension This idea of the relation between the logarithm of the number of copies of an object and the logarithm of the size of its copies suggested a generalization of the concept of dimension, which allows fractional values. This dimension is known as the self-similarity dimension. d = log N log1/ m where N is the number of copies of the object, and m is the reduction factor. 18
Example Cantor set: Reduction factor: m = 1/3 Number of copies: N = 2 d = log 2 log1/(1/ 3) = log 2 log3 = 0.63 19
Scope of Fractal Geometry Fractal geometry can and has been used to model and study dynamical systems, plant growth, molecular dynamic, ecological systems, forest fires, immune systems, economy, extinct plant species, plant responses and pest attacks, seashells, branching patterns, fungal growth, data compression, clouds, rivers, coastlines, movement of particles in liquids or gaseous environment, computer graphics, fireworks, fluid dynamics, explosions, forest scences, etc. 20
Relation to Dynamical Systems P c ( z) = z 2 + Let where z is a complex variable c = a+bi a complex constant M = {c: 0 is not in the basin of infinity for the map } P c ( z) = z 2 + c M is called Mandelbrot set. c c colored black if it belongs to the set. 21
Fractal Settlement Structures Berlin Paris 22