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Random Fractals

Peter Mörters

in New Perspectives in Stochastic Geometry

This chapter expounds the theory of random fractals, using tree representation as a unifying principle. Applications to the fine structure of Brownian motion are discussed.

From Cantor to Mandelbrot: Self-similarity and fractals

Iwo Białynicki-Birula and Iwona Białynicka-Birula

in Modeling Reality: How Computers Mirror Life

Objects that look similar after magnification are said to have fractal structure. Perfect fractals can exist only in mathematics but in the real world we have plenty of approximate fractals, from ... More

Objects that look similar after magnification are said to have fractal structure. Perfect fractals can exist only in mathematics but in the real world we have plenty of approximate fractals, from cauliflowers and ferns to internet networks. A characteristic feature of fractals is their dimension, different from a natural number. Mathematical fractals are obtained by a precise prescription, but surprisingly, they can be generated by a random walk.Less

Introducing Fractals

David P. Feldman

in Chaos and Fractals: An Elementary Introduction

This chapter focuses on fractals and the role of iteration in their generation. It first considers three familiar shapes from geometry: a circle, a line segment, and a rectangle. It then describes a ... More

This chapter focuses on fractals and the role of iteration in their generation. It first considers three familiar shapes from geometry: a circle, a line segment, and a rectangle. It then describes a collection of minuscule line segments known as the Cantor set, which exhibits a property called self-similarity like a fractal. It also looks at the typical sizes of fractals and concludes by comparing mathematical fractals with real fractals.Less

Spatial Scale and Species Diversity: Building Species–Area Curves from Species Incidence

Jack J. Lennon and William Edward Kunin

in Biodiversity in Drylands: Toward a Unified Framework

This chapter is largely focused on the species–area relationship (SAR), although it may not seem so for much of the time. Bear with us; we will get there ... More

This chapter is largely focused on the species–area relationship (SAR), although it may not seem so for much of the time. Bear with us; we will get there in the end. Our aim is to provide insights into how the relationship works, and how it is built. This leads us to take a rather reductionist approach, and to break down the SAR into its component parts. We will spend a substantial section of this chapter examining these pieces and their properties. We will then explore the logic by which the parts are reassembled, and will explore how biological and biogeographical properties of a system may affect the SAR. Before attempting this feat, however, we should begin with a brief discussion of the SAR itself, to explain why it is worth making such a fuss over. The SAR is, after all, only a simple graph: a plot of the number of species found in a sample as a function of the area sampled. Ecologists being an argumentative lot, we cannot even all agree on what this plot should look like; Gleason (1922, see also Williams 1964) argued that the absolute number of species should be plotted as a function of the logarithm of area, whereas Arrhenius (1921, see also Preston 1960) suggested that both species and area should be plotted logarithmically. Connor and McCoy (1979) found cases that fit both models, and two others besides (log species by untransformed area, and neither variable transformed). However it’s plotted, the SAR is not even a particularly attractive or elegant graph—at its best (!) it is simply a straight diagonal line within a tight scatter of datapoints on a rectangular plot. Hardly something to set the pulse racing. Yet the SAR is exciting stuff; that simple line encapsulates a great deal of information about the diversity of biological systems across a wide range of scales. Less

Random walks on percolation structures

J. Klafter and I.M. Sokolov

in First Steps in Random Walks: From Tools to Applications

In this chapter another, experimentally widespread, situation is considered. The random walk takes place not on a homogeneous lattice, where each site in principle accessible to the walker, but on a ... More

In this chapter another, experimentally widespread, situation is considered. The random walk takes place not on a homogeneous lattice, where each site in principle accessible to the walker, but on a percolation structure where some sites are not accessible. Close to the percolation threshold, when the system of accessible sites disintegrates into finite clusters, and the connected way through the whole lattice does not exist anymore, the properties of the corresponding walks are related to the fractal structure of the infinite cluster and are quite unusual. The chapter discusses some basic notions of fractal geometry, the properties of random walks on such structures and their effects on the kinetics of simple reactions in percolation systems. The case when the walker can start at an infinite as well as at a finite cluster is also considered.Less

Single-species dynamics

Tim Coulson and H. Charles J. Godfray

in Title Pages

What determines the densities of the different species of plants, animals, and micro-organisms with which we share the planet, why do their numbers fluctuate and ... More

What determines the densities of the different species of plants, animals, and micro-organisms with which we share the planet, why do their numbers fluctuate and extinctions occur, and how do different species interact to determine each other’s abundance? These are some of the questions addressed by the science of ecological population dynamics, the subject that underpins all the chapters in this book. In this chapter we introduce some of the basic principles of the subject by concentrating on the dynamics of singlespecies systems. These are species whose population biology can be studied without also explicitly including the dynamics of other species in the community. The chief justification for this brutal abstraction is that it allows many of the underlying processes to be described simply and more clearly. Moreover, arguments based on the analysis of single-species population dynamics are often surprisingly useful in understanding real populations, especially those in relatively simple environments such as agro-ecosystems. At the core of population dynamics is a simple truism: the density or numbers of individuals in a closed population is increased by births, and decreased by deaths. If the population is not closed then we need also to include immigration and emigration in our calculation. A population in which births exceed deaths will tend to increase and one where the reverse is true will tend to decrease. But more significant is the mode of change. If birth and death rates remain constant then the consequent increase or decrease in population numbers occurs exponentially—population dynamics occurs on a geometric rather than an arithmetic scale. In the first section of this chapter we describe the calculation of exponential growth rates for different types of population, and explore how such calculations, even though they are based on the simplistic assumption of constant demographic rates, can be very useful for a variety of problems in applied population biology. The fact that populations persist over appreciable periods of time inescapably means that demographic rates—births and deaths, immigration and emigration—do not remain constant. In fact, population persistence in the long term requires that as populations increase in density the death rate must rise relative to the birth rate and eventually exceed it. Less

Unparticle

Jonathon Keats

in Title Pages

“All science is either physics or stamp collecting.” So claimed Ernest Rutherford, the British physicist who discovered the atomic nucleus in 1910, touting ... More

“All science is either physics or stamp collecting.” So claimed Ernest Rutherford, the British physicist who discovered the atomic nucleus in 1910, touting the explanatory power of physics over the busywork of classifying elements or planets or animals. One hundred years later, the endless variety of matter postulated by physics—within the nucleus and throughout the universe—has far surpassed the inventories of the periodic table and solar system, leading particle physicists to refer to their domain as a bestiary and one textbook to be aptly titled A Tour of the Subatomic Zoo. There are electrons and protons and neutrons, as well as quarks and positrons and neutrinos. There are also gluons and muons—the unexpected discovery of which, in 1936, led the physicist Isidor Rabi to quip, “Who ordered that?”—and potentially axions and saxions and saxinos. In this menagerie it’s not easy for a new particle, especially a hypothetical one, to get attention. The unparticle, first proposed by American physicist Howard Georgi in 2007, is therefore remarkable for garnering worldwide media attention and spurring more than a hundred scholarly papers, especially considering that there’s no experimental evidence for it, nor is it called for mathematically by any prior theory. What an unparticle is, exactly, remains vague. The strange form of matter first arose on paper when Georgi asked himself what properties a “scale-invariant” particle might have and how it might interact with the observable universe. Scale invariance is a quality of fractals, such as snowflakes and fern leaves, that makes them look essentially the same at any magnification. Georgi’s analogous idea was to imagine particles that would interact with the same force regardless of the distance between them. What he found was that such particles would have no definite mass, which would, for example, exempt them from obeying special relativity. “It’s very difficult to even find the words to describe what unparticles are,” Georgi confessed to the magazine New Scientist in 2008, “because they are so unlike what we are familiar with.” For those unprepared to follow his mathematics, the name evokes their essential foreignness. Less

Computer Graphics Modeling for Virtual Environments

Mark Green and Hanqiu Sun

in Title Pages

Modelling is currently one of the most important areas in virtual environments research (Bishop et al., 1992). Of all the software areas, this is the area ... More

Modelling is currently one of the most important areas in virtual environments research (Bishop et al., 1992). Of all the software areas, this is the area that we know the least about. Modelling has been an active research area in computer graphics for many decades, and is still a major research area. Many of the modelling issues addressed in computer graphics and virtual environments are also of concern to researchers in robotics, mechanical engineering and biomechanics, so progress in modelling can have an impact on many fields. Modelling is difficult since most of the objects that we would like to model, such as people, animals, and airplanes, are quite complex. They have a large amount of geometrical detail and move in complex ways. This difficulty is compounded by the different fields that use modelling techniques, since each field has its own requirements and priorities. For example, it is highly unlikely that the same model of a human figure would be optimal for applications in both virtual environments and biomechanics. The following criteria can be used as a basis for evaluating different modelling techniques. Accuracy. The model should be an accurate representation of the real-world object. Ideally, we would like all of our models to be precise representations of the real-world objects, and not simply approximations to them. But, in reality accuracy comes with a price, usually increased display time or memory usage. The amount of accuracy required often depends upon the application. For example, in some applications it is acceptable to approximate a sphere with a large number of polygons, but for a large number of computer-aided design (CAD) applications the precise mathematical representation of the sphere is required. Since the polygonal representation can be drawn faster, it is often used in applications where display speed is more important than accuracy. Display speed. Many applications place restrictions on the time available to display individual objects. In the case of interactive applications, short display times increase the level of interaction between the user and the model. In large CAD applications there may be many objects, therefore, the time required to display individual objects becomes an important consideration in the usability of the application even if a high level of interactivity is not required. Less

Hierarchies, Complex Fractal Dimensions, and Log-Periodicity

Didier Sornette

in Why Stock Markets Crash: Critical Events in Complex Financial Systems

This chapter describes the concept of fractals and their self-similarity, including fractals with complex dimensions. It shows how these geometric and mathematical objects enable one to codify the ... More

This chapter describes the concept of fractals and their self-similarity, including fractals with complex dimensions. It shows how these geometric and mathematical objects enable one to codify the information contained in the precursory patterns before large stock market crashes. The chapter first considers how models of cooperative behaviors resulting from imitation between agents organized within a hierarchical structure exhibit the announced critical phenomena decorated with “log-periodicity.” It then examines the underlying hierarchical structure of social networks, critical behavior in hierarchical networks, a hierarchical model of financial bubbles, and discrete scale invariance. It also discusses a technique, called the “renormalization group,” and a simple model exhibiting a finite-time singularity due to a positive feedback induced by trend following investment strategies. Finally, it looks at scenarios leading to discrete scale invariance and log-periodicity.Less

The geometry of a single fracture

Pierre M. Adler, Jean-François Thovert, and Valeri V. Mourzenko

in Fractured Porous Media

This chapter addresses the geometry of a single fracture with a double objective which is to characterize it and to reproduce it. The geometrical quantities which characterize the structure of a ... More

This chapter addresses the geometry of a single fracture with a double objective which is to characterize it and to reproduce it. The geometrical quantities which characterize the structure of a fracture are the mean aperture, the probability density of the fluctuations of the fracture surfaces, the autocorrelation function of these fluctuations along each surface and the intercorrelation between the two surfaces. There are two major classes of autocorrelation functions, namely the Gaussian and the self-affine autocorrelation. A schematic presentation of the generation of random fractures which possess these characteristics, is provided. Then, the resulting geometrical properties are analysed. The concepts of percolation, fractals, universal exponents and self-similarity are briefly presented. Finally, the generation of correlated fields and the properties of self-affine fields are summarized.Less

Painting By Numbers

Nicholas Mee

in Celestial Tapestry: The Warp and Weft of Art and Mathematics

Fractals have a property of self-similarity. They are similar to themselves in different regions, and similar to themselves on different length scales, so an enlargement of part of a fractal looks ... More

Fractals have a property of self-similarity. They are similar to themselves in different regions, and similar to themselves on different length scales, so an enlargement of part of a fractal looks similar to the whole fractal. Benoit Mandelbrot pointed out that many objects, including coastlines, have a fractal structure. The mathematics of fractals dates back to Gaston Julia, who was seriously injured during the First World War and completed much of his work while recovering in hospital. Computer-generated imagery (CGI) is widely used in the video game and film industries. One technique for generating images of virtual scenes is known as ray-tracing. Filmed footage can be combined with computer-generated imagery using a technique known as chroma key. One of the most creative examples of computer art is the beautiful panoramic video in Pursuit of Venus by Lisa Reihana.Less

Extreme Outcomes, Connectivity, and Power Laws: Towards an Econophysics of Organization

Max Boisot and Bill McKelvey

in Knowledge, Organization, and Management: Building on the Work of Max Boisot

Organization science to study extremes more rigorously. For a new perspective, we turn to an emerging new physics expressly aimed at dealing with complexity dynamics, Econophysics. It incorporates ... More

Organization science to study extremes more rigorously. For a new perspective, we turn to an emerging new physics expressly aimed at dealing with complexity dynamics, Econophysics. It incorporates biological concepts of organization, ones in which organizational adaptation occurs in the context of a surrounding rank/frequency (R/F) distribution of millions of tiny start-up firms at one end vs. one giant firm at the other end of the two long tails of a Pareto distribution. This chapter distinguishes between the old simplicity of reductionism, equations, linearity and predictions of old physics, and the new simplicity of insignificant initiating ‘butterfly-events’—nonlinearity, similar causal dynamics at multiple levels, power laws, and scale-free theory. Cognitive representations of the real-world and follow-on creation of response schema are shifted from Gaussian to Paretian simplicity. Organizational schema formation and adaptation within the Gaussian and Paretian ontologies are framed in terms of adaptive responses subject to Ashby’s Law of Requisite Variety and our Ashby Space. An organization science based on researching rank-frequencies, scale-free dynamics, and fractal structures is outlined.Less

Opening Remarks

David P. Feldman

in Chaos and Fractals: An Elementary Introduction

This chapter starts by introducing the terms ‘chaos’ and ‘fractal’ and talks briefly about their history. Since the 1970s, these two concepts — chaos and fractals — have become an important element ... More

This chapter starts by introducing the terms ‘chaos’ and ‘fractal’ and talks briefly about their history. Since the 1970s, these two concepts — chaos and fractals — have become an important element of modern science and they have been borrowed and appropriated by other academic disciplines. They have also found themselves often mentioned in popular culture, including film and literature. What do these two terms mean? Why have they become so popular in scientific and popular discourse? This chapter outlines the aims of this book as a whole.Less

Dimensions

David P. Feldman

in Chaos and Fractals: An Elementary Introduction

This chapter explains how to characterise fractals by means of the dimension. It defines dimension in terms of the scaling properties of a shape in order to describe fractals quantitatively. The ... More

This chapter explains how to characterise fractals by means of the dimension. It defines dimension in terms of the scaling properties of a shape in order to describe fractals quantitatively. The discussion begins with a simple geometric exercise that leads to the definition of the self-similarity dimension. It then considers a non-integer dimension and the dimensions of the snowflake, the Cantor set, and the Sierpiński triangle.Less