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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

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

Unparticle

Jonathon Keats

in Virtual Words: Language on the Edge of Science and Technology

“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 ... 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

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

Random Fractals

David P. Feldman

in Chaos and Fractals: An Elementary Introduction

This chapter explores ways of generating fractals other than a deterministic procedure. In particular, it considers fractal-generating mechanisms that involve randomness or irregularity. The ... More

This chapter explores ways of generating fractals other than a deterministic procedure. In particular, it considers fractal-generating mechanisms that involve randomness or irregularity. The discussion begins by describing what happens when a little bit of randomness or noise is added to an otherwise deterministic process. It looks at the Koch curve, a classic fractal, and its self-similarity, as well as irregular fractals such as the Sierpiński triangle. It then explains how random and irregular fractals can be extended and refined to produce images that bear a striking resemblance to real landscapes. The chapter concludes by discussing the long-term fate of the orbit in the chaos game, an affine transformation, and the collage theorem.Less

The Box‐Counting Dimension

David P. Feldman

in Chaos and Fractals: An Elementary Introduction

There are several examples of fractals that are not exactly self-similar, as is the case with small parts of the random Koch curve which exhibit statistical self-similarity but not identicality. The ... More

There are several examples of fractals that are not exactly self-similar, as is the case with small parts of the random Koch curve which exhibit statistical self-similarity but not identicality. The dimension of a handful of fractal objects, including the Cantor set, the Sierpiński triangle and carpet, and the Koch curve, can be determined. This chapter considers the box-counting dimension, which extends the concept of dimension to objects that are not exactly self-similar. Instead of focusing on how many small copies of an object are contained in a large copy, it explains how the volume or size of the overall shape changes as measurement scales change.Less

When do Averages Exist?

David P. Feldman

in Chaos and Fractals: An Elementary Introduction

Since fractals are self-similar, it is often not useful to describe them in terms of an average size. Stating an average size does not capture what is interesting or noteworthy about the shape of a ... More

Since fractals are self-similar, it is often not useful to describe them in terms of an average size. Stating an average size does not capture what is interesting or noteworthy about the shape of a fractal object. This chapter considers a situation in which stating an average property is useless and mathematically ill-defined, and which shows that fractals are not just geometric objects but can also be used to describe processes that unfold in time. As a first example, it presents a simple game of tossing a coin, to illustrate what it means for something to possess an average. It then describes the St. Petersburg paradox, also known as the St. Petersburg game or lottery, and calculates the average winnings for this game.Less