Wilfrid S. Kendall and Ilya Molchanov (eds)
- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199232574
- eISBN:
- 9780191716393
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199232574.001.0001
- Subject:
- Mathematics, Geometry / Topology
Stochastic geometry is a subject with roots stretching back at least 300 years, but one which has only been formed as an academic area in the last 50 years. It covers the study of random patterns, ...
More
Stochastic geometry is a subject with roots stretching back at least 300 years, but one which has only been formed as an academic area in the last 50 years. It covers the study of random patterns, their probability theory, and the challenging problems raised by their statistical analysis. It has grown rapidly in response to challenges in all kinds of applied science, from image analysis through to materials science. Recently, still more stimulus has arisen from exciting new links with rapidly developing areas of mathematics, from fractals through percolation theory to randomized allocation schemes. Coupled with many ongoing developments arising from all sorts of applications, the area is changing and developing rapidly. This book is intended to lay foundations for future research directions by collecting together seventeen chapters contributed by leading researchers in the field, both theoreticians and people involved in applications, surveying these new developments both in theory and in applications. It will introduce and lay foundations for appreciating the fresh perspectives, new ideas, and interdisciplinary connections now arising from stochastic geometry and from other areas of mathematics now connecting to this area.Less
Stochastic geometry is a subject with roots stretching back at least 300 years, but one which has only been formed as an academic area in the last 50 years. It covers the study of random patterns, their probability theory, and the challenging problems raised by their statistical analysis. It has grown rapidly in response to challenges in all kinds of applied science, from image analysis through to materials science. Recently, still more stimulus has arisen from exciting new links with rapidly developing areas of mathematics, from fractals through percolation theory to randomized allocation schemes. Coupled with many ongoing developments arising from all sorts of applications, the area is changing and developing rapidly. This book is intended to lay foundations for future research directions by collecting together seventeen chapters contributed by leading researchers in the field, both theoreticians and people involved in applications, surveying these new developments both in theory and in applications. It will introduce and lay foundations for appreciating the fresh perspectives, new ideas, and interdisciplinary connections now arising from stochastic geometry and from other areas of mathematics now connecting to this area.
Iwo Bialynicki-Birula and Iwona Bialynicka-Birula
- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198531005
- eISBN:
- 9780191713033
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198531005.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This book covers a wide range of subjects concerning the use of computer modeling to solve a diverse set of problems. The book covers some advanced topics (cellular automata, Shannon measure of ...
More
This book covers a wide range of subjects concerning the use of computer modeling to solve a diverse set of problems. The book covers some advanced topics (cellular automata, Shannon measure of information content, dynamical systems, deterministic chaos, fractals, statistical linguistics, game theory, neural networks, genetic algorithms, Turing machines, and artificial intelligence). These advanced subjects are explained in terms of well known simple concepts such as the Game of Life, probability and statistics, Galton's board, Shannon's formula, game of twenty questions, game theory, and a format similar to a television quiz. Twenty-five programs written specifically for the book greatly enhance its pedagogical value and the enjoyment of learning. These can be found at http://www.modelingreality.net/.Less
This book covers a wide range of subjects concerning the use of computer modeling to solve a diverse set of problems. The book covers some advanced topics (cellular automata, Shannon measure of information content, dynamical systems, deterministic chaos, fractals, statistical linguistics, game theory, neural networks, genetic algorithms, Turing machines, and artificial intelligence). These advanced subjects are explained in terms of well known simple concepts such as the Game of Life, probability and statistics, Galton's board, Shannon's formula, game of twenty questions, game theory, and a format similar to a television quiz. Twenty-five programs written specifically for the book greatly enhance its pedagogical value and the enjoyment of learning. These can be found at http://www.modelingreality.net/.
Robert C. Hilborn
- Published in print:
- 2000
- Published Online:
- January 2010
- ISBN:
- 9780198507239
- eISBN:
- 9780191709340
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507239.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This book introduces the full range of activity in the rapidly growing field of nonlinear dynamics. Using a step-by-step introduction to dynamics and geometry in state space as the central focus of ...
More
This book introduces the full range of activity in the rapidly growing field of nonlinear dynamics. Using a step-by-step introduction to dynamics and geometry in state space as the central focus of understanding nonlinear dynamics, this book includes a thorough treatment of both differential equation models and iterated map models (including a detailed derivation of the famous Feigenbaum numbers). It includes the increasingly important field of pattern formation and a survey of the controversial question of quantum chaos. Important tools such as Lyapunov exponents, fractal dimensions, and correlation dimensions are treated in detail. Several appendices provide a detailed derivation of the Lorenz model from the Navier-Stokes equation, a summary of bifurcation theory, and some simple computer programs to study nonlinear dynamics. Each chapter includes an extensive, annotated bibliography.Less
This book introduces the full range of activity in the rapidly growing field of nonlinear dynamics. Using a step-by-step introduction to dynamics and geometry in state space as the central focus of understanding nonlinear dynamics, this book includes a thorough treatment of both differential equation models and iterated map models (including a detailed derivation of the famous Feigenbaum numbers). It includes the increasingly important field of pattern formation and a survey of the controversial question of quantum chaos. Important tools such as Lyapunov exponents, fractal dimensions, and correlation dimensions are treated in detail. Several appendices provide a detailed derivation of the Lorenz model from the Navier-Stokes equation, a summary of bifurcation theory, and some simple computer programs to study nonlinear dynamics. Each chapter includes an extensive, annotated bibliography.
Robert M. Mazo
- Published in print:
- 2008
- Published Online:
- January 2010
- ISBN:
- 9780199556441
- eISBN:
- 9780191705625
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199556441.001.0001
- Subject:
- Physics, Condensed Matter Physics / Materials
Brownian motion is the incessant motion of small particles immersed in an ambient medium. It is due to fluctuations in the motion of the medium particles on the molecular scale. The name has been ...
More
Brownian motion is the incessant motion of small particles immersed in an ambient medium. It is due to fluctuations in the motion of the medium particles on the molecular scale. The name has been carried over to other fluctuation phenomena. This book treats the physical theory of Brownian motion. The extensive mathematical theory, which treats the subject as a subfield of the general theory of random processes, is touched on but not presented in any detail. Random or stochastic process theory and statistical mechanics are the primary tools. The first eight chapters treat the stochastic theory and some applications. The next six present the statistical mechanical point of view. Then follows chapters on applications to diffusion, noise, and polymers, followed by a treatment of the motion of interacting Brownian particles. The book ends with a final chapter treating simulation, fractals, and chaos.Less
Brownian motion is the incessant motion of small particles immersed in an ambient medium. It is due to fluctuations in the motion of the medium particles on the molecular scale. The name has been carried over to other fluctuation phenomena. This book treats the physical theory of Brownian motion. The extensive mathematical theory, which treats the subject as a subfield of the general theory of random processes, is touched on but not presented in any detail. Random or stochastic process theory and statistical mechanics are the primary tools. The first eight chapters treat the stochastic theory and some applications. The next six present the statistical mechanical point of view. Then follows chapters on applications to diffusion, noise, and polymers, followed by a treatment of the motion of interacting Brownian particles. The book ends with a final chapter treating simulation, fractals, and chaos.
Peter Mörters
- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199232574
- eISBN:
- 9780191716393
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199232574.003.0008
- Subject:
- Mathematics, Geometry / Topology
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.
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.
Iwo Białynicki-Birula and Iwona Białynicka-Birula
- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198531005
- eISBN:
- 9780191713033
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198531005.003.0008
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
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
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.
David P. Feldman
- Published in print:
- 2012
- Published Online:
- December 2013
- ISBN:
- 9780199566433
- eISBN:
- 9780191774966
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199566433.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This book provides an elementary introduction to chaos and fractals. It introduces the key phenomena of chaos — aperiodicity, sensitive dependence on initial conditions, bifurcations — via simple ...
More
This book provides an elementary introduction to chaos and fractals. It introduces the key phenomena of chaos — aperiodicity, sensitive dependence on initial conditions, bifurcations — via simple iterated functions. Fractals are introduced as self-similar geometric objects and analysed with the self-similarity and box-counting dimensions. After a brief discussion of power laws, subsequent chapters explore Julia sets and the Mandelbrot set. The last part of the book examines two-dimensional dynamical systems, strange attractors, cellular automata, and chaotic differential equations.Less
This book provides an elementary introduction to chaos and fractals. It introduces the key phenomena of chaos — aperiodicity, sensitive dependence on initial conditions, bifurcations — via simple iterated functions. Fractals are introduced as self-similar geometric objects and analysed with the self-similarity and box-counting dimensions. After a brief discussion of power laws, subsequent chapters explore Julia sets and the Mandelbrot set. The last part of the book examines two-dimensional dynamical systems, strange attractors, cellular automata, and chaotic differential equations.
David P. Feldman
- Published in print:
- 2012
- Published Online:
- December 2013
- ISBN:
- 9780199566433
- eISBN:
- 9780191774966
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199566433.003.0016
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
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
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.
J. Klafter and I.M. Sokolov
- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199234868
- eISBN:
- 9780191775024
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199234868.003.0010
- Subject:
- Physics, Soft Matter / Biological Physics
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
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.
Didier Sornette
- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691175959
- eISBN:
- 9781400885091
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175959.003.0006
- Subject:
- Business and Management, Finance, Accounting, and Banking
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
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.
Pierre M. Adler, Jean-François Thovert, and Valeri V. Mourzenko
- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199666515
- eISBN:
- 9780191748639
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199666515.003.0002
- Subject:
- Physics, Condensed Matter Physics / Materials
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
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.
Max Boisot and Bill McKelvey
- Published in print:
- 2013
- Published Online:
- September 2013
- ISBN:
- 9780199669165
- eISBN:
- 9780191749346
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199669165.003.0004
- Subject:
- Business and Management, Organization Studies, Knowledge Management
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
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.
David P. Feldman
- Published in print:
- 2012
- Published Online:
- December 2013
- ISBN:
- 9780199566433
- eISBN:
- 9780191774966
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199566433.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
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
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.
David P. Feldman
- Published in print:
- 2012
- Published Online:
- December 2013
- ISBN:
- 9780199566433
- eISBN:
- 9780191774966
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199566433.003.0017
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
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
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.
David P. Feldman
- Published in print:
- 2012
- Published Online:
- December 2013
- ISBN:
- 9780199566433
- eISBN:
- 9780191774966
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199566433.003.0018
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
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
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.
David P. Feldman
- Published in print:
- 2012
- Published Online:
- December 2013
- ISBN:
- 9780199566433
- eISBN:
- 9780191774966
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199566433.003.0019
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
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
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.
David P. Feldman
- Published in print:
- 2012
- Published Online:
- December 2013
- ISBN:
- 9780199566433
- eISBN:
- 9780191774966
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199566433.003.0020
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
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
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.
David P. Feldman
- Published in print:
- 2012
- Published Online:
- December 2013
- ISBN:
- 9780199566433
- eISBN:
- 9780191774966
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199566433.003.0021
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter examines distributions known as power laws, which are fractal because they are scale-free. It first considers normal or Gaussian distributions to provide a useful contrast to power-law ...
More
This chapter examines distributions known as power laws, which are fractal because they are scale-free. It first considers normal or Gaussian distributions to provide a useful contrast to power-law or fractal distributions. Normal distributions are important because the fact of their ubiquity has influenced scientists' intuitions about statistics. The chapter also discusses the central limit theorem, the long tails of power-law distributions, and the self-similarity of both power-law distributions and fractals.Less
This chapter examines distributions known as power laws, which are fractal because they are scale-free. It first considers normal or Gaussian distributions to provide a useful contrast to power-law or fractal distributions. Normal distributions are important because the fact of their ubiquity has influenced scientists' intuitions about statistics. The chapter also discusses the central limit theorem, the long tails of power-law distributions, and the self-similarity of both power-law distributions and fractals.
David P. Feldman
- Published in print:
- 2012
- Published Online:
- December 2013
- ISBN:
- 9780199566433
- eISBN:
- 9780191774966
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199566433.003.0022
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
The Cantor set and the Koch curve are two examples of fractals may be considered not only as geometrical objects, but also as a means to explore apparent paradoxes and difficulties in set theory and ...
More
The Cantor set and the Koch curve are two examples of fractals may be considered not only as geometrical objects, but also as a means to explore apparent paradoxes and difficulties in set theory and analysis. This chapter focuses on the Cantor set and the properties of infinite sets. More specifically, it considers how big the Cantor set is. It discusses infinities of different sizes, first by thinking about finite sets, counting, countable infinities, rational numbers and irrational numbers, and binary. It then examines the cardinality of the unit interval and the Cantor set.Less
The Cantor set and the Koch curve are two examples of fractals may be considered not only as geometrical objects, but also as a means to explore apparent paradoxes and difficulties in set theory and analysis. This chapter focuses on the Cantor set and the properties of infinite sets. More specifically, it considers how big the Cantor set is. It discusses infinities of different sizes, first by thinking about finite sets, counting, countable infinities, rational numbers and irrational numbers, and binary. It then examines the cardinality of the unit interval and the Cantor set.
David P. Feldman
- Published in print:
- 2012
- Published Online:
- December 2013
- ISBN:
- 9780199566433
- eISBN:
- 9780191774966
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199566433.003.0033
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This concluding chapter highlights some of the key themes and lessons of chaos and fractals and reflects on how their impact can be characterised. It discusses how deterministic dynamical systems can ...
More
This concluding chapter highlights some of the key themes and lessons of chaos and fractals and reflects on how their impact can be characterised. It discusses how deterministic dynamical systems can yield unpredictable and seemingly random outcomes and shows that chaotic behaviour can be observed in the logistic equation, the Hénon map, the Lorenz equations, and many others. It also considers the bifurcation diagram for the logistic equation, Julia sets and the Mandelbrot set, cellular automata rule 110, and the Lorenz and Rössler attractors as instances of rich and complex images that can be achieved using very simple iterated rules or equations. Repeated application of simple deterministic procedures can generate fractals such as the Sierpiński triangle, the Koch curve, and the Cantor set.Less
This concluding chapter highlights some of the key themes and lessons of chaos and fractals and reflects on how their impact can be characterised. It discusses how deterministic dynamical systems can yield unpredictable and seemingly random outcomes and shows that chaotic behaviour can be observed in the logistic equation, the Hénon map, the Lorenz equations, and many others. It also considers the bifurcation diagram for the logistic equation, Julia sets and the Mandelbrot set, cellular automata rule 110, and the Lorenz and Rössler attractors as instances of rich and complex images that can be achieved using very simple iterated rules or equations. Repeated application of simple deterministic procedures can generate fractals such as the Sierpiński triangle, the Koch curve, and the Cantor set.
