Fractals Everywhere: The First Course in Deterministic Fractal Geometry

This volume is the second edition of the highly successful Fractals Everywhere. The Focus of this text is how fractal geometry can be used to model real objects in the physical world.This edition of Fractals Everywhere is the most up-to-date fractal textbook available today.Fractals Everywhere may be supplemented by Michael F. Barnsley's Desktop Fractal Design System (version 2.0) with IBM for Macintosh software. The Desktop Fractal Design System 2.0 is a tool for designing Iterated Function Systems codes and fractal images, and makes an excellent supplement to a course on fractal geometry A new chapter on recurrent iterated function systems, including vector recurrent iterated function systems. This Book Is Based On A Course Called 'fractal Geometry' Which Has Been Taught In The School Of Mathematics At Georgia Institute Of Technology For Two Years. 'fractals Everywhere' Teaches The Tools, Methods, And Theory Of Deterministic Geometry. Ti Is Useful For Describing Specific Objects And Structures. Models Are Represented By Succint 'formulas.' Once The Formula Is Known, The Model Can Be Reproduced. We Do Not Consider Statistical Geometry. The Latter Aims At Discovering General Statistical Laws Which Govern Families Of Similar-looking Structures, Such As All Cumulus Clouds, All Maple Leaves, Or All Mountains. In Deterministic Geometry, Structures Are Defined, Communicated, And Analysed, With The Aid Of Elementary Transformations Such As Affine Transformations, Scalings, Rotations, And Congruences. A Fractal Set Generally Contains Infinitely Many Points Whose Organization Is So Complicated That It Is Not Possible To Describe The Set By Specifying Directly Where Each Point In It Lies. Instead, The Set May Be Define By 'the Relations Between The Pieces.' It Is Rather Like Describing The Solar System By Quoting The Law Of Gravitation And Stating The Initial Conditions. Everything Follows From That. It Appears Always To Be Better To Describe In Terms Of Relationships. Metric Spaces; Equivalent Spaces; Classification Of Subsets; And The Space Of Fractals -- Spaces -- Metric Spaces -- Cauchy Sequences, Limit Points, Closed Sets, Perfect Sets, And Complete Metric Spaces -- Compact Sets, Bounded Sets, Open Sets, Interiors, And Boundaries -- Connected Sets, Disconnected Sets, And Pathwise-connected Sets -- The Metric Space (h (x), H): The Place Where Fractals Live -- The Completeness Of The Space Of Fractals -- Additional Theorems About Metric Spaces -- Transformations On Metric Spaces; Contraction Mappings; And The Construction Of Fractals -- Transformations On The Real Line -- Affine Transformations In The Euclidean Plane -- Mobius Transformations On The Riemann Sphere -- Analytic Transformations -- How To Change Coordinates -- The Contraction Mapping Theorem -- Contraction Mappings On The Space Of Fractals -- Two Algorithms For Computing Fractals From Iterated Function Systems -- Condensation Sets -- How To Make Fractal Models With The Help Of The Collage Theorem -- Blowing In The Wind: The Continuous Dependence Of Fractals On Parameters -- Chaotic Dynamics On Fractals -- The Addresses Of Points On Fractals -- Continuous Transformations From Code Space To Fractals -- Introduction To Dynamical Systems -- Dynamics On Fractals: Or How To Compute Orbits By Looking At Pictures -- Equivalent Dynamical Systems -- The Shadow Of Deterministic Dynamics -- The Meaningfulness Of Inaccurately Computed Orbits Is Established By Means Of A Shadowing Theorem -- Chaotic Dynamics On Fractals -- Fractal Dimension. Michael Barnsley. Includes Index. Bibliography: P. 381-384. "This book is based on a course called 'Fractal Geometry' which has been taught in the School of Mathematics at Georgia Institute of Technology for two years. 'Fractals Everywhere' teaches the tools, methods, and theory of deterministic geometry. Ti is useful for describing specific objects and structures. Models are represented by succinct 'formulas.' Once the formula is known, the model can be reproduced. we do not consider statistical geometry. The latter aims at discovering general statistical laws which govern families of similar-looking structures, such as all cumulus clouds, all maple leaves, or all mountains. In deterministic geometry, structures are defined, communicated, and analysed, with the aid of elementary transformations such as affine transformations, scalings, rotations, and congruences. A fractal set generally contains infinitely many points whose organization is so complicated that it is not possible to describe the set by specifying directly where each point in it lies. Instead, the set may be define by 'the relations between the pieces.' It is rather like describing the solar system by quoting the law of gravitation and stating the initial conditions. Everything follows from that. It appears always to be better to describe in terms of relationships." This edition also features additional problems and tools emphasizing fractal applications, as well as a new answer key to the text exercises.