[...]... journals; in particular the journal Fractals covers a wide range of theory and applications Part I FOUNDATIONS Fractal Geometry: Mathematical Foundations and Application Second Edition Kenneth Falconer  2003 John Wiley & Sons, Ltd ISBNs: 0-470-84861-8 (HB); 0-470-84862-6 (PB) Chapter 1 Mathematical background This chapter reviews some of the basic mathematical ideas and notation that will be used throughout... Addison (1997) and Lesmoir-Gordon, Rood and Edney (2000) The volume by Peitgen, J¨ rgens and Saupe (1992) is profusely illustrated with u diagrams and examples, and the essays collated by Frame and Mandelbrot (2002) address the role of fractals in mathematics and science education The books by Edgar (1990, 1998), Peitgen, J¨ rgens and Saupe (1992) and Le u M´ haut´ (1991) provide basic mathematical treatments... Preface to the second edition It is thirteen years since Fractal Geometry Mathematical Foundations and Applications was first published In the meantime, the mathematics and applications of fractals have advanced enormously, with an ever-widening interest in the subject at all levels The book was originally written for those working in mathematics and science who wished to know more about fractal mathematics... contains more advanced mathematical techniques for studying fractals Much of interest may be found in proceedings of conferences on fractal mathematics, for example in the volumes edited by Cherbit (1991), Evertsz, Peitgen and Voss (1995) and Novak (1998, 2000) The proceedings edited by Bandt, Graf and Z¨ hle (1995, 2000) concern fractals and probability, those by L´ vy V´ hel, Lutton a e e and Tricot (1997),... described as fractals—cloud boundaries, topographical surfaces, coastlines, turbulence in fluids, and so on None of these are actual fractals—their fractal features disappear if they are viewed at sufficiently small scales Nevertheless, over certain ranges of scale they appear very much like fractals, and at such scales may usefully be regarded as such The distinction between ‘natural fractals’ and the mathematical. .. mathematics and physics Topics include self-similar and self-affine sets, graphs of functions, examples from number theory and pure mathematics, dynamical systems, Julia sets, random fractals and some physical applications There are many diagrams in the text and frequent illustrative examples Computer drawings of a variety of fractals are included, and it is hoped that enough information is provided... written as Ø The integers are denoted by and the rational numbers by We use a superscript + to denote the positive elements of a set; thus + are the positive real numbers, Fractal Geometry: Mathematical Foundations and Application Second Edition Kenneth Falconer  2003 John Wiley & Sons, Ltd ISBNs: 0-470-84861-8 (HB); 0-470-84862-6 (PB) 3 4 Mathematical background and + are the positive integers Occasionally... points (if at all), and so on There are fractal analogues of such properties, usually with dimension playing a key rˆ le Thus we consider, for example, the local form of fractals, o and projections and intersections of fractals Part II of the book contains examples of fractals, to which the theory of the first part may be applied, drawn from a wide variety of areas of mathematics and physics Topics include... ndez, Gwyneth Stallard and Alex Cain a xiii xiv Preface to the second edition for help with this revision I am also very grateful for the continuing support given to the book by the staff of John Wiley & Sons, and in particular to Rob Calver and Lucy Bryan, for overseeing the production of this second edition and John O’Connor and Louise Page for the cover design Kenneth J Falconer St Andrews, January 2003... Dekking, L´ vy V´ hel, Lutton and Tricot (1999) address engie e neering applications Mandelbrot’s ‘Selecta’ (1997, 1999, 2002) present a wide range of papers with commentaries which provide a fascinating insight into the development and current state of fractal mathematics and science Edgar (1993) brings together a collection of classic papers on fractal mathematics Papers on fractals appear in many journals; .