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Fractals for the classroom: part one—introduction to fractals and chaos The title of this book reflects the authors' intent to bring chaos theory and fractal geometry to a classroom audience. The book does not assume much knowledge of technical mathematics, but rather gives readers a broad view of the notions behind fractals, chaos, and dynamics; their relation to other aspects of mathematics as well as to natural phenomena; and the visual and imaginative beauty in the structure and shapes of fractals and chaos. The book's forewo rd is by Benoit Mandelbrot, who discovered a mathematical object, the Mandelbrot set, that illustrates the deep kinship between chaos and geometry. The book is made attractive by the discussion of the history of mathematics and the explanation of different experiences in fractals. It is fully illustrated with images of fractals and is full of tables of values. The main ideas of each paragraph appear in notes in the left margin. Each chapter includes a simple program in BASIC to illustrate the computer's application in this field. Chapter 1 presents the main tools used in fractals: the principle of feedback (made concrete by a video feedback experiment), the multiple reduction copy machine (the reduction of similar images), basic types of feedback process (one-step machines, two-step machines, feedback machines, feedback machines with memory, and one-step machines with two variables), the quadratic iterator, the problem of error propagation, a population dynamic model, sensitive dependence on initial conditions, graphical iteration of feedback processes, chaos analysis, and a program for graphical iteration. Chapter 2 offers a gallery of historical fractals, with references to the concept of self-similarity: the Cantor set, Sierpinski's gasket and carpet, Pascal's triangle, the Koch curve, space-filling curves, Julia sets, and Pythagorean trees. Discussions of all of these fractals are enriched by short biographies of the people who created them and by graphical illustrations. The program in this chapter refers to the Sierpinski gasket by binary addresses. Chapter 3 deals with the concepts of limits and self-similarity. It discusses similarity and scaling in geometry, fractals, and solutions of equations, and considers basic self-similarity to grasp the limit of objects. The program addresses the Koch curve. In chapter 4 we are confronted with length, area, and dimension; measuring complexity; and scaling properties. We discover the impossibility of measuring the perimeter or area of an island exactly. The authors discuss the finite and infinite length of spirals and mention fractals with finite length (such as the Devil's Staircase). The program is about the Cantor set and the Devil's Staircase. Chapter 5 tries to introduce fractal geometry as a language that has primitive elements that may be combined into words and sentences. The authors create the concept of image coding by deterministic iterated function systems, which is based on a collection of contractions. The program in this chapter is for iterating the multiple reduction copy machine. Chapter 6 presents an interesting idea: how to arrive via the chaos game (a random game) at deterministic shapes like fractals. The authors illustrate how we can reach the Sierpinski gasket via the chaos game, variations of the chaos game, the pseudorandom generator of the computer, and a method of improved probabilities from a deterministic approximation of the attractor. The program presents the chaos game for the fern. The last chapter, chapter 7, discusses fractals in nature and the creation of irregular shapes by means of randomness. We find out about the introduction of some element of randomness into the otherwise rigorously organized fractals, an experiment that yields random fractal tree-like structures at an intermediate scale; a mathematical model based on the Brownian motion of particles implemented on a computer; the underlying scaling laws of Brownian motion; and one important generalization. With these tools in hand, fractal landscapes and coastlines can be simulated on a computer. The program in this chapter is about random midpoint displacement. The book is attractive and will entice readers to study or do research in this area. Fractals for the classroom: strategic activities (vol. 1) This book is aimed at students of mathematics and is intended for use with the first book. It contains a set of accessible hands-on exercises with fractals and discusses their underlying mathematical principles and characteristics. The book is designed for didactic purposes. It has three units, covering self-similarity, the chaos game, and complexity. The approach is through a series of activities that reveal ideas in a nonthreatening fashion. They involve readers directly in constructing, counting, computing, visualizing, and measuring. The book also focuses on a large number of mathematical connections between fractals and the contemporary mathematics curricula found in the United States. Each unit has the following sections: key objectives, notions, and connection to the curriculum; underlying notions used in the unit; mathematical background; information on using the activity sheets; and the illustrated activity sheets. The book ends with a section containing the answers for all the activities. Unit 1 involves students in constructing the Sierpinski gasket and carpet, the Sierpinski tetrahedron, Pascal's triangle, trees, and cellular automata. Unit 2 deals with the attractive chaos game connected to the Sierpinski triangle. Trees are used to tie addresses for finite sequences of plays in the chaos game to addresses for locations of subtriangles in related stages of the Sierpinski triangle. Trees are also connected to the Cantor set. This chapter includes a program for a Texas Instruments computer to simulate the chaos game, which is difficult using pencil and paper. Unit 3 is about complexity and familiarizes students with fractal dimension. The slide package that accompanies this first volume includes some of the highest-quality fractal images available anywhere. Students and educators alike will find these materials most stimulating. Using this book, students will enhance their enjoyment of mathematics as well as their awareness of the world around them. Their experiences with the fascinating and challenging topic of fractals will drive their curiosity, stretch their imagination, and pique their interest.