Fractals for the Classroom: Part Two: Complex Systems and Mandelbrot Set

ISBN-13: 9781461287582 / Angielski / Miękka / 2011 / 500 str.

Heinz-Otto Peitgen; Hartmut Jurgens; Dietmar Saupe

Opis

Fractals for the Classroom breaks new ground as it brings an exciting branch of mathematics into the classroom. The book is a collection of independent chapters on the major concepts related to the science and mathematics of fractals. Written at the mathematical level of an advanced secondary student, Fractals for the Classroom includes many fascinating insights for the classroom teacher and integrates illustrations from a wide variety of applications with an enjoyable text to help bring the concepts alive and make them understandable to the average reader. This book will have a tremendous impact upon teachers, students, and the mathematics education of the general public. With the forthcoming companion materials, including four books on strategic classroom activities and lessons with interactive computer software, this package will be unparalleled.

Spis treści

Introduction: Causality Principle, Deterministic Laws and Chaos.- 8 Recursive Structures: Growing of Fractals and Plants.- 8.1 L-Systems: A Language For Modeling Growth.- 8.2 Growing Classical Fractals with MRCMs.- 8.3 Turtle Graphics: Graphical Interpretation of L-Systems.- 8.4 Growing Classical Fractals with L-Systems.- 8.5 Growing Fractals with Networked MRCMs.- 8.6 L-System Trees and Bushes.- 8.7 Program of the Chapter: L-systems.- 9 Pascal’s Triangle: Cellular Automata and Attractors.- 9.1 Cellular Automata.- 9.2 Binomial Coefficients and Divisibility.- 9.3 IFS: From Local Divisibility to Global Geometry.- 9.4 Catalytic Converters or how many Cells are Black?.- 9.5 Program of the Chapter: Cellular Automata.- 10 Deterministic Chaos: Sensitivity, Mixing, and Periodic Points.- 10.1 The Signs of Chaos: Sensitivity.- 10.2 The Signs of Chaos: Mixing and Periodic Points.- 10.3 Ergodic Orbits and Histograms.- 10.4 Paradigm of Chaos: The Kneading of Dough.- 10.5 Analysis of Chaos: Sensitivity, Mixing, and Periodic Points.- 10.6 Chaos for the Quadratic Iterator.- 10.7 Numerics of Chaos: Worth the Trouble or Not?.- 10.8 Program of the Chapter: Time Series and Error Development.- 11 Order and Chaos: Period-Doubling and its Chaotic Mirror.- 11.1 The First Step From Order to Chaos: Stable Fixed Points.- 11.2 The Next Step From Order to Chaos: The Period Doubling Scenario.- 11.3 The Feigenbaum Point: Entrance to Chaos.- 11.4 From Chaos to Order: a Mirror Image.- 11.5 Intermittency and Crises: The Backdoors to Chaos.- 11.6 Program of the Chapter: Final State Diagram.- 12 Strange Attractors: The Locus of Chaos.- 12.1 A Discrete Dynamical System in Two Dimensions: Hénon’s Attractor.- 12.2 Continuous Dynamical Systems: Differential Equations.- 12.3 The Rössler Attractor.- 12.4 The Lorenz Attractor.- 12.5 The Reconstruction of Strange Attractors.- 12.6 Fractal Basin Boundaries.- 12.7 Program of the Chapter: Rössler Attractor.- 13 Julia Sets: Fractal Basin Boundaries.- 13.1 Julia Sets as Basin Boundaries.- 13.2 Complex Numbers — A Short Introduction.- 13.3 Complex Square Roots and Quadratic Equations.- 13.4 Prisoners versus Escapees.- 13.5 Equipotentials and Field Lines for Julia Sets.- 13.6 Chaos Game and Self-Similarity for Julia Sets.- 13.7 The Critical Point and Julia Sets as Cantor Sets.- 13.8 Quaternion Julia Sets.- 13.9 Program of the Chapter: Julia Sets.- 14 The Mandelbrot Set: Ordering the Julia Sets.- 14.1 From the Structural Dichotomy to the Potential Function.- 14.2 The Mandelbrot Set — A Road Map for Julia Sets.- 14.3 The Mandelbrot Set as a Table of Content.- 14.4 Program of the Chapter: The Mandelbrot Set.