Fractals for the Classroom Vol. 3 : Strategic Activities

by Peitgen, Heinz-Otto; Jurgens, Hartmut; Saupe, Dietmar; Maletsky, Evan; Perciante, Terry; Peitgen, Heinz-Otto

• ISBN13: 9780387984209
• ISBN10: 0387984208
• Format: Paperback
• Copyright: 1999-06-01
• Publisher: Springer Verlag

Summary

This third and final volume of Strategic Activities on fractal geometry and chaos theory focuses upon the images that for many people have provided a compelling lure into an investigation of the intricate properties embedded within them. By themselves the figures posses fascinating features, but the mechanisms by which they are formed also highlight significant approaches to modeling natural processes and phenomena. The general pattern and specific steps used to construct a fractal image illustrated throughout this volume, comprise an iterated function system. The objective of this volume is to investigate the processes and often surprising results of applying such systems. These strategic activities have been developed from a sound instructional base, stressing the connections to the contemporary curriculum as recommended in the National Council of Teachers of Mathematics' Curriculum and Evaluation Standards for School Mathematics. Where appropriate, the activities take advantage of the technological power of the graphics calculator. The contents of this volume joined with the details contained in the prior two books. Together they provide a comprehensive survey of fractal geometry and chaos theory. The dynamic nature of the research and the experimental characteristics of related applications provides an engaging paradigm for classroom activity.

Table of Contents

Preface	vii
Authors	viii
Table of Contents	ix
Connections to the Curriculum	x
Foreword by Jana Wallace	xii
Unit 7 IFS IN TWO DIMENSIONS
Key Objectives, Notions, and Connections	1	(1)
Mathematical Background	2	(3)
Using the Activities Sheets	5	(32)
The Sierpinski Curve	9	(2)
Why the Chaos Game Works	11	(4)
Addressing the Sierpinski Gasket and Binary Numbers	15	(4)
Combinations	19	(2)
Sierpinski's and Pascal's Triangle via Combinations	21	(4)
Kummer's Criterion	25	(2)
Mappings	27	(2)
Copying Machine Gone Wild	29	(6)
The Sierpinski Triangle, Sierpinski Curve, and Self-Similarity	35	(2)
Unit 8 IFS AND GEOMETRIC GENETIC CODES
Key Objectives, Notions, and Connections	37	(1)
Mathematical Background	38	(2)
Using the Activities Sheets	40	(47)
Symmetries	47	(2)
Compositions	49	(4)
Algebraic Mappings	53	(4)
Fractal Images	57	(4)
Building Fractals	61	(6)
Summarizing the Process	67	(6)
Chaos Game Variations	73	(4)
Families of Fractals	77	(4)
Originals and Relatives	81	(6)
Answers	87