Fractals for the Classroom: Strategic Activities Volume Two by Heinz-Otto Peitgen

By Heinz-Otto Peitgen

The comparable components that encouraged the writing of our first quantity of strategic actions on fractals persevered to motivate the meeting of extra actions for this moment quantity. Fractals supply a atmosphere in which scholars can take pleasure in hands-on reports that contain vital mathematical content material attached to quite a lot of actual and social phenomena. The remarkable photo photos, unforeseen geometric homes, and engaging numerical techniques supply unheard of chance for enthusiastic scholar inquiry. scholars experience the power found in the growing to be and hugely integrative self-discipline of fractal geom­ etry as they're brought to mathematical advancements that experience happened over the last half the 20th century. Few branches of arithmetic and machine technology provide this kind of contem­ porary portrayal of the wonderment to be had in cautious research, within the notable discussion among numeric and geometric methods, and within the vigorous interplay among arithmetic and different disciplines. Fractals proceed to provide an unusual environment for lively educating and examine­ ing actions that spotlight upon basic mathematical suggestions, connections, problem-solving recommendations, and plenty of different significant themes of user-friendly and complex arithmetic. It continues to be our desire that, via this moment quantity of strategic actions, readers will locate their delight in arithmetic heightened and their appreciation for the dynamics of the realm in­ creased. we'd like reviews with fractals to brighten up interest and to stretch the imagination.

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Example text

In order for the sequence of iterates to stay within the range, f(x) :::; 1, the maximum value possible for a is 4. As a approaches 4, this surprising, chaotic phenomenon occurs. 2. 11A The programmable calculator provides a quick way to numerically iterate the function f(x) = ax(1 - x). In this activity, we use a program to study the issues of predictable and unpredictable behavior in the iteration. We will see how small errors in the graphical or numerical evaluation of the function mayor may not lead to chaos.

Thus, the small drawing errors in the function f(x) = ax(1 - x) will be most severe when the parameter a is the greatest. In order for the sequence of iterates to stay within the range, f(x) :::; 1, the maximum value possible for a is 4. As a approaches 4, this surprising, chaotic phenomenon occurs. 2. 11A The programmable calculator provides a quick way to numerically iterate the function f(x) = ax(1 - x). In this activity, we use a program to study the issues of predictable and unpredictable behavior in the iteration.

Other results occur when a < 1 or a > 4. 1. 5. Where is the attractor? y : I : : -----t---~--1--- . 5 --L----i---- : I I I 2. __-l+- x for all the points in the interval 1 < Xo < 2· -1 essentially the same as that shown? • What about the pOints in the two intervals, -1 < Xo < 0 and 0 < Xo < 1? What are . . the iteration behaviors for the individual points -1, 0, and 1? y 3. 9. Describe the long term behavior. Is there an attractor or can a point in the interval o S Xo S 1 escape to negative infinity?

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