Spiral Formations from Iterated Exponentiation

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This Demonstration facilitates visual exploration of the sequence , , , , , etc. in the complex plane. Change the value of by moving the locator in the image pane. The first 150 terms of the sequence are then displayed, with line segments joining the successive values. For most values of in the domain, spiral patterns emerge, suggesting a convergent sequence. For some initial values , the iterates converge to a stable "orbit," rather than to a single limit point. The controls allow you to explore the space of iterations while creating visually stimulating patterns.

Contributed by: Shafer Busch and Bruce Torrence (March 2011)
After work by: Jaime Rangel-Mondragon
Open content licensed under CC BY-NC-SA


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