WOLFRAM|DEMONSTRATIONS PROJECT

Spiral Formations from Iterated Exponentiation

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black background
color
thickness
Spiral Formations
This Demonstration facilitates visual exploration of the sequence
1
,
z
,
z
z
,
(
z
z
)
z
,

(
z
z
)
z

z
, etc. in the complex plane. Change the value of
z
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
z
in the domain, spiral patterns emerge, suggesting a convergent sequence. For some initial values
z
, 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.