The Power Tower Fractal

We can construct a beautiful fractal set by defining an operation of iterating exponentials and applying it to the numbers in the complex plane. The operation is tetration and the fractal is called the power tower fractal or sometimes the tetration fractal. A detail of the set is shown in the figure here.

Detail of the power tower fractal.

The Operation of Tetration

We define an operation known as tetration, so-called because it is the fourth in a sequence of operations starting with addition, multiplication and exponentiation. For any complex number ${z\in\mathbb{C}}$ we define

$\displaystyle ^{0}z = 1 \qquad ^{1}z = z \qquad ^{2}z = z^{z} \qquad ^{3}z = z^{z^{z}} \qquad ^{n}z = \underbrace{ {z^{z^{z^{.^{.^{.^{z}}}}}}}}_{n\ \mathrm{terms}} \,.$

We call ${^{n}z}$ the n-th tetration of ${z}$ . It is an exponential stack with ${n}$ levels. It may be defined recursively:

$\displaystyle ^{0}z = 1 \,, \qquad ^{n+1}z = z^{(^{n}z)} \,.$

Exponential towers like this grow with spectacular rapidity. Thus, even with ${z=2}$ ,

$\displaystyle ^{2}2 = 4 \qquad ^{3}2 = 2^{2^{2}} = 16 \qquad ^{4}2 = 65{,}536 \qquad ^{5}2 \approx 2\times 10^{19{,}728} \,.$

With ${z=10}$ we find that ${^{2}10 = 10^{10}}$ , so that ${^{3}10}$ is ${1}$ followed by ten billion zeros and ${^{4}10}$ is vastly greater than a googolplex (ten to the power of a googol).

If the process is continued indefinitely, we can write the limit as

$\displaystyle ^{\infty}z = \lim_{n\rightarrow\infty}{^{n}z}$

Euler showed that this limit is finite for real ${z}$ in the interval ${[e^{-e},e^{1/e}]}$ . For example, if ${z=\sqrt{2}}$ then ${^{\infty}z = 2}$ . More generally, ${^{\infty}z}$ is related to the Lambert W-function: ${^{\infty}z = -W(-\log z)/\log z}$ .

There is no commonly agreed extension of the tetration operation to the case where ${n}$ is replaced by a real or complex number, although several definitions have been proposed.

The Power Tower Fractal

In previous posts, we have considered the real power-tower ${^{\infty}x}$ , and also the case where ${z=i}$ yielding the imaginary power tower. We now come to the behaviour of the sequence ${\{^{n}z\}}$ for complex ${z}$ . For some values of ${z}$ this converges to a finite value; for other values it is periodic; for others it varies erratically but remains finite; and for yet others, it “escapes” to infinity.

The boundary separating the values in the ${z}$ -plane for which the sequence is finite from those for which it escapes to infinity is fractal. We define ${\mathbf{\Pi}}$ to be the set of points for which ${^{\infty}z}$ is finite. Then the `escape’ set is the complement of this set
in the complex plane.

The figure below shows the escape region in light yellow and the region where the sequence is bounded in dark blue. This image was generated using code acquired from the website of Peter Young (see URL below). It was a simple matter to replace the equation ( ${z_{n+1}=z_{n}^2 + c}$ ) defining the Mandelbrot set by the equation ( ${z_{n+1} = z^{z_{n}}}$ ) defining the power tower sequence. The boundary of the set ${\mathbf{\Pi}}$ is exquisitely complex. It appears to hold riches comparable to those of the much-studied Mandelbrot set.

Power tower fractal for |x| and |y| less than 10 (left) and less than 4 (right).

The right panel of the figure is a zoom from ${[-10,10]\times[-10,10]}$ to ${[-4,4]\times[-4,4]}$ .

In the figure below further zooms are shown. On the left, we zoom in to the structure to the left of the origin ( ${-3.25\le x\le 0.25}$ , ${-1.75\le y\le 1.75}$ . A further zoom (right panel ${-0.525\le x\le 0.225}$ , ${-0.375\le y\le 0.375}$ ) shows a structure reminiscent of a marine creature. Let’s call it the Crab. The small white spot in the right panel marks the origin.

Zoom to the region left of the origin (left) and further zoom to the Crab (right).

In the next figure (left panel) we zoom in to the antenna of the Crab ( ${-0.23\le x\le -0.13}$ , ${+0.2\le y\le 0.3}$ ) and in the right panel blow up the spiral structure visible in the left ( ${-0.193\le x\le -0.183}$ , ${+0.23\le y\le 0.24}$ ).

Zoom to the Crab antenna (left) and further zoom to a spiral (right).

The process of zooming in can be continued indefinitely, revealing more structure as the resolution increases. It must be noted that the fine details at any resolution are not reliable, as they depend on the convergence criteria. It is necessary to set the escape radius to a very large value (e.g. ${r_{max}=10^{48}}$ ) and to allow a large number of iterations. Structures that appear to be disjoint may be connected by fine filaments that are visible only at high resolution.

Conclusion

Much more may be said about the power tower fractal. Amongst the matters we have not considered are fixed points, for which ${^{\infty}z=z}$ . Clearly, ${z=1}$ and ${z=-1}$ are fixed points. Then there are periodic orbits. The site http://www.tetration.org/ discusses periodic orbits, finding orbits of period three and noting that Sarkovskii’s Theorem implies that a map containing period three must contain all periods from one to infinity.
Sources

Paul Bourke’s web pages:
http://paulbourke.net/fractals/tetration/index.html.
The Power Tower. Blog post on thatsmaths.com:
https://thatsmaths.com/2013/01/10/the-power-tower/.
A website dealing with many aspects of tetration, including the power tower fractal: http://www.tetration.org/
Young, Peter, UCSC: The Mandelbrot set: an example of a fractal.
http://young.physics.ucsc.edu/115/mandelbrot.pdf.

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