Let’s tetrate

A Julia set fractal

I don’t know if tetrate is a real word, but tetration is.
Also known as hyper-4 (because it’s the fourth hyperoperation, the first being addition, the second being multiplication, and the third being exponentiation) it might be more familiar to you when I say ‘power tower‘.

Let’s look at an example of a power tower or a tetration.

    \[2{{{^2}^2}^2}\]

How do you calculate is? The tetration is not associative, so

    \[2{{{^2}^2}^2}\neq(((2{{{^2})^2})^2})\]

The latter evaluates to

    \[(((2{{{^2})^2})^2})=((4{{^2})^2})=16^2=256\]

but the former (power tower) has to be caculated (by definition or agreement if you like) from above to below or right to left. So it’s

    \[2{{{^2}^2}^2}=2{{^2}^4}=2^{16}=65,536\]

There are a few ways to describe a tetration, like

    \[^4 2\]

or

    \[2\uparrow \uparrow 4\]

There are more variants of the power tower, like

    \[2{{{{^2}^2}^2}^0}\]

which we call iterated exponentials. You can calculate it yourself, and it will end up being 16 (remember: from top to bottom!).

A bonus is that tetration and fractals are related. If for ^nz with z in the complex plane and n approaching infinity the tetration will exhibit fractal behavior. No surprise given the repeating behavior of a power tower.