# Why does this fractal approximate so many others?

The Mandelbox is a three-dimensional fractal (definition below); the recursive sequence in its definition uses folding space in place of the squaring in the definition of the Mandelbrot set. It can approximate different fractal structures, appearing to contain approximations to many well-known fractals. In particular, the scale $-1.5$ Mandelbox (see below for scale) contains structures looking like Appollonian gaskets, Kleinian spheres, Maskit fractals, Mandelbrot sets, Koch snowflakes, Cantor dust, hyperbolic tilings, Menger sponges, Sierpinski triangles, and IFS and L-system fractals (images below). What I am wondering about is why this fractal is so 'general' (this generality is mentioned here, and compared with the generality of other 'kaleidoscopic' fractals).

Thus my question is: Can anyone give a mathematical explanation as to why the Mandelbox contains approximations to so many well known fractals, or an explanation of why a particular one of the listed fractals appears?

Images:

(Note: these 4 images were posted by Tglad on fractalforums (I have no idea about copyright; I assume this is fair use); they all show one fractal, the Mandelbox with $s=-1.5$; see below for $s$)

Koch snowflake and Cantor dust (see this); larger versions here and here:

Kleinian spheres (the shadows of the boxes are reminiscent of...) and Maskit fractal (see upper border of linked animations); larger versions here and here:

Also see Cantor dust, IFS and L-system fractals, Sierpinski triangle, Poincare disk hyperbolic tiling (see this), and Menger sponge (these examples were documented at a website that appears to have been removed; cached version here) and Cantor dust (from here). Here are other examples with slightly different scale; note the example with branches in the Fibonacci sequence.

I have encountered in the past the fractal formed by plotting the roots of polynomials with coefficients $\pm1$, in particular that it contains approximations to the Dragon curve, but for this there is a mathematical explanation. I am wondering if there is also an explanation for the fractal approximations in the Mandelbox.

Definition: For all $x\in\mathbb{R}$ define:

$$f(x)=\begin{cases}-2-x&&x<-1\\x&&-1\le x\le1\\2-x&&x>1\end{cases}$$

For all $\bar{x}=(x_1,x_2,x_3)\in\mathbb{R}^3$ let $\operatorname{boxFold}(\bar{x})=\left(f(x_1),f(x_2),f(x_3)\right)$; also define for all $\bar{x}\in\mathbb{R}^3$:

$$\operatorname{ballFold}(\bar{x})=\begin{cases}4\bar{x}&&||\bar{x}||<\frac{1}{2}\\\frac{\bar{x}}{||\bar{x}||^2}&&\frac{1}{2}\le||\bar{x}||<1\\\bar{x}&&||\bar{x}||\ge1\end{cases}$$

(simple explanation and illustration here). For any $s\in\mathbb{R}$ and $\bar{c}\in\mathbb{R}^3$ define $T_{s,\bar{c}}(\bar{x})=s\operatorname{ballFold}\left(\operatorname{boxFold}(\bar{x})\right)+\bar{c}$; the Mandelbox is the set $M_s=\left\{\bar{c}\in\mathbb{R}^3\;|\;\left\{T_{s;\bar{c}}^{\;\;n}(0)\right\}_{n=0}^{\infty}\nrightarrow\infty\right\}$ (i.e. the set of all $\bar{c}$ such that the sequence $T_{s,\bar{c}}^{\;\;n}(0)$ does not diverge); $s$ is called the scale of $M_s$.

• I don't know much about fractals, but here is a similar topic: how one simple machine can simulate any other machine. And it's not restricted to Turing machines either - for example Rule 110 cellular automaton can simulate any other cellular automaton (it's been proven) – Yuriy S Feb 6 '17 at 7:59
• @YuriyS Rule 110 certainly is very interesting. I'm not sure whether the methods that can be used for describing cellular automata (and which might extend to fractals described as limits of simple sequences, e.g. the Koch snowflake) will extend to fractals described like the Mandelbrot set or Mandelbox as a set of points satisfying a particular recurrence property. I do wonder this does have anything to do with that sort of simulation or whether perhaps the fractals that are simulated here might all be described by some general rule which the Mandelbox happens to be a more general form of. – Anon Feb 6 '17 at 21:38