I have a question which has been interesting me for some time, namely when a Mandelbox has a non-empty interior. A definition of the Mandelbox may be found here. Essentially, it is defined by a Mandelbrot set-like iteration where each step is a two-step process of the form (taken from the above link):
$v_n=s\times ballFold(r,f\times boxFold(v_{n-1}))+c$
where we take any point $c$ in the plane and subject some point $v_0$ to this iteration repeatedly (I assume we can just take $v_0=0$ or $v_0=c$ for instance as for the Mandelbrot set, although I have not seen this proved). The ball fold and box fold are defined well at the linked website, and a good video that illustrates the folds is also found there. A paper with a succinct description of the Mandelbox (and some interesting further work) is here.
Essentially the box fold consists of seeing if one of the coordinates $x_i$ of the point $(x_1,x_2,...,x_n)$ under consideration is greater than 1, and if so reflecting it about the plane $x_i=1$, and doing the same for all of the $x_i$, and then doing a similar process for each plane $x_k=-1$ (checking whether points are less than $-1$). The ball fold consists of scaling the point by a certain factor if it is close enough to the origin, and also reflecting it in the unit circle if it is close enough to the origin. The Mandelbox is the set of points in $R^n$ ($R^3$ in the case of the images) for which this iteration does not diverge. One can see from this how the $c$ term in the equation will cause complicated behaviour. Typically $r$ is taken as 0.5 and $f$ is taken as 1. For large scale factors (e.g. $s=3$) the result looks quite disconnected, whereas e.g. for $s=2$ the Mandelbox looks quite solid from simulations (see images below).
This website has a comprehensive mathematical definition of the Mandelbox, and presents proofs that the interior of a Mandelbox with scale $1<s<2$ or $-2<s<-1$ is nonempty, but my question is whether it is known for any other scales (e.g. $s=2$), so specifically is it known for exactly which scales the standard Mandelbox has a nonempty interior?
(My interest in this arose from the idea that if the Mandelbox has non-empty interior then there must by the Banach-Tarski theorem be some number of finite pieces into which a cube for instance could be decomposed and reassembled to form the Mandelbox - or any other fractal with nonempty interior. Please correct me if I'm wrong. I assume that the minimum number of such pieces is not known, although I would love to be proved incorrect.)
Images to illustrate the apparent connectedness and disconnectedness mentioned above, although note that even the $s=3$ Mandelbox looks pretty connected:
$s=2$ Mandelbox:
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(source: wikimedia.org)
$s=3$ Mandelbox:
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