On the connected components of fractal cubes

We show that a fractal cube $F$ in ${\mathbb{R}}^3$ may have an uncountable set $Q$ of connected components $K_{\alpha }$ neither of which is contained in any plane, whereas the set $Q$ is a totally disconnected self-similar subset of the hyperspace $C({\mathbb{R}}^3)$, isomorphic to a Cantor set.