Banach-Tarski Paradox for the unit interval? Does there exist such a map $\phi: [0, 1] \rightarrow S^2$ which determines some sort of a Banach-Tarski decomposition for the unit interval $[0, 1]$? I did read through Stan Wagon's The Banach-Tarski Paradox but the language is somewhat terse. I did read somewhere that Felix Hausdorff proved that one can chop up the unit interval into countably many pieces, slide the pieces around, and fit them together into the interval $[0, 2]$. Not sure of any journals that describe the exact proof. Is it possible to find $\phi$ explicitly perhaps in terms of the middle-third Cantor set?

One cannot expect an explicitly constructible example. By an old result of Solovay, if ZF (Zermelo-Fraenkel Set Theory without Choice) plus "there exists an inaccessible cardinal" is consistent (which is generally believed to be the case), then so is ZF plus "every set of reals is Lebesgue measurable." 
The sets involved in the paradoxical decomposition cannot all be Lebesgue measurable. Indeed thinking of them as measurable is what makes the decomposition seem paradoxical. Thus some version of the Axiom of Choice is needed, and an explicit set-theoretic construction from just ZF is not possible. 
As for "one dimensional" paradoxical decompositions into a finite number of sets, these are not possible. For it can be shown that there is a finitely additive translation-invariant "measure" that extends Lebesgue measure and is defined on all sets of reals. Using such a  "measure" we can translate the intuitive notion of paradoxicalness into a proof that there is no paradoxical decomposition. The Banach-Tarski Paradox can be thought of as a proof that there is no translation invariant finitely additive "measure" defined on all subsets of $\mathbb{R}^3$.
