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RE: Stefan Banach — The Prince of Polish mathematicians
In relation to the Banach-Tarski paradox it is maybe important to state that it concerns translation and rotation transformation but not scaling transformations. If you are allowed to use scaling transformations then the statement would be trivial.
So the singularity of the Banach-Tarski paradox lies in not concerning scaling transformation. And despite of this it is possible to get such paradoxical duplication of figures because of considering 1-dimensional or non-measurable parts which are not subject for scaling operations. Something like that? :)
Yup that is pretty much it. In addition, Banach-Tarski specifies explictely what type of transformations you are allowed to use to construct the two spheres.
Yeah! Thank you @mathowl, your suggestions are always valuable :). I am not quite good in this field of mathematics so I decided to not write extensive content ;)