Explanation of example of Banach Tarski . I am reading Probability with Martingales by David Williams. In the book he gives an example where Banach and Tarski showed that if the Axiom of Choice is assumed, as it is throughout conventional mathematics, then there exists a subset $F$ of the unit sphere $S^2$ in $R^3$ such that for $3 < k < \infty$ (and even for $ k =\infty$), $S^2$ is the disjoint union of $k$ exact copies of $F$:$$S^2=\bigcup_{i=1}^{k} \tau^{(k)}_iF$$ Where each $\tau_{i}^k$ is a rotation. If $F$ has an 'area', then that area must simultaneously be $4\pi/3,4\pi/4,..., 0$. The only conclusion is that the set $F$ is non-measurable (not Lebesgue measurable): it is so complicated that one cannot assign an area to it. Banach and Tarski have not broken the Law of Conservation of Area: they have simply operated outside its jurisdiction.Question{what i am thinking is they are taking a a Subset $F$  of the unit sphere ,and i can't think how are they rotating this sphere is it about an axis and trying to attach these set  by rotating and making a whole new sphere.}
 A: Although Williams doesn't say explicitly, the rotations $\tau_i$ are rotations about distinct axes through the origin.
Williams's statement of the Banach-Tarski paradox says there is a subset $F\subset S^2$ and a sequence of rotations $\tau_1,\tau_2,\tau_3,\cdots$ such that
$$\begin{array}{ll}S^2&=\tau_1^3F \bigcup\tau_2^3F\bigcup\tau_3^3F\\
&=\tau_1^4F \bigcup\tau_2^4F\bigcup\tau_3^4F\bigcup\tau_4^4F\\
&=\tau_1^5F \bigcup\tau_2^5F\bigcup\tau_3^5F\bigcup\tau_4^5F\bigcup\tau_5^5F\\
&=\cdots
\end{array}$$
where the sets in each right-hand-side line are disjoint.
That is, $S^2$ can be made by 3 non-overlapping copies of $F$, by 4 non-overlapping copies of $F$, by 5 non-overlapping copies of $F$ etc. where each copy of $F$ is a rotation of $F$ about an axis through the origin.
The subset $F$  is not an area in any normal sense: it's more like a cluster of scattered points. When reassembled in $S^2$, any open disk of $S^2$ will contain points from each of the copies of $F$. That is, no open disk of $S^2$ lies completely inside $F$.
For the proof of the paradox, you'd need some elementary group theory - at least up to the level of understanding group actions, orbits and free groups. Then you should be able to work through the proof sketch in Wikipedia.
