I have a question regarding a visualization in Tristan Needham's Visual Complex Analysis.
Now let us return to [29]. By arbitrarily picking one of the three values of $\sqrt[3]{p}$ at $z=p$, and then allowing $z$ to move, we see that we obtain a unique value of $\sqrt[3]{Z}$ associated with any particular path from $p$ to $Z$. However, we are still dealing with a multifunction: by going round the branch point at 0 we can end up at any one of the three possible values of $\sqrt[3]{Z}$.
On the other hand, the value of $\sqrt[3]{Z}$ does not depend on the detailed shape of the path: if we continuously deform the path without crossing the branch point then we obtain the same value of $\sqrt[3]{Z}$. This shows us how we may obtain a singlevalued function. If we restrict $z$ to any simply connected set $S$ that contains $p$ but does not contain the branch point, then every path in $S$ from $p$ to $Z$ will yield the same value of $\sqrt[3]{Z}$, which we will call $f_{1}(Z)$. Since the path is irrelevant, $f_{1}$ is an ordinary, single-valued function of position on $S$; it is called a branch of the original multifunction $\sqrt[3]{z}$
Figure [31] illustrates such a set $S$, together with its image under the branch $f_{1}$ of $\sqrt[3]{z}$.Here we have reverted to our normal practice of depicting the mapping going from left to right. If we instead choose $\sqrt[3]{p}=b$ then we obtain a second branch $f_{2}$ of $\sqrt[3]{z}$, while $\sqrt[3]{p}=c$ yields the third and final branch $f_{3}$. Notice, incidentally, that all three branches display the by now ubiquitous (yet mysterious) preservation of small squares.
Why does the image of this set split up this keyhole shaped set into three sectors? Moreover, why is the grid part seen on the left hand side mapped to each sector?