# Visualizing the domain of the square root

I would like to show someone the domain of the complex square root function (the 2-sheeted riemann surface). Is there a good interactive visualization software for this?

I would like some sort of GeoGebra style app where there is a forbidden point or disk, but outside of that you can drag a point around. As you drag around the pole, it changes from red to blue smoothly, so that on the bottom sheet it is red, and on the top sheet it is blue (or some reasonable periodic color scheme where 50% apart is always very distinct).

It would be doubly nice if one could have simple geometric shapes do the same. Basically I want a nice double covering of a plane symmetry group that is still very geometric.

It would be n-tuply nice if one could handle nth roots and the n-sheeted Riemann surface, but n=2 suffices for me, I think.

I've used other people's geogebra apps, but never made my own, and have no idea how to keep track of what sheet the point is on (or honestly how to animate the color, though I assume once I have a 0…4π valued argument function, I should be fine).

Some pretty images from wikipedia:

-
Unlike the log surface, I'm not sure it is even possible to embed the square-root surface in 3-space. The third picture you posted, while a decent representation, seems to suggest that the surface is singular along a line, whereas the truth is that it is singular only at one point (namely $0$). – Zhen Lin Sep 7 '11 at 1:40
@Zhen, since every point in the range of the function (except, arguably, zero) is hit exactly once, the Riemann surface is homeomorphic to $\mathbb C$, which is trivially embeddable in 3-space. The embedding is not smooth at the origin, but that would be a bit much to expect. – Henning Makholm Sep 7 '11 at 2:18