Can a Helicoid Surface be Extended to All of S^3 in a Natural Way?

[Spinnor]

Surface like helicoid in S^3?

Consider the surface of a helicoid in cylindrical coordinates:

z = phi , see for example:

http://images.google.com/images?hl=en&q=helicoid&btnG=Search+Images&gbv=2

Now say I'm sitting in the space S^3 whose radius is much larger then my height. I hold the truncated surface of a helicoid in my hand.

Does this surface "extend" to all of S^3 in some natural way? Is there a simple function for this helicoid like z = phi?


Thanks for any help.

 

[Spinnor]

So I think I have a construction which I think I can state fairly clearly which might extend the helicoid in a "natural" way in S^3.

Pick two nearby points in the space S^3. These two points define a great circle. Let this great circle be the axis of our helicoid. Now pick a point on our great circle and from it construct a short line segment which is perpendicular to our axis. Now advance a short distance along the axis and construct another short line segment also perpendicular to our axis and rotated slightly about the axis with respect to the first line segment. Now continue this process until you come back to where you started. Make sure that the rotation was such that when you get back to the starting point that an integral (or half integral?) number of rotations was completed. Now for the final step. The short line segments define a unique great circles which form closed loops, so extend the line segments "straight" until they come back to the axis. In a smooth way fill in the entire surface.

Did I construct something like a helicoid in S^3?

Does the fact that S^3 is parallizable help us in any way?


Thanks for any help.