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

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In summary, the conversation discusses the possibility of extending a surface like a helicoid in the space S^3 and whether or not it can be done in a natural way. The suggested construction involves using two points on a great circle as the axis and constructing perpendicular line segments along the axis. It is also mentioned that the fact that S^3 is parallizable may be helpful.
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Spinnor
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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.
 
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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.
 

1. What is a surface like helicoid in S^3?

A surface like helicoid in S^3 is a type of surface that can be formed by rotating a straight line in 3-dimensional space around a fixed point in 3-dimensional space, with the added constraint that the line must remain perpendicular to the fixed point. This results in a spiral-like surface that is embedded in a 3-dimensional sphere (S^3).

2. What are some properties of a surface like helicoid in S^3?

One important property of a surface like helicoid in S^3 is that it is a constant mean curvature surface, meaning that its curvature is the same at every point. It also has a non-zero Gaussian curvature, which is a measure of how much the surface curves in different directions.

3. How is a surface like helicoid in S^3 different from a regular helicoid?

A regular helicoid is a surface that can be formed by rotating a straight line in 3-dimensional space around a fixed point, without the constraint of the line remaining perpendicular to the fixed point. This results in a spiral-like surface that is not embedded in a 3-dimensional sphere. In contrast, a surface like helicoid in S^3 is embedded in a 3-dimensional sphere and has the added constraint of the line remaining perpendicular to the fixed point.

4. What are some real-world examples of surfaces like helicoid in S^3?

One example of a surface like helicoid in S^3 is the shape of a DNA molecule, which has a helical structure that is similar to a helicoid. Another example is the shape of a spiral staircase, which has a helical structure that is similar to a helicoid in S^3.

5. What is the significance of studying surfaces like helicoid in S^3?

Studying surfaces like helicoid in S^3 has important applications in mathematics, physics, and engineering. It can help us better understand the properties of different surfaces and how they can be used in various fields. Additionally, it can lead to the development of new technologies and innovations in areas such as material science and structural design.

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