- #1

Spinnor

Gold Member

- 2,123

- 317

## Main Question or Discussion Point

Invariant of a helicoid, like an electron but not quite.

Consider the surface of a helicoid whose axis extends to infinity, see for example:

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

This surface has an interesting geometrical invariant. Consider a helicoid whose axis intersects an infinite plane. Let a perpendicular to this plane be arbitrarily labeled up. Consider the "change in phase" for a loop that lies in the plane and that goes once around the axis of the helicoid. I posit the "change in Phase" is 2*pi regardless of the angle the axis of the helicoid makes with respect to the plane? I have not defined the "phase" of a helicoid, let me try to do that now. In cylindrical coordinates (of proper orientation) the surface of a helicoid is :

z = phi

If we add 2*pi to phi we have in effect rotated the helicoid by an angle of 2*pi. The surface is invariant to rotation by any multiple of 2*pi. Because of this I suspect the above is true, namely:

"the "change in Phase" is 2*pi regardless of the angle the axis of the helicoid makes with respect to the plane?"

So in one way a helicoid is like an electron but not quite. Can you modify this picture to make the analogy with an electron more exact?

Thanks for any help.

Consider the surface of a helicoid whose axis extends to infinity, see for example:

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

This surface has an interesting geometrical invariant. Consider a helicoid whose axis intersects an infinite plane. Let a perpendicular to this plane be arbitrarily labeled up. Consider the "change in phase" for a loop that lies in the plane and that goes once around the axis of the helicoid. I posit the "change in Phase" is 2*pi regardless of the angle the axis of the helicoid makes with respect to the plane? I have not defined the "phase" of a helicoid, let me try to do that now. In cylindrical coordinates (of proper orientation) the surface of a helicoid is :

z = phi

If we add 2*pi to phi we have in effect rotated the helicoid by an angle of 2*pi. The surface is invariant to rotation by any multiple of 2*pi. Because of this I suspect the above is true, namely:

"the "change in Phase" is 2*pi regardless of the angle the axis of the helicoid makes with respect to the plane?"

So in one way a helicoid is like an electron but not quite. Can you modify this picture to make the analogy with an electron more exact?

Thanks for any help.