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

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

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

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 "change in phase" for a loop that lies in the plane and that goes once around the axis of the helicoid
Unless I'm misunderstanding you (which is quite possible!), isn't it impossible for such a loop of the helicoid to actually lie within the plane?

Gold Member
Unless I'm misunderstanding you (which is quite possible!), isn't it impossible for such a loop of the helicoid to actually lie within the plane?

Yes. The loop both lies in the plane that the axis of the helicoid intersects and encircles the axis of the helicoid. All points not on the surface of the helicoid can be given a phase greater then 0 and less then 2*pi in a well defined way, namely by what angle must the helicoid be rotated about its axis so the surface coincides with the point in question.