Are planes one-sided or two-sided?

Question

I'm looking for clarity here - because, although it seems that planes are treated normally as one-sided, as I understand it, when I read of 'projection onto a plane', it seems that the projection is going to be on a single side of the plane. But, for instance, if we were to use both sides of the plane, then a spherical projection onto a plane could use 'the other side' for the upper hemisphere, but it doesn't seem to be done that way?

A simple, but well-illustrated and intuitive answer would be welcome.

Answer 1

It depends on what you consider two-sided. Let's think of your simple example of the surface of the unit sphere in $\mathbb R^3$. We fix the norh pole $x=(0,0,1)$.

If you want to move from $x$ out of the plane, you can move either inside or outside the sphere. This is due to the fact, that there exist two possible normal vectors. The distinction of these normal vectors is important for example in Gauss' Thorem.

If you think of sides similar to sides of a cube i.e. surfaces that enclose your set, or boundaries, then a plane consists only of one boundary. If you consider again the north pose $x$ it does not lie on the inside or outside of the surface of the sphere, and the same holds for every point on any plane. In some cases, like the projection on the plane the distinction between normal vectors as mentioned before is just not relevant.

Answer 2

A plane in its self does not have any side. It is just 2D. The concept of 2-sidedness asks for an embedding of your plane into a 3D environment.

This is just like a line, which by itself does not have any sides. Only its embedding into a 2D world will provide an upper an lower side of that line.

But be aware, when you'd embed that very line into a 3D world, then you'd get a continuous one-parametric plentitude of "sides"! - The same would hold for your plane, when embedded within a 4D setup.

Even worse for a single point within a 3D world. That one would have a continuous 2-parametric plentitude of "sides". Same holds for the line in a 4D setup and also for your plane within a 5D environment.

--- rk