For some reason, Ciro Santilli is mildly obsessed with understanding and visualizing the real projective plane.
To see why this is called a plane, move he center of the sphere to , and project each line passing on the center of the sphere on the x-y plane. This works for all points of the sphere, except those at the equator . Those are the points at infinity. Note that there is one such point at infinity for each direction in the x-y plane.
It good to think about how Euclid's postulates look like in the real projective plane:
- two parallel lines on the plane meet at a point on the sphere!Since there is one point of infinity for each direction, there is one such point for every direction the two parallel lines might be at. The parallel postulate does not hold, and is replaced with a simpler more elegant version: every two lines meet at exactly one point.One thing to note however is that ther real projective plane does not have angles defined on it by definition. Those can be defined, forming elliptic geometry through the projective model of elliptic geometry, but we can interpret the "parallel lines" as "two lines that meet at a point at infinity"
- points in the real projective plane are lines in
- For every two projective points there is a single projective line that passes through them.Note however that not all lines in the real plane correspond to a projective line: only lines tangent to a circle at zero do.
Unlike the real projective line which is homotopic to the circle, the real projective plane is not homotopic to the sphere.
The topological difference bewteen the sphere and the real projective space is that for the sphere all those points in the x-y circle are identified to a single point.
One more generalized argument of this is the classification of closed surfaces, in which the real projective plane is a sphere with a hole cut and one Möbius strip glued in.
This is the standard model.
Ciro Santilli's preferred visualization of the real projective plane is a small variant of the standard "lines through origin in ".
Take a open half sphere e.g. a sphere but only the points with .
Each point in the half sphere identifies a unique line through the origin.
Then, the only lines missing are the lines in the x-y plane itself.
For those sphere points in the circle on the x-y plane, you should think of them as magic poins that are identified with the corresponding antipodal point, also on the x-y, but on the other side of the origin. So basically you you can teleport from one of those to the other side, and you are still in the same point.
Ciro likes this model because then all the magic is confined just to the part of the model, and everything else looks exactly like the sphere.
It is useful to contrast this with the sphere itself. In the sphere, all points in the circle are the same point. But this is not the case for the projective plane. You cannot instantly go to any other point on the by just moving a little bit, you have to walk around that circle.
Spherical cap model of the real projective plane
. On the x-y plane, you can magically travel immediately between antipodal points such as A/A', B/B' and C/C'. Or equivalently, those pairs are the same point. Every other point outside the x-y plane is just a regular point like a normal sphere.To see that the real projective plane is not simply connected space, considering the lines through origin model of the real projective plane, take a loop that starts at and moves along the great circle ends at .
Note that both of those points are the same, so we have a loop.
Now try to shrink it to a point.
There's just no way!
