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<sphere> with two <Möbius strips> stuck into it as per the <classification of closed surfaces>.


Klein bottle

<Ciro Santilli>'s preferred visualization of the real projective plane is a small variant of the standard "lines through origin in <\R^3>".

Take a open half <sphere> e.g. a sphere but only the points with $z > 0$.

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 $z=0$ 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 $z = 0$ are the same point. But this is not the case for the <projective plane>. You cannot instantly go to any other point on the $z=0$ by just moving a little bit, you have to walk around that circle.

\Image[https://raw.githubusercontent.com/cirosantilli/media/master/spherical-cap-model-of-the-real-projective-plane.svg]
{title=Spherical cap model of the real projective plane}
{description=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>.}


Spherical cap model of the real projective 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 <\R^3>
* lines in the real projective plane are planes in <\R^3>.

  For every two projective points there is a single projective line that passes through them.

  Since it is a plane in <\R^3>, it always intersects the real plane at a line.

  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.


Ciro Santilli @cirosantilli 36

 Incoming links: Sphere