A Closer Look at Projective Space

Someone asked me to explain the real projective plane a bit better, so here's a post on RP^n. Once again, for a general audience, as this one is pretty geometric. So we'd defined the real projective plane as the set of lines through the origin, but what does this mean?First the formal definition, but don't worry about it too much: n-dimensional projective space (RP^n) is the set of all lines through the origin in (n+1)-dimensional Euclidean space. Let's go through that bit by bit, because it's a lot.

So what's n-dimensional Euclidean space (R^n)? This is just normal, flat space. Examples should help:

-R^1 is a line

-R^2 is a plane, i.e. an infinitely big piece of paper

-R^3 is everyday space

Note that the dimension, n, is the number of independent directions you can move.

Now, why would we think the set of lines through the origin would be a nice shape (manifold), or even a shape at all? Well let's look at the first few cases. The set of all lines through a line is, well, just the one line. So RP^0 is just a point.

RP^1, the real projective line, is the set of lines through the origin in a plane, the picture should help. How do we see this is a manifold? Well if we draw a circle around the origin each line hits exactly two points of the circle.

We'd really like a shape where each point represents one line, so we make one! Just fold the circle in half and glue it together to get a semi-circle. We also need to glue the two ends together, these represent a horizontal line. The result is.. a circle!

So we have a shape which describes RP^1: each point represents exactly one line. Even better, this shape is a circle! (topologically at least, remember we're allowed stretch and squish) So in some way RP^1 is actually a circle, which we know is a manifold!

So that was a lot, have a think about it for a little while. As a break I should say the motivation from this comes from perspective and vanishing points. When we look down train tracks the bars seem to get closer to each other, even though they don't. There's a construction which says that these parallel lines do actually converge at a point, infinity! To go into this properly have a look at stereographic projection, I unfortunately won't get to it this time. It should hopefully make the name "projective space" make sense.

Okay now that we know how the projective line works, lets try the same with RP^2! Just like before we notice each line intersects a sphere around the origin at two points. So as before we cut the sphere in half, making a hemisphere. Now we also have to glue opposite points of the circle on the boundary, but when we do this we have to twist the circle! The picture might help, but the best thing to do is try with a piece of paper. Now hopefully the original picture makes sense! We want to stretch the shape back out to a sphere like we did with the glued circle before, but we messed it up! The sphere has a twist, it looks like it intersects itself.

Here's some different views. We really can't draw it properly: we say it can't be embedded in R^3. This means we can't represent it without it seeming to intersect itself. Turns out it can be embedded in R^4, but that doesn't help us see it. Something interesting about the twist is that it actually makes RP^2 unorientable - the shape doesn't have an inside and outside.

Still, I hope that you can believe that this is a manifold. The best way to formally show this for all projective spaces is using stereographic projection, but I really just want you to get intuition.

Here's some further reading:

-An excellent video showing how we can view RP^2 as a Möbius band and a disk. The Möbius band is a strip of paper with its ends glued together with a half twist.

-An explanation of the above video

-Here's post by math3ma which talks about that video and uses it to compute the fundamental group of RP^2. As a warning this one is fairly complicated.

That's probably enough for now! Higher projective spaces are a nightmare to visualise, at that stage I just rely on the equations. Maybe someone else can pitch in a neat interpretation?