We love dualities in math, and fortunately for us, there are many of them. One of them — a duality between unions and intersections in set theory — is something we've already written about and designed about. We're now going to talk about another duality. Actually, we're going to talk about an "almost duality". Our desire to make it an actual duality will lead us into the amazing world of projective geometry, for which we also have a design.

#### An "Almost Duality"

We all probably remember from school that "two points make a line". Namely, given two points, we can define the line that connects them:

**assumes**that the two lines intersection. And unfortunately, parallel lines exist.

#### What to do?

We're now faced with three options. First, we could give up on the duality altogether. I don't much like that option. Second, we could rephrase the duality by saying "2 points, 1 line, and 2 non-parallel lines, 1 point." But that's ugly. Third, we could change our geometry so that this duality becomes manifest. Let's choose this option, because it means we get to keep doing math.

#### How to do it?

Let's take our inspiration from the following incredible drawing of a desert road romantically trailing off into the horizon, avoiding some very realistically-drawn mountains:

#### Lines Through The Origin

Our goal is to add some points "out at infinity" to our 2-dimensional plane in such a way that the geometry within our normal 2-d plane stays the same, but where parallel lines (in our 2-d plane) intersect out at infinity. It is likely not obvious how to do this, but fortunately for us a number of smart mathematicians have already done this work, so we'll just relay it here.

The key first step is to think about 3-d space — normal, Euclidean, 3-d space. The crucial second step is to think of all the **lines in 3-d space that go through the origin. **Namely, an entire, infinitely long line in 3-d space that **goes through the origin** will be a **single point** in our 2-d "projective space". Here is a rough schematic of how this works:

To reiterate: our 2-dimensional space is the set of all **lines through the origin **in 3 dimensions. Roughly speaking, we get only 2-dimensions worth of things because a line is 1-dimensional, so collapsing each line down to a point in our space effectively takes a 3-d space and collapses one of the dimensions. This can be, and is, made more precise in a field called "differential geometry," about which we'll have more to say elsewhere.

#### Identifying Our Plane And The Points At Infinity

So where is our normal 2-d plane hiding, and where are our points at infinity? To see this, we need to find a more manageable way of discussing these "lines through the origin." Namely, these lines through the origin are **individual points** in our projective space, so let's try to find a way to manipulate them **as individual points.**

To do this, draw a plane horizontal to the x-y plane, but sitting above the x-y plane, like so:

**don't**lie in the x-y plane

**will intersect our plane in exactly one point:**

**are still in our projective space.**That's because

**all lines through the origin**are in our projective space.

**projective space**corresponds to an infinite collection of lines through the origin in 3-d space:

**in projective space**flies off to the right in that picture, the

**lines through the origin**get closer and closer to a particular line that lies entirely in the x-y plane (in this picture it happens to be approaching the y-axis itself). This line that lies entirely in the x-y plane — the line that all these other lines-through-the-origin are approaching — is simply the "point at infinity" corresponding to this line:

**in our projective space**that is parallel to the line

**in our projective space**that we just drew. That line will also correspond to a whole collection of lines-through-the-origin, and those lines-through-the-origin will approach

**the same**line-in-the-x-y-plane that we just described. Thus, these two lines

**in our projective space**truly do intersect at a point — the point in our projective space corresponding to the line in the x-y plane that these lines-through-the-origin approach.

#### Just The Beginning

This is just the beginning of the whole story of projective space. Projective spaces give us a great set of examples for interesting things in differential geometry, algebraic geometry, sheaf cohomology, and so much more. They're also the natural first step on the way to objects called "Grassmannians".

In short (we're about to get technical for a second), we can view projective space as "the set of all 1-dimensional subspaces of an N-dimensional vector space." Grassmannians are the obvious generalization: the Grassmannian G(k,N) is the set of all k-dimensional subspaces of an N-dimensional vector space. This object is amazing to study and has actually found recent and amazing applications in theoretical physics. We'll have more to say about all of this in the future.

For now, we'd like to remind you that we have a Projective Geometry design (albeit, one that focuses on 3-dimensional projective space (so, lines through the origin of a 4-dimensional space!)) that hopefully makes a little more sense now. And, as always, if you have any questions or things you'd like to discuss, you know where to find us!