Infinity and the Projective Plane
You may wonder where the projective plane gets it's name from. To answer this question, we must first look to projective geometry. One way to think of projective geometry is to imagine taking a photograph of something. Obviously the photograph is not the same thing as the object it was taken of; in fact it is not even the same shape. The photograph is flat whereas the object probably was not. Nevertheless, when we look at the photograph, we can "see" what the photograph was taken of. How is this possible? The answer is that the human eye works in much the same way as a camera. We don't actally "see" points in space as they actually are, but only the light from them that gets projected onto our retina. Because of this, two very different objects can look the same so long as the light reflected from them results in the exact same pattern of light falling onto our retina. This also means that we cannot judge the distance to an object, at least not with only one eye. (With two eyes the brain is able to gauge the depth of objects, so long as they aren't more than a dozen or so meters away. But let's suppose for the moment that we are looking at a scene from a large enough distance that having a second eye doesn't help us.) Let us consider for a moment the pinhole camera. Even though the pupil of our eye or the aperture of a camera isn't quite a pinhole, they work in approximately the same way. For objects which are far enough away, we can assume for all practical purposes that our eye functions like a pinhole camera.
Notice in the figure on the left that the red points A and B in space are both projected onto the same point, C, on the film surface. This means that points A and B will look like the same point in the photograph. They will be indistinguishable. Notice also that the image on the film will actually be upside down. This is easily corrected by just turning the film around, so photographers need not be concerned about this. In fact the image on our retina is upside down too, but this apparently causes no problem for our brain. The projection process can also be made to work in reverse, by placing a human observer where the pinhole was and displaying an image on a screen (such as a movie screen, computer monitor, or photograph). This is shown in the figure on the right. Since the image that falls on the retina will be the same as if it had come from real life, the human will see the object that the image was taken of, rather than just a flat screen. If the light coming from the screen could somehow be made to perfectly match the color, intensity, etc. of the light that had come from the original object, the oberver would not be able to tell it wasn't real. Obviously technology has not yet been able to do this, but in principle it is possible.
This leads to a model, called Projective Geometry, where two points in space are considered equivalent if they will be seen as the same point. To be more precise, suppose I put the pinhole of the camera or a human's eye at the point (0,0,0) in space. Then two points in space will be considered the same if they are on the same line going through (0,0,0). If they are both in front of the observer, they will be seen as the same point. If they are on opposite sides of (0,0,0), one can think of one point as an object, and the other one as the image of that object on a film surface or retina. Two points behind the observer can be thought of as two possible locations for the image of the same point with the retina or film surface simply placed at different locations. One advantage to using this model is that we can then use this exact same model for all three situations.
Having done this, one natural question to ask is whether each point in space can be assigned to some sort of "representative point", so that points which are equivalent both have the same representative point. One possible idea is to put a plane in front of the observer (sort of like an infinite "movie screen"). Then every point in space has a representative on the screen somewhere, well almost!
Unfortunately, points which lie at a 90 degree angle to the observer will not be projected anywhere on the screen, since the line going through (0,0,0) and one of these points will be parallel to the screen. Another idea might be to put a sphere around the observer. This is the idea behind the Omnimax theaters, although they are more of a hemisphere than a full sphere. A full sphere is somehow "too much". Any two diametrically opposite points on the sphere would be equivalent, since the line through them goes through (0,0,0). So now we seem to have the opposite problem. Even cutting the sphere down to a hemisphere would not quite solve the problem, since the circular edge of the hemisphere would still contain twice as many representative points as necessary.
What we do instead is to define the Projective Plane, which is the original idea of an infinite plane, but we add what are called "points at infinity". A point at infinity may be a difficult thing to grasp, but in fact we've all seen them. Just look for a really straight railroad track. The place where the tracks appear to meet is a point at infinity. But the tracks don't actually meet, you say! You are right, they don't meet. This is why a plane, even though going on infinitely, does not actually contain points at infinity. To imagine how we could possibly add such points, we must first squish down the plane into a finite sized region. Now you may wonder "How is it possible to take something infinite and make it finite?". The answer is that even a finite sized region is still in some sense infinite. Just as you can keep dividing the numbers between 0 and 1 infinitely (so in fact there are infinitely many numbers between 0 and 1, a finite distance), so also we can divide space up infinitely. One way to compress the infinite plane into a finite sized region is with a wide angle mirror, as shown below.
What you're seeing is the image of an infinitely large checkerboard in a spherical mirror. Notice the entire infinite plane has been crammed into an open disc (in mathemtics, an open disc is the set of all points inside of, but not on, a circle, and a closed disc includes the points inside of and on the circle). This gives us some idea of "where" to put the new so-called "points at infinity". We should place them on the boundary circle! Even this picture is still not quite right, since opposite points at infinity should be considered the same (Remember the idea of a line through (0,0,0) parallel to the movie screen? Both "ends" of the line are points at infinity that should be considered the same, since there is a line connecting them going through (0,0,0). If you have trouble imagining the "ends" of the line, just think of what the line would look like in the spherical mirror.) So what we really need to do is to take the (now closed) disc and stitch together the opposite points on the circle.
Actually doing this is a lot harder that it may seem. In fact it has been proven that in 3-dimensional space there's no way to do this unless you either
a.) don't mind cutting holes in the disc, or
b.) stitch together other points that really shouldn't be stitched together.
Of course the reason you would have to cut holes in the disc is because in our experience materials cannot simply pass through one another. But knit fabric is different than this. While it approximates a planar surface, in reality it is already full of little holes. This makes it possible for one piece of knitting to literally "cross through" another one, and this is what is done in the Projective Plane Hat. Below are some pictures of the projective plane shaped into Boy's Surface. The line at infinity (shown in red) is shaped like a rounded triangle, and as you can see the surface crosses through itself. It's a little hard to see from these pictures, but there is a 3-leaf clover shape where the surface crosses through itself, and in the center of this "clover" there is a triple point where three pieces of surface cross through it. From left to right the pictures are: front view, cutaway view (inside tinted blue), back view, and side view
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