 In the previous video, we introduced the three parallel alternatives for which their names are the following. We have the Euclidean parallel posture, which we're going to typically call that EPP for short. We have the hyperbolic parallel postulate, HPP. And then the last one was the elliptic parallel posture. Well, in terms of an acronym, that would also be EPP. So we're going to call that one LPP for short. It seems a little bit weird, but it's elliptic. Well, okay, I get an elliptic sound like it starts with an L. So LPP elliptic parallel posture, EPP. Euclidean parallel postulate, HPP, was the hyperbolic parallel postulate. LPP says that there are no parallel lines in your geometry. EPP says that parallel lines are unique. And HPP essentially says that there's a multiplicity in parallel lines. Now, when we listed those parallel postulates in the previous video, I stated that these parallel postulates are mutually exclusive. If you have one, you cannot have the others. You can't have two of them. But you might have the false impression that you must have at least one of them. That's not true. You can't have a geometry which doesn't satisfy any of these parallel postulates because the parallelism of the geometry might be a little bit more mixed, less uniform. And so I want to provide you an example of a five point incidence geometry that doesn't satisfy any of the three parallel alternatives. So you have these five points right here, this five point geometry. This is not the five point handshake geometry you saw in the previous lecture. While this geometry does have four handshake lines, it also has two lines which contain exactly three points. There's this horizontal line which contains ABC. There's this vertical line that contains BED. These are three point lines. So this is a different geometry than the handshake geometry we saw before. This geometry does not satisfy the elliptic parallel postulate because it has parallel lines. The lines AD and EC are parallel to each other. So this rules out the elliptic parallel postulate. I also want to show you that that same counter example basically also kills the hyperbolic parallel postulate because if I take this line AD and I take the point E right here, there does exist a parallel line to the line AD that passes through E but there's not another one, right? If you take any other line through E, it intersects, it intersects, and this line doesn't even go through E so I'm not even gonna consider it. So in this situation, we see that there's only one line parallel to AD that contains the point E. That's in violation of the hyperbolic parallel postulates. What about the Euclidean parallel postulate because this line with this point had a unique parallel line. If I took this point, you'd also have a unique parallel line, okay? What if I take this point right here? How many lines that pass through B are parallel to AD? Well, there's only two lines that pass through B. You have the line ABC which intersects AD and you have the line BED which also intersects AD at the point D. So in this situation, there's no line parallel to the given line that passes through B. So this would then violate EPP. And so I want you to notice what happened here. When I looked at this line at this point, it looked Euclidean. But when I looked at this line at this point, it looked elliptic. All the lines intersect. So the local behavior with this line at this point was elliptic, but this line at this point was Euclidean. So this geometry is sort of like a hybrid between elliptic and Euclidean geometries. Locally it could be Euclidean, locally it could be elliptic. Now in this geometry, there's really nowhere where it looks like it's hyperbolic. At least I don't think so because if you look at this line right here and you take this point right here again, any lines go through it intersect that line. This, depending on the line of the point, it could look elliptic, it could looked Euclidean. It's never hyperbolic. So that one's definitely, definitely not on our list. But the point is if locally it's elliptic, but locally at a different place, at a different locale, it's Euclidean, that means it's not Euclidean, it's not elliptic, because elliptic parallel postulate is a global condition. That needs to be true for any point in any line. Same thing with Euclidean geometry. And so because of that, the parallel alternatives are not, it's not the only choices. There is a choice where you can be locally one of these things, but globally that behavior is absent.