It can only move in two directions and that’s what we’re considering.
Today I am going to tell you about the parallel postulate in geometry. Originally appearing in Euclid’s Elements (published 300 BC), it was taken to be the example of what it means to know something, as it is deductive - a set of rules from which truths are derived. The parallel postulate states that if we are given a line in a plane and a point that is not in the line, there is one and only one line that does not intersect the line and goes through the point. This seems to be the most obvious thing in the world, as any other line drawn would have one side tipping towards the line. Although this is not how it appears in Euclid’s Elements, he states that if the sum of two angles is less than 180 degrees, then the lines, once extended to infinity, will intersect.
Not only can geometry be done in a plane, but also in a sphere (S2). This is a 2-dimensional space, as we are only dealing with the shell of the sphere, not with the filling. We can think of it in terms of meridians and latitude, like in Google Maps where a precise location is given with two coordinates. So it’s kind of a surprise that we can do that. It wasn’t until the 19th century that there was an upheaval in geometry when it was realized that the first four postulates of Euclid could be applied to other models. It was then that hyperbolic geometry was considered to show that the parallel postulate was not true in general. In this geometry, there are parallel lines, but they are not unique and there are infinitely many. This is because the way distances are measured in this new geometry is such that two points that are close together are far away. This means that the fifth postulate is not a consequence of the first four, which had been a long-standing question. Additionally, some have suggested that Euclid’s hesitation to use the fifth postulate in his Elements was a clue that he was uneasy about it. I think of the fifth one as more important than an ugly aberration, as it allows us to think of more than one way of doing geometry in two dimensions. The angle is less than two right angles, and while there are a lot of words to describe it, it is fascinating.