![]() Then this statement is clearly false in elliptic geometry, where there are no parallel lines. Through a point not on a line, there is exactly one parallel to the given line. Uniqueness of Parallels Theorem (Playfair's Parallel Postulate): If we take Playfair as stated in Section 13-6: Once again, let's look at how we can interpret the fifth postulate. Notice my claim that even the fifth postulate can be considered true in elliptic geometry. This is the only way to make (5) the correct answer - perhaps this is what Pappas had in mind. In fact, one could even imagine counting all of the arcs between two points that differ by multiples of 360 degrees as distinct segments. In this case, this would make the second postulate true. But one could consider continuing to go around the circle, forming arcs that are greater than 360 degrees. This postulate could be considered false, since a great circle is finite. A line segment can be extended indefinitely along a line. This would make the first postulate false, and (2) the correct answer.Ģ. And so, given two points on the sphere, there are at least two such arcs between the two points - the short way and the long way around the sphere. If a line is a great circle, then a segment is just an arc of that circle. But let's look at how the U of Chicago interprets this postulate in Section 13-6:ġ. Nearly any of the postulates can be said to hold or fail in elliptic geometry (including the fifth!), based on how one interprets them.įor example, Euclid's first postulate can be interpreted as "two points determine a line." My argument above, in which (1) is the correct answer, implies that this postulate is true. Hyperbolic geometry satisfies the first four postulates of Euclid, and the fifth fails - there are infinitely many lines parallel to a given line through a given point.īut with elliptic geometry, it's more difficult to state which of Euclid's postulates fail. Of the two types of non-Euclidean geometry, hyperbolic geometry is actually more straightforward than elliptic geometry - despite it being much easier to visualize a model of the latter (a sphere) than the former. In elliptic geometry, there is exactly one line passing through any two distinct points - just like in Euclidean geometry! The second problem is that elliptic geometry differs slightly from spherical geometry in that antipodal points count as a single point! And so technically, the correct answer is (1). If the points aren't antipodal, then there's only one great circle through the two points. The first is that there are infinitely many great circles passing through two points if and only if these points are antipodal - that is, if these two points are directly opposite on the sphere, like the North and South Poles. Now there are indeed infinitely many great circles passing through the North and South Poles - these are the meridians. There are not infinitely many lines passing through two points in the version of non-Euclidean geometry known as elliptic geometry! Usually, we model elliptic geometry using a sphere, with great circles as the lines. This time, I had an entire three-day weekend to prepare the worksheet for today.)īut choice (5), under the usual interpretation, is wrong. (I would've actually posted this last week on the 5th, except that I wanted to spend more time preparing a worksheet for my geometry student. In elliptic geometry the number of lines passing through two distinct points is:īecause this question appeared on February 5th, the intended answer must be choice (5). Last week, there was a question on my Mathematical Calendar - you know, the one that Theoni Pappas publishes nearly every year - about non-Euclidean geometry: But I admit that this is still a very confusing concept. I've briefly discussed the concept of non-Euclidean geometry a few times before. Therefore, we're already done with this part of Section 13-6. We used this postulate to prove Playfair, and then used Playfair to prove the Parallel Consequences. On this blog, we began by accepting Perpendicular to Parallels as our version of the Fifth Postulate. Notice that we've already covered this topic during the first semester. It then refers to non-Euclidean geometry, in which Playfair doesn't hold. It discusses the five geometric postulates of Euclid, and states and proves the Uniqueness of Parallels Theorem - also known as Playfair's Parallel Postulate. ![]() ![]() The section, titled "Uniqueness," begins by discussing Euclidean geometry. ![]() Let's look at what we are missing in this section. With all of our jumping around the U of Chicago text, one section that's been torn apart is 13-6. ![]()
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