@OverlySarcasticProductions

5 years ago

Hey gang! Can't help but notice the comment section is a little bit on fire. That's all good with me, but one recurring complaint I've noticed has started to get under my skin - namely that my explanation of non-euclidean geometry was insufficient, or even - dare I say - inaccurate. Now this is a fair complaint, because after a lifetime of experience finding that people's eyes glaze over when I talk math at them, I concluded that interrupting a half-hour horror video with a long-winded explanation of a mathematical concept wouldn't go over too well. I put it in layman's terms and used a simple example to illustrate the point. However, since some of the more mathematically-inclined of you took offense, I now present in full a short (but comprehensive) explanation of what exactly non-euclidean geometry is. First, we axiomatically establish euclidean geometry. Euclidean geometry has five axioms: 1. We can draw a straight line between any two points. 2. We can infinitely extend a finite straight line. 3. We can draw a circle with any center and radius. 4. All right angles are equal to one another. 5. If two lines intersect with a third line, and the sum of the inner angles of those intersections is less than 180º, then those two lines must intersect if extended far enough. Axiom #5 is known as the PARALLEL POSTULATE. It has many equivalent statements, including the Triangle Postulate ("the sum of the angles in every triangle is 180º") and Playfair's Axiom ("given a line and a point not on that line, there exists ONE line parallel to the given line that intersects the given point"). Euclidean geometry is, broadly, how geometry works on a flat plane. However, there are geometries where the parallel postulate DOES NOT hold. These geometries are called "non-euclidean geometries". There are, in fact, an infinite number of these geometries, and because the only defining characteristic is "the parallel postulate does not hold", they can be all kinds of crazy shapes. (As you can see, my explanation of "this is just how geometry works on a curved surface" is quite reductive, but at the same time serves to get the general impression across without going into too much detail.) An example of a non-euclidean geometry is "Elliptic geometry", geometry on n-dimensional ellipses, which includes "Spherical geometry" as a subset. Spherical geometry is, predictably enough, how geometry works on the two-dimensional surface of a three-dimensional sphere. In spherical geometry, "points" are defined the same as in euclidean geometry, but "line" is redefined to be "the shortest distance between two points over the surface of the sphere", since there is no such thing as a "straight line" on a curved surface. All "lines" in spherical geometry are segments of "great circles" (which is defined as the set of points that exist at the intersection between the sphere and a plane passing through the center of that sphere). The axiom that separates spherical geometry from euclidean geometry and replaces the parallel postulate is "5. There are NO parallel lines". In spherical geometry, every line is a segment of a great circle, and any two great circles intersect at exactly two points. If two lines intersect when extended, they cannot be parallel, and thus there are no parallel lines in spherical geometry. Since the Parallel Postulate is equivalent to Playfair's Axiom, the fact that no parallel lines exist in spherical geometry negates Playfair's Axiom, which thus negates the Parallel Postulate and defines spherical geometry as a non-euclidean geometry. Also, since the Triangle Postulate is another equivalent property to the Parallel Postulate, it is thus negated in spherical geometry. Hence, my use in-video of an example of a triangle drawn on the surface of a sphere whose inner angles sum greater than 180º. Hope that cleared things up (and helped explain why I didn't want to say "see, non-euclidean geometry is just a geometry where Euclid's Parallel Postulate doesn't hold - hold on, let me get the chalkboard to explain what THAT is-" in the video) Peace! -R ✌️

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