A Problem with the Parallel Postulate - Numberphile

Featuring Juanita Pinzón Caicedo from University of Notre Dame.
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NUMBERPHILE

Videos by Brady Haran

[nicksamek]

An extra video explaining how distance is calculated in hyperbolic geometry might be interesting

[CorrectHorseBatteryStaple]

I think what Brady was maybe trying to get at around 6:20 and in the preceding discussion, is that there are many "little" circles that you could also draw that go through two points, but someone on the surface of the sphere would feel like they're "turning" while following that circle, whereas the great circles are the only ones where it "feels" like you're moving in a straight line

[Gunbudder]

by FAR the most useful fact about parallel lines was never taught to me in schools! that fact is, for any line segment that is between two parallel lines, the middle point of that line segment is exactly the middle point between the two parallel lines, regardless of the angles involved. in practical terms this means you can measure the exact center of any straight edges object (like a 2x4 piece of wood) by putting a ruler across it at any angle and marking the middle of the ruler. this makes it incredible easy to mark the center if you angle the ruler such that 0 inches is on one end and 6 inches (for example) is on the other. if you mark with a pencil at 3 inches it will be EXACTLY in the center of the board. if the board is wider than 6 inches, you can use any other even number larger than the width of the board.

it may sound a bit complicated, but the second you do it one time, you will understand and then finding the center of a board becomes trivial and you will be able to do it in seconds. the proof of why this works relies on the postulates, but the postulates themselves are very rarely ever useful in real word applications (in other words never useful)

[theadamabrams]

12:43 I heard that because the Parallel Postulate was much longer to write than the others, geometers felt it was less fundamental or less deserving of being a building block of geometry. They _hoped_ that it could be derived from the other four, and they tried to do as much geometry as possible using only the first four. From a modern perspective, it's nice that any results proved with just 1-4 must be true in flat and spherical and hyperbolic geometries.

[tremkl]

This story is one of my favorite math stories, but I’m not a fan of how it’s structured. The ending needs to be the beginning.

Euclid really wanted to build all of geometry from completely self evident axioms, and the first for are much, much simpler statements that are much easier to see why they must be true. The fifth axiom was always needed to derive geometry, but also feels like something that would be much better if it could be proved, rather than being a starting axiom.

A few options were attempted to fix this. As referenced in this video, people attempted to prove the fifth axiom using the first 4, which if it had been possible, it could be removed as an axiom. This proved impossible. People attempted to break down the fifth axiom, I.e. perhaps replace it with a much simpler, more self evident axiom, from which the fifth axiom could be derived. Nothing was found.

Ultimately, one option remained: proof by contradiction. If you assume the fifth axiom is false, and develop the rules of geometry this would follow, and it lead to a contradiction, then the fifth axiom must be true. This lead to exploring geometry without parallel lines and with multiple parallel lines. Neither lead to contradictions.

Where the story becomes amazing to me, is that the non-Euclidean geometries present in these two examples actually describe curved spaces, and therefore all the attempts to prove the fifth axiom, while they didn’t accomplish what they set out to do, proved incredibly useful in their own right.

[connorcriss]

I just realized - I’m pretty sure Brady already knew about how non-euclidean geometry was defined. He’s asking the questions he’s asking not for himself but for the viewer. Appreciate you Brady!

[likenem]

I have never seen someone draw a sphere that quick

[mazza]

i always enjoy your non euclidean geometry videos

[samrichardson]

I’ve been teaching geometry tutorials at my university and we had the students see how this version, Playfair’s axiom, is an equivalent statement to the parallel postulate as Euclid wrote it! We cover a bit of spherical geometry too but the students don’t tend to like it 😅

[jakebourdages]

Why aren't the longitudinal east to west lines on a sphere not be considered "parallel" to the equator if they do not intersect with one another? Would this not be considered an exception to the rule that there are no parallel lines on the 2 sphere mentioned @8:00?

[reallifeistoflat]

What I don't understand about this is why we're limited to great circles? There doesn't seem to be a reason. Great circles aren't inherently similar to lines on a flat plane.

Also strictly speaking Euclids postulate is still true here as the angles don't (can't) add up to 180° when using great circles. unless I'm missing something.

[topilinkala]

It defines on which type of surface you are on: a negative, positive or no curvature.

[andrewharrison]

To get an appreciation of distance near the edge of the hyperbolic disk try some of the M C Escher "Circle Limit" engravings.

[EebstertheGreat]

The way she describes spherical geometry, it doesn't only violate Euclid's fifth postulate but also the first postulate. Because between any two antipodal points there is not a unique line segment but infinitely many line segments (all of them great semicircles). In particular, the Saccheri-Legendre theorem does not hold on the sphere, showing that it cannot be a model of absolute geometry.

[alveolate]

wait why do lines on a sphere HAVE to be great circles? the lattitudes are parallel... they're literally called "parallels" when they line up with borders! so for each point P, you DO get a parallel line, it's just not a great circle.

[Happy_Abe]

On the point of line segments on the sphere being the shorter one, what if the line segments would be equal? Which one is the segment?

[stevenfullman]

This is fascinating and Juanita is a great teacher

[FirstLast-gwmg]

A great circle for 2 points on a sphere is just the extension of the shortest arc that can be drawn to connect them. I like this definition better since it more closely corresponds with the ordinary cartesian 2D definition of a line segment... a straight line will always follow the shortest distance between any 2 points it intersects. A straight line in 2D planar coordinates corresponds to a great circle in 2D spherical coordinates. It's (somewhat counterintuitively) both the longest possible path around the whole sphere, but it's the shortest path between any 2 distinct points on it (unless the 2 points are exact polar opposites, in which case there are infinitely many possible paths, they're just all the same distance).

This is not true for the intersection of a plane that does not go through the origin point. If you identify any 2 points on the circle, the circle is _not_ the shortest path between them, it will curve away from that shortest path (which would be part of a great circle).

[Veptis]

what surprised me, is that every "space" can be defined by their measure of distance. And that matters for stuff like hyperbolic word embeddings. How these might efficiently implemented to work with our modern computers (floating points instead of bits) could be a topic for an undergraduate project.

[jespermikkelsen]

1:34: I've never heard this before, and I think it's VERY interesting indeed, because "the parallel postulate" as it were told the 1st time is NOT equivalent to the 2nd version. The 2nd version (apparently how Euclid wrote it) does not postulate the line to be unique and he does not postulate any existence. The 2nd version just say, that if the sum of the those angels < 180, then the lines will intersect. In other words: Sum(angles) < 180 => lines intersect somewhere. But false => true is a true implication too. In fact, the 2nd version of the postulate does not imply anything if sum(angles) >= 180. So some one messed up with the logic that the old man came up with.

Correction. it seems that I was a bit to fast. A => B is equivalent to not B => not A. A is Sum(angles) < 180 (or > 180), B is lines intersect. So version 2 gives
If lines do not intersect => sum(angles) = 180.