A Problem with the Parallel Postulate - Numberphile 

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

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)

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!

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

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 😅

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

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

The original Euclid's V postulate holds in sphaerical geometry. It's the "playfair" version of it that does not hold. And the playfair version is equivalent to the V postulate only if you take an extra axiom to prove the exterior angle theorem (which excludes elliptical geometry).

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.

Ahhhh I’m learning this in school as a freshman, so cool that I can understand this!

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.

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).

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.

I really enjoyed her presentation of this topic!

Is there a geometry in which we have more than one, but a finite number of parallel lines going through a given point P?

When I leari circular inversion and hyperbolic geometry it absolutely blew my mind

If you have a point in the hyperbolic disc all "straight lines" going through that point are circles with centers at the same line. If you have two points that are not in the opposite sides of the origin those lines are not parallel and they will intersect at some point that is the center of a circle ("straight line") going through both points. And if those points are in the opposite sides of the center you can just draw a line through them and the origin. So you can always connect two points in the hyperbolic disk with a "straight line".

If you force all euclidean 2d flat lines to go through the origin, there are also no parallel lines. You can make parallel lines on a sphere if they don't have to lie on a plane that passes through the origin.

This is a book I will order tonight, such an old book so full of flavourful mathematics.

But the 5th postulate states that if there is a third line which intersects the two lines in a way that the inner angles on one side are less then 2x90° the lines will intersect. Which is true for the sphere, because the lines do intersect and it is true for the disk as the angles on neither side are less than 180°. So the postulates holds? So, this is only showing that saying that the postulate is equivalent to "there is one and only one parallel line" is wrong. The 5th postulate doesn't say anything about whether there is a non-intersecting line at all and if so how many?

Its interesting how parralel lines can be different in amounts in different geometric shapes. I wonder which ones are like plains,spheres, or the circle

When you define that it must be a "great circle" ( or longitutde and latitude) then that automatically cuts off the possibility you'll have two parallel great circles (because obviously longitutde and latitude are always perpendicular). But it's not clear to me why you can't use a great circle and also a little circle (not centered through the origin). Wouldn't that create parallel lines on a circle?

Thanks , from an amatore math person.

Are parallel lines equidistant from each other in all geometries? Seems they would have to be by definition.

probably one of the best introductions to manifolds without ever mentioning the term😃

1:07 That is not the exac Euclid expresion, that is more the simplified version known as Playfair axiom; Euclid said " If a line segment intersects two straight lines forming two interior angles on the same side that are less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles. " It is understandable that this 5th postulate caused controversy over the centuries; it was tried to be deducted from the other postulates as a logical consequence. A lot of things had to be consider in this historical discussion: the lack of clarity of the nature of mathematical axioms in relation to theoretical thinking and reality and the search for truth; the nature of geometry and its object; the foundations of mathematical theories as formal theories. Now we think euclidean geometry as one possible geometry that accepts the validity of the 5th postulate, without reflecting any physical space. Also we need to consider that the term ' line ' , undefined in Euclid can be defined in modern theories with stronger and more precise conceptual frames of differential geometry, metric spaces, topological spaces, and mathematical logic formal systems.

She's wearing the same watch as me! It is also broken in the same place :)

I don't think it really makes sense to describe it as a problem. Defining a particular form of the parallel postulate essentially creates Euclidean geometry. A different form of the parallel postulate, or no parallel postulate, creates a different geometry. That's exactly what you should expect. "When I use different rules I get a different game." Well ... yeah. How could it be otherwise?

Euclid: If a line segment intersects two straight lines forming two interior angles on the same side that are less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles. Time traveler: Well sure, in Euclidean geometry. Euclid: Excuse me?

For the sphere, it's pretty simple. The big circle cannot be larger. And the second circle cannot cross it so imagine we cut the sphere into 2 hemispheres, and we keep only one. How can you draw a big circle on that hemisphere? Any circle you draw will be smaller than the border. So it's not possible.

Perfect Sphere ⭕️

Great video! Thanks a lot for that insight. I have a few questions, though which i would love being answered maybe in another video ;) Is there a function that transists every point in "normal" euklidean space to that hyperbolic disc space and back? If so, how do those infinite numbers of parallel lines on the hyperbolic disc translate back to euclidean space? Are they one and the same line? But they include different points in hyperbolic space, so how can that be? Secondly: If there are more than one line parallel to L in hyperbolic space. Call them K1, K2, ..., Kn...; Are K1, K2 etc. parallel _to each other_ as well? don't they intersect in Point P? Thanks again for this great geometry video. Love to see more of that interresting topic. Greetings from Germany.

I don't understand if you take a parallel longitudinal cut through a point it will be parallel to the first line

thanks for the explanation. I finally got it

why are great circles considered as lines? what about shorter circles created by planes that do not intersect the origin of the sphere?

the main problem is that treating it as a postulate to be proven is anachronistic. the vocabulary Euclid used to talk about it doesn't support the way it's been taken up by mathematics. for instance, Euclid himself never indicates at all that the preceding postulates should suffice to derive it. further, Euclid specifically mentions straight lines, and this 'postulate' only fails in a curved space, where there are no straight lines. its basic purpose was for doing work like masonry and laying out devices like the Antikythera Mechanism. in such a setting, the goal is to establish a planar surface to work on, and one of the way to check that is to lay out two seemingly parallel straight lines and then actually check if they are indeed the same distance apart all along. so you're just fundamentally wrong about everything you're saying. and it should be noted that modern mathematics is subject to Incompleteness anyway, which means that it's known and accepted that not all axioms/definitions can be proven from within the system anyhow. which means you're even wrong to really just note that this 'postulate' hasn't been proven.

The problem was never "Is parallel postulate true?" as all postulates are true by definition. The problem was "Is it necessary?". The answer is: "It is not, but then you don't get Euclidian but absolute or neutral geometry." or "It is necessary for Euclidian geometry."

Hasn’t there been a video about this before? I think the video was called “The Fifth Postulate.”

¡Una colombiana en Numberphile! 🇨🇴

That sphere noise reminds me of that video which explains how you can turn a sphere inside out

Great Circle Navigation, simple

So, what was the "problem"? Getting click-baity in the titles. Just because you can define a surface with rules where there are no parallel lines doesn't disprove the postulate. You could easily change the definition of a line on the sphere to allow parallel lines... Like lines of latitude rather than longitude.

Poor Euclid, people keep breaking his rules.

Why can't parallel lines be defined as intersections at infinity, infinity being defined as the sum distance of each line's domain.

Do the other 4 hold in spherical geometry or not?

Yes but with another definition of a line in those spaces, the theorem would still hold, right ? If we say that a line in the spherical space is a line when its projection from an angle onto a 2D plane is also a line, then the theorem holds ! In that way, latitudes and longitudes in geography are lines and they are parallel. What I mean is how can we say that one definition of a line in a space is the rigt one ?

Euclidean geometry still has something hyperbolic and spherical geometry don't have. You can use something called the "turn" of a line. When two lines are parallel, that just means those two lines have the same turn. Two lines lines intersect in a unique point if and only is the turn of the lines is different. Also, through every point, there is a unique line with a given turn. The angle between line k and line l, can be defined to be equal to the turn of line k minus the turn of line l. This is not a normal angle, but this type of angle is a lot more useful than other types of angles. They are called measured angles, and are used for Olympiad training. This makes it trivial to prove that the sum of the angles of a triangle is 0°=180°, because those are the same in this setting. My favourite theorem is that given a (non-degenerate) circle with points A and B, that the turn of chord AB is equal to A̅+B̅, where X̅ is a function of X (I hate Unicode). If A=B, the chord is of course the tangent. This implies that the angle between chord AB and BC doesn't depend on B, which is the most important Olympiad theorem, at least for geometry. I have been in Olympiad training for three years, and went to the IMO the last two years, but I never looked at it this way. It's definitely in my top 10 favourite findings, but probably my favourite finding that wasn't found earlier, as far as I know.

But, if one postulates a "flat" surface, can the "parallel theorem" then be derived from _those_ five postulates?

So yer saying to me that Euclid’s postulate is only true in Euclidean space?

So the problem with one of Euclidean postulate is that it doesn’t apply in non-Euclidean geometry How does a math channel click bait so often?

Wait… you’re saying the surface of a sphere is two dimensional. Are euclidding me?

Can't you hust create a line parallel from the sphere? Like a little circle? It will not work as a parallel line since it has a smaller circumference?

well explained

Is it just me or does the sound not work on this video?

And where is the follow up video?

Is it just me or is the sound very out of sync for parts of the video?

this emphasizes that often, humans make assumptions without realizing they are making assumptions

It seems like a cheat to consider a sphere a "real" 2D space because it is inherently tied to the (3D) center of the volume AND the 3D space around it which gives meaning to a normal line to the surface. If a sphere truly was 2D then latitude lines would be parallel (because you wouldn't have to be relying on the center of the sphere or the normal to the surface). I think sometimes we have to step back and look at what the postulate (of the S2 sphere) is saying here. If it really is "2D" then you could only measure it by the "left/right" and "forward/backward" dimensions on it's surface and it wouldn't have any measurable relationship to the center of volume or the lines of normal to the surface.

She didn't explain what hyperbolic geometry is like she did for spherical geometry.

It was taken to mean "what it means to know something"... So you could say, it was quite.... elementary

Brady is 5am I can't do math rn 😭😭

I love non Euclidean geometry because in school i have was doubt to assuming his postulate 👻😅

11:40 I just thought about that. Have anyone proposed a 3D model inspired on the Pointcare's Disk where the depth of the 3D model corrects the distance on the Euclidian 3D space? So, we would have like a 3D hyperbole, but, in a logarythmic way so that the distances corrected stays logarythmic? And if we look at the 3D model from above, it would be exactly like the Pointcare's Disk. I think it would be interesting to visualize. Please, let me know if there exist such an object. Edit: I think I just described the Hyperboloid model... Which... Is pretty cool! Numberphile could make a video about it!

"I wanted to get from 4th street to 8th... Then I remembered Einstein postulating that parallel lines eventually meet. They're dredging my car from Lake Michigan as we speak." -Emo Philips

0, 1, or infinitely many parallel lines.. 🤔 This must involve linear algebra but I'm not sure what exactly this connection is!

Why can you only use the great circle on the sphere? Who says that. Seems an arbitrary definition because there are parallel lines that or not great circles.

Am I understanding the hyperbolic parallel line example correctly? Her 2nd and 3rd lines were both parallel to the 1st (center) line, but not to each other (since they intersected). In fact it appears as if there are an infinite number of lines that would be parallel to a given line, but not to each other, through a single point.

To Juanita: I think this was a bit unfair to ancient spherical geometers, especially Menelaus and later translators/commentators writing in Arabic. You should take a closer look at _Spherics,_ which does quite a bit of axiomatic geometry intrinsic to the sphere. The only surviving manuscripts are Arabic translations of Menelaus's original, but there’s a newly published English translation of _Spherics_ by Rashed and Papadopoulos which also has some nice front matter putting the work in historical context.

That is definitely NOT the parallel postulate. That might be a fact derived from the parallel postulate but it is in fact NOT the parallel postulate

tldw; what's the "problem" with it, exactly?

Spherical geometry does not follow the 1st postulate; antipodal points on a sphere do not define a line/great circle.

I felt it could have been mentioned that the representation of hyperbolic geometry here is just that, a represntation - unlike spherical/elliptic geometry, hyperbolic geometry cannot be embedded in euclidian space, which is why distances look funny to us, but otherwise a nice introduction!

5:31 I am not sure who made a mistake here. Juanita or Brady. She says "line" where is animation shows segment.

She must be Colombian. Tatiana Toro, Federico Ardila and now Juanita. That's cool

this feels less like a problem with the parallel postulate and more a reasoning as to why looking at the context around theorems and postulates is necessary to get to an accurate conclusion

These models where just outside of Euclid's sphere of influence.

That is playfair's axiom

and what can you do with this information? real world applications?

wait a minute. if I take a tennis ball and chop it like a tomato, this is going to be pretty "parallelly", isn't it? I always thought this way when it came to parallel lines on a sphere

I'm right down the road from Notre Dame. Maybe I can go do some maths with them.

Now we need the geometry in which all lines are parallel and none ever intersect.

why not mention the first 4 at least once

Can anyone explain why it took thousands of years to realize that any two straight lines drawn on the surface of a sphere, such as the surface of the earth, intersect?

I have the same watch, and the strap broke in exactly the same way.

i wonder whether there wille be a mathmatician someday, that has no problem.

All this parallel line stuff is just a matter of perspective.

I would want to know if euclidean space is the only space where all of his 5 postulates hold. And if the other 2 space (spherical and hyperbolic) are the only other 2 spaces where the remaining 4 postulates hold.

I hear Euclid, I listen

Didnt Numberphile already make this video like 7 years ago with "Ditching the 5th Axiom"? I guess this one goes more into explaining the non-euclidean geometries and the other goes more into euclids axioms, but its funny how brady seems to have forgotten the stuff he learned from that video.

There are infinite many parallel lines from "equator" to "pole"... noone says it has to be great circle or has to lay on plane which goes thru the center of sphere, all of them satisfy euclidian definition... not the best numberfile video, sorry...

Waaaaaay too short 😢

* Lobachevsky joined the chat *

I can not recommend the VR game Hyperbolica enough

Yes we know. Every math youtuber in existence has covered the 5th postulate and non-Euclidean space. Find a different topic.

It gets interesting when one accepts that no _physical_ space of any interest is actually Euclidean. It would have to be an empty space away from everything, to begin with.

From the thumbnail, I thought this was NileRed

Spheres are three dimensional objects. Circles are two dimensional objects.

Hyperbolica

chthulu aproves this video

It is a weird and wacky problem. Extra Credits got five episodes out of it.