Roughly speaking, it is the length of the shortest straight line between the two points. How we calculate that will depend on our model. In the poincare disk model, it's a little awkward, but you can find a formula on Wikipedia. In the half plane model it is the natural log of the cross-ratio of the two points (plus the two end points of the "straight line" through them thought of as points om the x-axis)
Surely you can do it with a basic sigmoid function. When r is the apparent (on a disc of radius s) distance from the center, and R is the hyperbolic distance from the center, you can make a formula describing their relationship as: r = s/(1-e^R) (describing f(R)) R = ln(1 - s/r) (describing f^(-1)(r))
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
@@maitland1007 No, this is not how lines are defined in non-planar geometry, specially because contrary to what is suggested in this video, the notion of distance is not necessary to define what lines are. Actually, in Minkowski space-time, straight segments are the longest possible path between two points, not the shortest.
@@topilinkala1594 It saves fuel because great circles are shortest paths, which is not the definition of straight lines (as mentioned in another comment above, in Minkowski space, they are the longest).
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.
Exactly so. But while Euclid is best known for his writing about plane geometry, which we now call "Euclidean" geometry, the fact is he also wrote about the geometry of a spherical surface. So it seems reasonable he could have seen the need to include a postulate that would distinguish the geometry of a plane from that of a sphere's surface, and that the parallel postulate was included for that reason. So at least one form of "non-Euclidean" geometry was in fact known to Euclid.
If you can prove something with the four first axioms that contradicts some (different) fifth axiom, then your theory is inconsistent and anything can be proved in it
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 knew this fact about parallel lines but never really thought about it in a practical context until a few months ago where I saw someone demonstrate this trick
In English scientific literature this is commonly called “Intercept theorem”. It basically states the following: If a line A passing though several parallel lines P1,P2,… is cut (by P-lines) into segments a12, a23, a34,… Then any other line B passing though P1,P2,… will be cut in segments proportional to a12,a23,… |\ -----P1 | \
I just want to add that it should be "for any line segment [whose endpoints lie on each respective parallel line]" When I read for any line segment between two parallel lines, the first example my mind conjured looked something like this: ----------------------------------- which wouldn't have a midpoint lying on the midpoint line of the parallel lines.
As a side note, this will only get you the center with respect to 1 axis. If you do this twice, in slightly different locations, and then connect those points with a line, and then switch to the other axis and do the same thing, the 2 lines you created will intersect at the exact middle of the board with respect to both axes.
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!
@@poppyseedsnuranium so he flat out lies? in the sin(x)cos(x) integration video he flat out said he didn't remember calculus or integration. humility is fine. false humility is deception and i hate it. far more than pride.
Not a sphere, but when drawing a circle, the secret is to move your whole arm in a circular motion and put the pen on paper while already in motion. That way you can get a nice circle very fast.
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.
As someone who wrote his masteral thesis on differential geometry, this topic means so much. It’s the reason I learned to love geometry, and eventually choosing to study geometry over other mathematical disciplines.
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 😅
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.
I have seen so many times people think they can object to what Euclud claimed, by suddenly taking the claims out of context and applying them to some kind of curved space, and then expect people to say "wow, you are smarter than Euclid, who must have been a very primitive thinker". I am not saying this happens in this video (have only watched a few minutes yet), but it is a bit on the border.
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.
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.
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?
Do you mean latitudinal lines? (ex. 30°N, the path that crosses through Iraq, India, Nepal, Houston, Morocco, and Algeria) Those aren't great circles; the plane that circle defines doesn't intersect the center of the sphere. On a more practical note, if you were to try to walk (or fly) along it, you would need to turn rather than going in a straight line. Think of the line of latitude right next to the north pole (89.9999°N); that's clearly just a circle on the ground, and not a line.
@@jakebourdages3445 So, I was looking for a comment like this. Like Hayls said, you probably meant latitudinal lines (think like the rungs of a LATter). Anyways, it's a quick video and they played it _fast and loose_ . But, there can be some communication error since a lot a definition wasn't given to the objects used. (I'm gonna rant) So, typically I like to define a line merely as the path of translation of a point. That's a joining quality giving us a fair class of related objects. So, skipping a bunch of hooda for now. Those longitudinal lines mirrored across the equatorial great circle (as defined by your choice of pole) could (I'd say should) be "parallel". Given that the actual quality of parallel we're particularly looking for here should be "a fixed distant greater than 0 of two or more congruent lines" (between "positionally" (
@@poppyseedsnuraniumMathematicians usually don't use the word 'lines' for the objects you are describing, but instead 'curves' or 'paths'. You could define a notion of 'parallel' for two curves to be remaining a fixed distance from each other, if you like, and then curves of constant latitude would be parallel.
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).
@@BenMakesGames think about standing on the equator, and trying to make a “small” circle. You would have to constantly turn to make it back to your starting point. But if you follow the equator itself, you are always walking straight ahead. Basically, if the world wasn’t round, the great “circle” would just go on forever. This isn’t true for the small circles (which could always be drawn on a flat map that shows strictly less than half of the world)
@@Muhahahahaz And now realise that the sphere is fully symmetrical and you pick and choose your equator as any big circle. (Does not work so well for the earth because it is rotating and there is a reason why we have fixed the north and south poles.)
It’s important to note that shortest distance works for finding these lines in some geometries, but not others. A simple and very important example is Minkowski Spacetime, where the straight-line path between two points in spacetime is actually the _longest path_
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As soon as you started talking about other spaces, I predicted pretty much the entire rest of the video. I feel smart now. ;) BTW, googling "hyperbolic grid" gives some nice visualisations. Remembering how those look helped me figure out in my head that there are infinite "parallel" lines in hyperbolic space.
Euclid fully realized they were assumptions. That's why he called them postulates. They were never meant to be proven; they were to be accepted as true to prove other things.
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".
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?
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.
I think the simplest way to see this is that the shortest distance between two points is along a great circle. If you walk along the Earth in what you feel is a straight line, you will be walking along a great circle
Actually, right angles still exist on the sphere, so you can construct the line that the postulate would say is parallel, (the line where the interior angles are the sum of two right angles) and it would not be parallel.
@@Aetheraev but there is no requirement to make the shortest path in the original proof. You could clearly make a parallel line by creating a plane intersection that cuts off the top third of any hemisphere you create. I accept that the angles may not equal 180, I thought the error would cancel, however parallel lines are very clearly possible.
@@reallifeistoflat No, those would not be parallel lines because one of them is not a line at all. Only great circles are geodesics, the generalization of straight lines. "straight lines" are in fact defined as the shortest path between two points. To walk on your second "line", you would have to constantly steer to one side, like a circle in the euclidian plane. You have shown that there is some curved path on a sphere which does not intersect a given line. In the same way there are infinitly many curved paths in the euclidian plane which do not intersect a given line. But that's not what euclids postulate is about.
@@reallifeistoflat You could use "at a constant distance away" as a definition of parallel but you lose the straight lines. Only in euclidean geometry does a curve at a constant distance from a straight line have to be straight itself. In your example we see that a constant distance away from our great circle we get just a regular circle. But a regular circle in spherical geometry is distinctly curved. If you don't see why this is true consider one further away from the equator. As we get closer to the pole a regular circle "parallel" to the equator becomes a tighter and tighter circle until it collapses to a point at the pole itself.
postulate 2 says something like: "Any straight line segment can be extended indefinitely in a straight line." - this also seems to break in the spherical plane, as the segment becomes a full line quite quickly... (this applying the "uniqueness" definition - if the line defined by the same points is always the same line, then the points from a line are always the same points, so after you extend the segment to reach it's own start, you cannot extend anymore as the line is already there, no more/other points available => no more extending)
I remember talking about this with my maths teacher at secondary school in England in the late 1960s. My argument with him was that he had a very limited definition of a 'line'. I argued that it is possible to draw parallel lines on a sphere but none of them are great circles (except the equator). I asked him why lines of latitude are not lines. He did not have an answer except one which he freely admitted he'd simply been taught. Can anyone explain this to someone who has not studied formal geometry for over 50 years? What is the definition of a line? Brady: where do you find these wonderful people to talk with?
That's funny I JUST finished a long rant to the commenter above you about this. In short, in the video they were specifically talking about the "shortest path between points" which is a subcategory of a general "line". People might just call lats a "closed curve". But, meh.
You could maybe call them lines, but they are not "straight lines." They have differing curvature to the ambient curvature of the geometry. If you walk along a line of latitude you must be continuously turning.
@@PeterGaunt Because those are great circles. Their curvature matches the ambient one. Remember the curvature of a circle (as a curve in euclidean space) is 1/r where r is the radius, so smaller circles are more curved. Lines of longitude all have the same radius and thus the same curvature whereas lines of latitude do not. Lines of latitude are (except for the equator) more curved.
With euclid geometry, a line is a primitive object : it is something that follow the axioms of the euclidian geometry. In particular, a "line of latitude" can't be a line, because then you would have more than one line that pass between two points.
FYI, Eulid has a fifth postulate which he had defined because he was well aware of the Parallel Postulate during his time. Widely regarded as unnecessary and redundant for its original purpose, until people started using his theories for other things. I believe that the most egregious understanding was the term "plane". Fairplay's Axiom is a better representation of the Parallel Postulate. This video actually uses Fairplay's Axiom but with its own variation of "one and only one" line while Fairplay's states that there is "at most one".
"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
I think his postulate does scale up one dimension, but you have to scale up everything for it to work properly. Instead of being constrained to a sphere, picture one spoke with two freely rotating wheels in empty space. These wheels have to be exactly parallel to each other in order to not intersect if you were to extend them infinitely, and there's no other orientation where that is true. Of course, this is only true if space is flat, but the same can be said of the original postulate - if either is on a curved plane, you get different results. I have no idea whether this would scale up to further dimensions, though. I'm sure an actual mathematician could tell me where a fault in this logic is, but as a mere math enthusiast, I don't see any problems with this being a scaled up version of the original.
I mean yeah, euclid’s postulates scale well in Euclidean geometry, because Euclidean spaces can be broken down into lower dimensional Euclidean spaces- in every plane, a line is defined uniquely by two points, and through any third point, one unique parallel line exists. In very 3d space, three points define a unique plane, and for any fourth point, one unique parallel plane exists, etc…
imho, the point here is nonEuclidian geometry is limited in usage. No way to scale, coordinates not corresponding to volume. It is hard to calc volume. What volume formula for spherical geometry?
How interesting it is that the definition of "parallel" in spherical geometry necessitates that the full circumference always be invoked. Yet, there are an infinite number of lines that pass through any two polar opposite points, and there are an infinite number of latitude lines that are effectively parallel, but are also curved in that geometry. How is it that these latitude lines escape inclusion in some concept of "being parallel"?
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."
Historically speaking-in fact, right from Euclid's time itself-the problem that the geometers were trying to solve was _"Can Euclid's 5th 'postulate' be proved from his first 4?"_ (That is, whether it's actually a theorem rather than an independent postulate.) For a long time, it was suspected to be so. But they weren't able to prove it. While attempting to prove it, they started playing around with alternative postulates. Thereby discovering valid geometries that differed from Euclid's. And thus, they realized that the 5th is indeed independent of the first 4. I'm not sure what you mean by "necessary." By definition, geometry that uses only Euclid's postulates-and nothing contradicting them-is called Euclidean geometry. Absolute (aka neutral) geometry is Euclidean, but the converse isn't always true, thanks to the 5th postulate-as you pointed out.
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.
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.
Hi Of course you can do like you proposed, but without the requirement to be great circles it is not longer true that there exists only one line between two points
It's all about definitions. You could call the latitudes on a sphere 'parallel circles', but I won't call them straight lines (except the equator).They're called 'parallels' because in most projections they're neat, parallel lines.
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
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.
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.
In hyperbolic there is a way to partition the infinite parallel lines so that each partition has that property. Create an euclidian line with a given turn using the center of the circle (of the hyperbolic model used here). each hyperbolic 'line' that has an euclidean center on that line (including the infinite case) are in the same partition.
@@ingiford175 Never thought about that this way. Using projective transformations, I noticed earlier that in projective geometry, the turn of a line k can be seen as the intersection point between line k and a default line, which is normally the line at infinity. But I never considered this in any other geometry, but this even works in spherical geometry.
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.
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?
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.
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?
Also in regard to lines being parallel, in case of a 2D plane parallel also means that the shortest distance between any point on L1 to L2 is constant. So if parallel would be a) non-intersecting and b) shortest distance for any point on a line to the other line is constant that would eliminate "unlimited number of parallel lines" for a sphere and a disk? In 3D space (3D orthogonal space at least) the number of lines that satisfy a) and b) is only a subset of the parallel lines that just satisfy a) (don't intersect). Are there two different kinds or qualities of being parallel?
Well, really, spherical geometry is the wrong setting here. We should move to elliptical geometry and identify opposite points on the sphere. Then every straight line intersects every other in exactly one point, meaning there are no parallel lines. You could argue this subtly avoids the original formulation of the parallel postulate, but it certainly does not avoid its converse.
@Gunnar Frenzel the definition of parallel here is usually taken to be your a). Thus spherical/elliptic geometry had no parallel lines and hyperbolic has infinite families of them. The constant distance version produces something different and is only equivalent in Euclidean geometry
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.
Some of the other postulates do not hold on a hemisphere, e.g. any straight line segment cannot be extended indefinitely in a straight line on a hemisphere.
I prefer to look at conic sections as an explanation for hyperbolic geometry. It makes more intuitive sense than the circle from above. And it's more in line with the S2 geometry as far as moving your intuition one atomic step forward.
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.
'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.' 'If a line segment intersects two straight lines...' '...two straight lines...' you're wrong. stop this nonsense. you should also be capable of noting that your spherical example disallows extension to infinity. thus not only do you lack straight lines, you lack explicitly mentioned requirement: '...then the two lines, if extended indefinitely...' so really you're just illiterate.
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
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).
So Euclid was possibly the first philosopher to think about Non-Euclidean Geometry. A Greek using the opposite of irony, a term invented by the Greeks. I love that.
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?
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!
Now I find all the comments I'm looking for. _Practically_ speaking they _should_ be. Else, you get a lot of wacky things all under the same class of objects.
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...
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.
I guess the problem is that of these five famous axioms, one of them fails in other geometries (while the others hold). Maybe it's not a "problem" as such, but no need to get angry about it. ;) We're just having fun.
The postulate isn't "disproved". The point of this is that you can replace the fifth postulate with others and arrive at addition non-Euclidian geometries that apply to different kinds of plane. Unlike the other postulates, which are universal across all geometries, the fifth postulate is actually a definition of the kind of plane.
The postulate is not something to be disproved. It is basically an axiom. The classical "problem" with the parallel postulate is that people thought it should be possible to deduce it from the other postulates, but it is not. Hence, you can define other geometries with the other rules the same but that don't have a parallel postulate
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
Mathematicians want unique line segments and therefore they choose the shortest path connecting two points on the surface of a sphere. The shortest path is a segment of the great circle going through both points. (If you have two points on the equator, then "clearly" the shortest path between them is along the equator, not first going north and then south. Now realise that any great circle can be called the equator, since it's all symmetric.)
I don' get it. Euclid said a Line is those points the shortest distance between 2 given points, extending in both directions forever. Your line on a globe meets back on itself. I don't get it. Your middle part. Given a "line" on the 40th parallel of the globe, and a point on the 30th parallel, could one not construct a line on that 30th parallel? Would they not be parallel, especially since they're called parallels?
Great circles do give the shortest distance between two points on a sphere. And you can think of them extending infinitely if you like, it just so happens that you get back to where you started after a certain distance. The parallels are not great circles. They are not the shortest path between any points and to follow one, you would have to continously steer in one direction. So they are not straight lines, they are curved. And this postulate is not about any curved paths.
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.
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 ?
If it requires turning in one direction or the other, it's not straight. A great circle is the only path on a sphere like Earth that you'll get from travelling in a perfectly straight line relative to Earth's surface. Latitudanal lines look straight on many flat projections, but those projections are not the actual globe.
The Euclid axioms say that it pass exactly one line between two points, but with your construction, you have infinitely many lines that pass between two points.
@@Tryss86 Elliptic geometry is the way to resolve this. It only covers one hemisphere rather than the full sphere, so there are no antipodal points except around the equator, which seems to typically be treated as the horizon. The parallel postulate still doesn't hold, but it does regain the property of a unique line between two points.
@ certainly, but you might need to add some new axioms/postulates to preserve some of what you removed. To do "geometry", I would say you need points and lines, with each two points connected by a line, and at least two points on each line. There is a whole area of combinatorics concerned with finite geometries where you have just a finite set of points and line. The game "Set", which I think was in a recent video, was basically such a finite geometry with the cards being the points and the "sets" the lines.
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.
The tangent space to a general Riemannian manifold is Euclidean (which is, for example, where velocity vectors live), so Euclidean spaces are of physical relevance.
Suppose there is no problem left to solve in mathematics. Then I construct a problem: "Prove that there is no problem left to solve in mathematics", contradicting that there was no problem left to solve.
If it's two-dimensional, then what's the difference between a sphere and a circle in this non-Euclidean geometry? Are there definitions? Axioms? Theorems?
@@DrEMichaelJones Is the line which defines the z-axis 3-dimensional, because it requires a third dimension to picture? Dimensions are a very intuitive concept which actually need a very specific definition. I won't give that specific definition here, but I can give an analogy to explain it. A sphere can be pictured in 3-dimensions, but the sphere itself is 2-dimensional. The reason we say this is best explained by imagining that you are an ant on the surface of that sphere. If the sphere is suitably large enough, you can't actually tell that you're not on a plane (See flat earthers). Another way to say this was in the video, which is something like 'the dimension of the space you're in is how many coordinates you need to define your position'. For a circle, I need just one coordinate: a single angle! Similarly an "ant" on a circle large enough might believe it is on a line. A circle is 1-dimensional, specifically if the circle is just defined as the circumference, and not its area inclusive. Usually in maths we refer to the circle with its interior as a "disk" to distringuish these two things. I hope this answers your question.
@@smoughlder5549 thanks. I just wanted the definition for a sphere in spherical geometry. A circle isn't defined as just a circumference and what an ant might or might not "believe" doesn't sound like the axiomatic system that Euclidean geometry is.
@@DrEMichaelJones i don't really understand your question. Im not clued up on axiomatic geometry so perhaps im missing something. My answer was to address what I thought was your confusion about the dimensions of things (from the standpoint of topology/geometry).
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!
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An extra video explaining how distance is calculated in hyperbolic geometry might be interesting
Roughly speaking, it is the length of the shortest straight line between the two points. How we calculate that will depend on our model. In the poincare disk model, it's a little awkward, but you can find a formula on Wikipedia. In the half plane model it is the natural log of the cross-ratio of the two points (plus the two end points of the "straight line" through them thought of as points om the x-axis)
There is a Numberphile video with Dick Canary that does actually talk about this.
Surely you can do it with a basic sigmoid function.
When r is the apparent (on a disc of radius s) distance from the center, and R is the hyperbolic distance from the center, you can make a formula describing their relationship as:
r = s/(1-e^R) (describing f(R))
R = ln(1 - s/r) (describing f^(-1)(r))
Here's a Numberphilic video on just that topic.
th-cam.com/video/u6Got0X41pY/w-d-xo.html
Any extra video with Juanita would be interesting. Cheers.
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
Yes, or another way of thinking about that is that segments of the smaller circles wouldn't represent the shortest distance between the points.
@@maitland1007 No, this is not how lines are defined in non-planar geometry, specially because contrary to what is suggested in this video, the notion of distance is not necessary to define what lines are.
Actually, in Minkowski space-time, straight segments are the longest possible path between two points, not the shortest.
@@fuxpremier Tell that to airlines which use great circle paths whenever possible because it saves fuel.
@@fuxpremierThat is entirely due to the metric and not relevant to this discussion
@@topilinkala1594 It saves fuel because great circles are shortest paths, which is not the definition of straight lines (as mentioned in another comment above, in Minkowski space, they are the longest).
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.
Exactly so. But while Euclid is best known for his writing about plane geometry, which we now call "Euclidean" geometry, the fact is he also wrote about the geometry of a spherical surface. So it seems reasonable he could have seen the need to include a postulate that would distinguish the geometry of a plane from that of a sphere's surface, and that the parallel postulate was included for that reason. So at least one form of "non-Euclidean" geometry was in fact known to Euclid.
* if spherical and hyperbolic geometries don't rely on any additional axiom
If you can prove something with the four first axioms that contradicts some (different) fifth axiom, then your theory is inconsistent and anything can be proved in it
My understanding was that everyone who tried to derive it just failed.
It's not because it was longer to write, per se, but just that it felt like it should follow from the other postulates.
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 knew this fact about parallel lines but never really thought about it in a practical context until a few months ago where I saw someone demonstrate this trick
In English scientific literature this is commonly called “Intercept theorem”.
It basically states the following:
If a line A passing though several parallel lines P1,P2,… is cut (by P-lines) into segments a12, a23, a34,…
Then any other line B passing though P1,P2,… will be cut in segments proportional to a12,a23,…
|\
-----P1
| \
I just want to add that it should be "for any line segment [whose endpoints lie on each respective parallel line]"
When I read for any line segment between two parallel lines, the first example my mind conjured looked something like this:
-----------------------------------
which wouldn't have a midpoint lying on the midpoint line of the parallel lines.
As a side note, this will only get you the center with respect to 1 axis. If you do this twice, in slightly different locations, and then connect those points with a line, and then switch to the other axis and do the same thing, the 2 lines you created will intersect at the exact middle of the board with respect to both axes.
@@Dziaji Or you could put the endpoints of the connecting line segment in corners of the board and need just 1 measurement.
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!
Yeah, he's just awesome like that.
@@poppyseedsnuranium so he flat out lies?
in the sin(x)cos(x) integration video he flat out said he didn't remember calculus or integration.
humility is fine. false humility is deception and i hate it. far more than pride.
considering how busy he is and how much he learns all the time, I wouldn't be surprised if he forgot
@@sharpnova2 I think that you care way too much about this
I have never seen someone draw a sphere that quick
Not a sphere, but when drawing a circle, the secret is to move your whole arm in a circular motion and put the pen on paper while already in motion. That way you can get a nice circle very fast.
Quickly
If you tap on the settings cog, you can change the video playback speed to 2x and see it even more swiftly
I dunno it looked to me like it was staying still.
Kudos for making a compliment to Juanita. Cheers ❤
i always enjoy your non euclidean geometry videos
That's nice to hear. Thank you for watching.
Juanita rules 🤟
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.
This is fascinating and Juanita is a great teacher
As someone who wrote his masteral thesis on differential geometry, this topic means so much. It’s the reason I learned to love geometry, and eventually choosing to study geometry over other mathematical disciplines.
Juanita is the best, looking forward to followups. Cheers ❤
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 😅
Show them this video, maybe some of them will change their opinion 😉
😅
To get an appreciation of distance near the edge of the hyperbolic disk try some of the M C Escher "Circle Limit" engravings.
I really enjoyed her presentation of this topic!
It defines on which type of surface you are on: a negative, positive or no curvature.
Ahhhh I’m learning this in school as a freshman, so cool that I can understand this!
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.
I have seen so many times people think they can object to what Euclud claimed, by suddenly taking the claims out of context and applying them to some kind of curved space, and then expect people to say "wow, you are smarter than Euclid, who must have been a very primitive thinker". I am not saying this happens in this video (have only watched a few minutes yet), but it is a bit on the border.
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.
I think later he proves there is a unique line at a point, but its been a while since I glanced at Elements.
This is a book I will order tonight, such an old book so full of flavourful mathematics.
Great video!
Was surprised you haven’t mentioned Nikolai Lobachevskiy when speaking about hyperbolic geometry.
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.
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?
Do you mean latitudinal lines? (ex. 30°N, the path that crosses through Iraq, India, Nepal, Houston, Morocco, and Algeria) Those aren't great circles; the plane that circle defines doesn't intersect the center of the sphere. On a more practical note, if you were to try to walk (or fly) along it, you would need to turn rather than going in a straight line. Think of the line of latitude right next to the north pole (89.9999°N); that's clearly just a circle on the ground, and not a line.
@@nothayley Thanks that makes sense, i appreciate it
@@jakebourdages3445 So, I was looking for a comment like this. Like Hayls said, you probably meant latitudinal lines (think like the rungs of a LATter). Anyways, it's a quick video and they played it _fast and loose_ . But, there can be some communication error since a lot a definition wasn't given to the objects used. (I'm gonna rant)
So, typically I like to define a line merely as the path of translation of a point. That's a joining quality giving us a fair class of related objects. So, skipping a bunch of hooda for now. Those longitudinal lines mirrored across the equatorial great circle (as defined by your choice of pole) could (I'd say should) be "parallel". Given that the actual quality of parallel we're particularly looking for here should be "a fixed distant greater than 0 of two or more congruent lines" (between "positionally" (
*ladder
@@poppyseedsnuraniumMathematicians usually don't use the word 'lines' for the objects you are describing, but instead 'curves' or 'paths'. You could define a notion of 'parallel' for two curves to be remaining a fixed distance from each other, if you like, and then curves of constant latitude would be parallel.
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).
thanks for this explanation! I was wondering about smaller circles, as those could obviously be parallel to other circles!
@@BenMakesGames think about standing on the equator, and trying to make a “small” circle. You would have to constantly turn to make it back to your starting point. But if you follow the equator itself, you are always walking straight ahead. Basically, if the world wasn’t round, the great “circle” would just go on forever. This isn’t true for the small circles (which could always be drawn on a flat map that shows strictly less than half of the world)
I agree! Shortest distance is the key here.
@@Muhahahahaz And now realise that the sphere is fully symmetrical and you pick and choose your equator as any big circle. (Does not work so well for the earth because it is rotating and there is a reason why we have fixed the north and south poles.)
It’s important to note that shortest distance works for finding these lines in some geometries, but not others. A simple and very important example is Minkowski Spacetime, where the straight-line path between two points in spacetime is actually the _longest path_
As soon as you started talking about other spaces, I predicted pretty much the entire rest of the video. I feel smart now. ;)
BTW, googling "hyperbolic grid" gives some nice visualisations. Remembering how those look helped me figure out in my head that there are infinite "parallel" lines in hyperbolic space.
When I leari circular inversion and hyperbolic geometry it absolutely blew my mind
¡Una colombiana en Numberphile! 🇨🇴
this emphasizes that often, humans make assumptions without realizing they are making assumptions
Euclid fully realized they were assumptions. That's why he called them postulates. They were never meant to be proven; they were to be accepted as true to prove other things.
@@realitant I think that the poster was referring to how it was later proven that there were more axioms than Euclid wrote down.
Thanks , from an amatore math person.
probably one of the best introductions to manifolds without ever mentioning the term😃
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".
Perfect Sphere ⭕️
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?
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?
thanks for the explanation. I finally got it
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.
I think the simplest way to see this is that the shortest distance between two points is along a great circle. If you walk along the Earth in what you feel is a straight line, you will be walking along a great circle
Actually, right angles still exist on the sphere, so you can construct the line that the postulate would say is parallel, (the line where the interior angles are the sum of two right angles) and it would not be parallel.
@@Aetheraev but there is no requirement to make the shortest path in the original proof. You could clearly make a parallel line by creating a plane intersection that cuts off the top third of any hemisphere you create. I accept that the angles may not equal 180, I thought the error would cancel, however parallel lines are very clearly possible.
@@reallifeistoflat No, those would not be parallel lines because one of them is not a line at all. Only great circles are geodesics, the generalization of straight lines. "straight lines" are in fact defined as the shortest path between two points. To walk on your second "line", you would have to constantly steer to one side, like a circle in the euclidian plane. You have shown that there is some curved path on a sphere which does not intersect a given line. In the same way there are infinitly many curved paths in the euclidian plane which do not intersect a given line. But that's not what euclids postulate is about.
@@reallifeistoflat You could use "at a constant distance away" as a definition of parallel but you lose the straight lines. Only in euclidean geometry does a curve at a constant distance from a straight line have to be straight itself. In your example we see that a constant distance away from our great circle we get just a regular circle. But a regular circle in spherical geometry is distinctly curved. If you don't see why this is true consider one further away from the equator. As we get closer to the pole a regular circle "parallel" to the equator becomes a tighter and tighter circle until it collapses to a point at the pole itself.
Fascinating!
postulate 2 says something like: "Any straight line segment can be extended indefinitely in a straight line." - this also seems to break in the spherical plane, as the segment becomes a full line quite quickly... (this applying the "uniqueness" definition - if the line defined by the same points is always the same line, then the points from a line are always the same points, so after you extend the segment to reach it's own start, you cannot extend anymore as the line is already there, no more/other points available => no more extending)
I remember talking about this with my maths teacher at secondary school in England in the late 1960s. My argument with him was that he had a very limited definition of a 'line'. I argued that it is possible to draw parallel lines on a sphere but none of them are great circles (except the equator). I asked him why lines of latitude are not lines. He did not have an answer except one which he freely admitted he'd simply been taught. Can anyone explain this to someone who has not studied formal geometry for over 50 years? What is the definition of a line?
Brady: where do you find these wonderful people to talk with?
That's funny I JUST finished a long rant to the commenter above you about this. In short, in the video they were specifically talking about the "shortest path between points" which is a subcategory of a general "line". People might just call lats a "closed curve". But, meh.
You could maybe call them lines, but they are not "straight lines." They have differing curvature to the ambient curvature of the geometry. If you walk along a line of latitude you must be continuously turning.
@@Aetheraev And why aren't you continuously turning if you walk along a line of longitude?
@@PeterGaunt Because those are great circles. Their curvature matches the ambient one. Remember the curvature of a circle (as a curve in euclidean space) is 1/r where r is the radius, so smaller circles are more curved. Lines of longitude all have the same radius and thus the same curvature whereas lines of latitude do not. Lines of latitude are (except for the equator) more curved.
With euclid geometry, a line is a primitive object : it is something that follow the axioms of the euclidian geometry. In particular, a "line of latitude" can't be a line, because then you would have more than one line that pass between two points.
FYI, Eulid has a fifth postulate which he had defined because he was well aware of the Parallel Postulate during his time. Widely regarded as unnecessary and redundant for its original purpose, until people started using his theories for other things. I believe that the most egregious understanding was the term "plane". Fairplay's Axiom is a better representation of the Parallel Postulate. This video actually uses Fairplay's Axiom but with its own variation of "one and only one" line while Fairplay's states that there is "at most one".
"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
I think his postulate does scale up one dimension, but you have to scale up everything for it to work properly. Instead of being constrained to a sphere, picture one spoke with two freely rotating wheels in empty space. These wheels have to be exactly parallel to each other in order to not intersect if you were to extend them infinitely, and there's no other orientation where that is true. Of course, this is only true if space is flat, but the same can be said of the original postulate - if either is on a curved plane, you get different results. I have no idea whether this would scale up to further dimensions, though. I'm sure an actual mathematician could tell me where a fault in this logic is, but as a mere math enthusiast, I don't see any problems with this being a scaled up version of the original.
I mean yeah, euclid’s postulates scale well in Euclidean geometry, because Euclidean spaces can be broken down into lower dimensional Euclidean spaces- in every plane, a line is defined uniquely by two points, and through any third point, one unique parallel line exists. In very 3d space, three points define a unique plane, and for any fourth point, one unique parallel plane exists, etc…
imho, the point here is nonEuclidian geometry is limited in usage. No way to scale, coordinates not corresponding to volume. It is hard to calc volume. What volume formula for spherical geometry?
How interesting it is that the definition of "parallel" in spherical geometry necessitates that the full circumference always be invoked. Yet, there are an infinite number of lines that pass through any two polar opposite points, and there are an infinite number of latitude lines that are effectively parallel, but are also curved in that geometry. How is it that these latitude lines escape inclusion in some concept of "being parallel"?
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."
Historically speaking-in fact, right from Euclid's time itself-the problem that the geometers were trying to solve was _"Can Euclid's 5th 'postulate' be proved from his first 4?"_ (That is, whether it's actually a theorem rather than an independent postulate.)
For a long time, it was suspected to be so. But they weren't able to prove it. While attempting to prove it, they started playing around with alternative postulates. Thereby discovering valid geometries that differed from Euclid's. And thus, they realized that the 5th is indeed independent of the first 4.
I'm not sure what you mean by "necessary." By definition, geometry that uses only Euclid's postulates-and nothing contradicting them-is called Euclidean geometry. Absolute (aka neutral) geometry is Euclidean, but the converse isn't always true, thanks to the 5th postulate-as you pointed out.
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.
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.
Hi
Of course you can do like you proposed, but without the requirement to be great circles it is not longer true that there exists only one line between two points
@@andreadedomenico1479 antipodes on a sphere have infinitely many lines connecting them
It's all about definitions. You could call the latitudes on a sphere 'parallel circles', but I won't call them straight lines (except the equator).They're called 'parallels' because in most projections they're neat, parallel lines.
@@runeodin7237 they're called parallels because they are indeed parallels in the 3D space the Earth is in.
@@mbrusyda9437 of course they are not both counted in the model shown in the video...
Is there a geometry in which we have more than one, but a finite number of parallel lines going through a given point P?
That sphere noise reminds me of that video which explains how you can turn a sphere inside out
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
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.
It was taken to mean "what it means to know something"... So you could say, it was quite.... elementary
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.
In hyperbolic there is a way to partition the infinite parallel lines so that each partition has that property. Create an euclidian line with a given turn using the center of the circle (of the hyperbolic model used here). each hyperbolic 'line' that has an euclidean center on that line (including the infinite case) are in the same partition.
@@ingiford175 Never thought about that this way.
Using projective transformations, I noticed earlier that in projective geometry, the turn of a line k can be seen as the intersection point between line k and a default line, which is normally the line at infinity.
But I never considered this in any other geometry, but this even works in spherical geometry.
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.
But now you have points that are not connected by a line at all, violating the first postulate.
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?
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.
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?
Also in regard to lines being parallel, in case of a 2D plane parallel also means that the shortest distance between any point on L1 to L2 is constant. So if parallel would be a) non-intersecting and b) shortest distance for any point on a line to the other line is constant that would eliminate "unlimited number of parallel lines" for a sphere and a disk?
In 3D space (3D orthogonal space at least) the number of lines that satisfy a) and b) is only a subset of the parallel lines that just satisfy a) (don't intersect). Are there two different kinds or qualities of being parallel?
Well, really, spherical geometry is the wrong setting here. We should move to elliptical geometry and identify opposite points on the sphere. Then every straight line intersects every other in exactly one point, meaning there are no parallel lines. You could argue this subtly avoids the original formulation of the parallel postulate, but it certainly does not avoid its converse.
@Gunnar Frenzel the definition of parallel here is usually taken to be your a). Thus spherical/elliptic geometry had no parallel lines and hyperbolic has infinite families of them. The constant distance version produces something different and is only equivalent in Euclidean geometry
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.
Some of the other postulates do not hold on a hemisphere, e.g. any straight line segment cannot be extended indefinitely in a straight line on a hemisphere.
6:13 is such a satisfying sound! ASMR… 🥰
10:52 also lovely sound ha ha
Poor Euclid, people keep breaking his rules.
well explained
I prefer to look at conic sections as an explanation for hyperbolic geometry. It makes more intuitive sense than the circle from above.
And it's more in line with the S2 geometry as far as moving your intuition one atomic step forward.
What is the conic section model of hyperbolic geometry?
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.
'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.'
'If a line segment intersects two straight lines...'
'...two straight lines...'
you're wrong. stop this nonsense.
you should also be capable of noting that your spherical example disallows extension to infinity. thus not only do you lack straight lines, you lack explicitly mentioned requirement:
'...then the two lines, if extended indefinitely...'
so really you're just illiterate.
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
Great Circle Navigation, simple
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).
She's wearing the same watch as me! It is also broken in the same place :)
So Euclid was possibly the first philosopher to think about Non-Euclidean Geometry. A Greek using the opposite of irony, a term invented by the Greeks. I love that.
Euclids 5th shows us how and why geometry is important, it gives us the natural flat surface of our earth.
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?
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!
The hyperbolic plane cannot globally be embedded in Euclidean space. But small portions of it can.
Now we need another geometric space where there are multiple parallel lines through a given point - but only a finite number!
Are parallel lines equidistant from each other in all geometries? Seems they would have to be by definition.
Now I find all the comments I'm looking for. _Practically_ speaking they _should_ be. Else, you get a lot of wacky things all under the same class of objects.
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...
Hasn’t there been a video about this before? I think the video was called “The Fifth Postulate.”
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.
I guess the problem is that of these five famous axioms, one of them fails in other geometries (while the others hold). Maybe it's not a "problem" as such, but no need to get angry about it. ;) We're just having fun.
The postulate isn't "disproved". The point of this is that you can replace the fifth postulate with others and arrive at addition non-Euclidian geometries that apply to different kinds of plane. Unlike the other postulates, which are universal across all geometries, the fifth postulate is actually a definition of the kind of plane.
The postulate is not something to be disproved. It is basically an axiom. The classical "problem" with the parallel postulate is that people thought it should be possible to deduce it from the other postulates, but it is not. Hence, you can define other geometries with the other rules the same but that don't have a parallel postulate
Now we need the geometry in which all lines are parallel and none ever intersect.
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
Yeah, that was what I was thinking. And furthermore, it's only one such line. Although, it's not on a great circle. :)
Mathematicians don't cook, apparently. :)
Mathematicians want unique line segments and therefore they choose the shortest path connecting two points on the surface of a sphere. The shortest path is a segment of the great circle going through both points. (If you have two points on the equator, then "clearly" the shortest path between them is along the equator, not first going north and then south. Now realise that any great circle can be called the equator, since it's all symmetric.)
I don' get it. Euclid said a Line is those points the shortest distance between 2 given points, extending in both directions forever. Your line on a globe meets back on itself.
I don't get it. Your middle part. Given a "line" on the 40th parallel of the globe, and a point on the 30th parallel, could one not construct a line on that 30th parallel? Would they not be parallel, especially since they're called parallels?
Great circles do give the shortest distance between two points on a sphere. And you can think of them extending infinitely if you like, it just so happens that you get back to where you started after a certain distance.
The parallels are not great circles. They are not the shortest path between any points and to follow one, you would have to continously steer in one direction. So they are not straight lines, they are curved. And this postulate is not about any curved paths.
A parallel at a distance of one meter to the North Pole is about 6m28 long. If you walk along it, you will agree it is not a straight line :)
I've never been this early to a numberphile video.
Welcome!
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.
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 ?
If it requires turning in one direction or the other, it's not straight. A great circle is the only path on a sphere like Earth that you'll get from travelling in a perfectly straight line relative to Earth's surface.
Latitudanal lines look straight on many flat projections, but those projections are not the actual globe.
The Euclid axioms say that it pass exactly one line between two points, but with your construction, you have infinitely many lines that pass between two points.
@@Tryss86 Elliptic geometry is the way to resolve this. It only covers one hemisphere rather than the full sphere, so there are no antipodal points except around the equator, which seems to typically be treated as the horizon. The parallel postulate still doesn't hold, but it does regain the property of a unique line between two points.
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?
She didn't explain what hyperbolic geometry is like she did for spherical geometry.
Do the other 4 hold in spherical geometry or not?
Brady is 5am I can't do math rn 😭😭
I don't understand if you take a parallel longitudinal cut through a point it will be parallel to the first line
This is the axiom that if denied leads to non-euclidean geometry right?
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
why are great circles considered as lines? what about shorter circles created by planes that do not intersect the origin of the sphere?
Now I wonder: What happens if you leave out any of the other four axioms?
Then a lot of Euclid's proofs break down. You would be including systems that we don't consider to be "geometry".
@@ronald3836 Sure, but that's still mathematics. I bet interesting things can be found out about that system.
@ certainly, but you might need to add some new axioms/postulates to preserve some of what you removed.
To do "geometry", I would say you need points and lines, with each two points connected by a line, and at least two points on each line. There is a whole area of combinatorics concerned with finite geometries where you have just a finite set of points and line. The game "Set", which I think was in a recent video, was basically such a finite geometry with the cards being the points and the "sets" the lines.
I hear Euclid, I listen
can this generalized version, ie. that the behavior of parallels depends on curvature, be derived from the first four postulates?
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.
The tangent space to a general Riemannian manifold is Euclidean (which is, for example, where velocity vectors live), so Euclidean spaces are of physical relevance.
i wonder whether there wille be a mathmatician someday, that has no problem.
Then their problem would be to find one.
Suppose there is no problem left to solve in mathematics. Then I construct a problem: "Prove that there is no problem left to solve in mathematics", contradicting that there was no problem left to solve.
@@smoughlder5549 Nice one!
If it's two-dimensional, then what's the difference between a sphere and a circle in this non-Euclidean geometry? Are there definitions? Axioms? Theorems?
A circle is 1-dimensional.
@@JacobPlat no, a straight line is 1 dimensional. Book 1, definition 15: A circle is a plane figure.
@@DrEMichaelJones Is the line which defines the z-axis 3-dimensional, because it requires a third dimension to picture? Dimensions are a very intuitive concept which actually need a very specific definition. I won't give that specific definition here, but I can give an analogy to explain it. A sphere can be pictured in 3-dimensions, but the sphere itself is 2-dimensional. The reason we say this is best explained by imagining that you are an ant on the surface of that sphere. If the sphere is suitably large enough, you can't actually tell that you're not on a plane (See flat earthers). Another way to say this was in the video, which is something like 'the dimension of the space you're in is how many coordinates you need to define your position'. For a circle, I need just one coordinate: a single angle! Similarly an "ant" on a circle large enough might believe it is on a line.
A circle is 1-dimensional, specifically if the circle is just defined as the circumference, and not its area inclusive. Usually in maths we refer to the circle with its interior as a "disk" to distringuish these two things.
I hope this answers your question.
@@smoughlder5549 thanks. I just wanted the definition for a sphere in spherical geometry. A circle isn't defined as just a circumference and what an ant might or might not "believe" doesn't sound like the axiomatic system that Euclidean geometry is.
@@DrEMichaelJones i don't really understand your question. Im not clued up on axiomatic geometry so perhaps im missing something. My answer was to address what I thought was your confusion about the dimensions of things (from the standpoint of topology/geometry).
So no parallel lines on the sphere... What does it mean to have a ray? It would intersect itself at the point...
who is Juanita Pinzon Caicedo?
Wait… you’re saying the surface of a sphere is two dimensional. Are euclidding me?
If it wasn't, then nobody would've even tried to make a 2D map.
topologically...
I'm right down the road from Notre Dame. Maybe I can go do some maths with them.
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!