# parallel lines never meeting **URL:** \ **Category:** Factual Questions\ **Created:** [February 6, 2018, 10:16pm UTC](https://boards.straightdope.com/t/parallel-lines-never-meeting/808333 "2018-02-06T22:16:29Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [February 7, 2018, 7:34pm UTC](https://boards.straightdope.com/t/parallel-lines-never-meeting/808333/21 "2018-02-07T19:34:03Z")
> [@Velocity](#): > > I am a laymen of math laymen-laymen but isn’t the very definition of “parallel,” two lines that never meet? If they could meet, then they wouldn’t be “parallel.” That is what I said. But now it seems we are considering Riemannian instead of Euclidean geometry. If you have a Riemannian manifold, the Riemannian metric determines geodesics, two which could, under some circumstances, intersect each other even when they start out “parallel” (as in a line crosses them forming equal angles), like two meridians on a sphere. Note that on a (non-compact) Riemannian manifold you cannot always extend an arbitrary geodesic indefinitely, and even on a geodesically complete manifold things can happen like a geodesic looping back around to intersect itself, so Euclidean geometry no longer applies, except locally. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [February 7, 2018, 7:41pm UTC](https://boards.straightdope.com/t/parallel-lines-never-meeting/808333/22 "2018-02-07T19:41:55Z")
> [@DSYoungEsq](#): > > It is not a given that a “right” angle must measure 90 degrees. It depends upon your geometry and your measuring methodology. To some extent, that question is identical to the one being presented. I am not sure what you mean by this statement, but a Riemannian metric, if we are given one, determines the angle between any two tangent vectors based at the same point. So you know if it measures 90 degrees or not. ---
**Author:** ![RTFirefly](https://avatars.discourse-cdn.com/v4/letter/r/c77e96/32.png) [@RTFirefly](https://boards.straightdope.com/u/RTFirefly)\ **Post date:** [February 7, 2018, 7:55pm UTC](https://boards.straightdope.com/t/parallel-lines-never-meeting/808333/23 "2018-02-07T19:55:30Z")
> [@Exapno\_Mapcase](#): > > I’ve told you and told you and told you. Never ask a question about math. Never. Especially a simple one. You won’t like the answers. But does anybody listen to me? Noooooooooo. Someone asks one and suddenly non-compact higher-dimensional manifolds are introduced and everyone’s running screaming for the door and there’s grade juice everywhere. And then what happens? Somebody asks another math question. > > Who’s going to clean up all this grape juice, I ask, but nobody listens to that either. It’s enough to drive a person non-Euclidian. Let’s not argue parallel lines, let’s argue about whether .9999… = 1. 😃 :runs for the hills: ---
**Author:** ![Lumpy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lumpy/32/446_2.png) [@Lumpy](https://boards.straightdope.com/u/Lumpy)\ **Post date:** [February 7, 2018, 9:00pm UTC](https://boards.straightdope.com/t/parallel-lines-never-meeting/808333/24 "2018-02-07T21:00:19Z")
> [@Exapno\_Mapcase](#): > > Never ask a question about math. Never. Especially a simple one Or philosophy. The simpler the question, the thornier the answer. ---
**Author:** ![Lumpy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lumpy/32/446_2.png) [@Lumpy](https://boards.straightdope.com/u/Lumpy)\ **Post date:** [February 7, 2018, 9:03pm UTC](https://boards.straightdope.com/t/parallel-lines-never-meeting/808333/25 "2018-02-07T21:03:09Z")
> [@RTFirefly](#): > > let’s argue about whether .9999… = 1. 😃 Yes; because I say so. And if you disagree you’re a counterrevolutionary traitor who will be shot. Now, see how easy that was? 😛 ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [February 7, 2018, 9:06pm UTC](https://boards.straightdope.com/t/parallel-lines-never-meeting/808333/26 "2018-02-07T21:06:57Z")
> [@Lumpy](#): > > Yes; because I say so. And if you disagree you’re a counterrevolutionary traitor who will be shot. Oh, if we could only enforce that. What a wonderful world it would be. 😃 ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [February 7, 2018, 9:45pm UTC](https://boards.straightdope.com/t/parallel-lines-never-meeting/808333/27 "2018-02-07T21:45:00Z")
To me the question makes most sense in the context of Riemannian manifolds as that’s where it makes most sense to talk about “zero curvature”. The example of a cone is just a simple example to show that some degree of caution is needed before assuming the parallel postulate always applies, but also a red herring as the parallel postulate fails at the apex precisely because the Riemannian manifold structure fails at the apex. Flat (zero curvature) Riemannian manifolds have the property of being locally isometric to the Euclidean n-space, or in other words for every point on such a manifold there exists a neighbourhood that is geometrically equivalent to a neighbourhood of Euclidean n-space. From this a local definition of parallel can be taken from the Euclidean definition of parallel. I would guess that the parallel postulate may come unstuck when considering flat non-orientable manifold, but I’m waiting for someone with a better grasp of the finer points of geometry and topology to fill in the details. For example a parallel transport of a vector along a flat mobius strip can change it to an antiparallel vector. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [February 8, 2018, 4:22am UTC](https://boards.straightdope.com/t/parallel-lines-never-meeting/808333/28 "2018-02-08T04:22:50Z")
If we had a triangle on our mysterious (but known to be flat) surface, and two of the angles were right angles, what could the third angle possibly be? If anything like the Gauss-Bonnet theorem held, it would have to be 0 (or 2π), which seems problematic. ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [February 8, 2018, 4:27am UTC](https://boards.straightdope.com/t/parallel-lines-never-meeting/808333/29 "2018-02-08T04:27:30Z")
On the subject of analytical algebraic topology of locally Euclidean metrization of infinitely differentiable Riemannian manifolds, wasn’t it proved somewhere that 0.9999… = π which certainly isn’t = 1. ---
**Author:** ![crowmanyclouds](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/crowmanyclouds/32/19884_2.png) [@crowmanyclouds](https://boards.straightdope.com/u/crowmanyclouds)\ **Post date:** [February 8, 2018, 1:26pm UTC](https://boards.straightdope.com/t/parallel-lines-never-meeting/808333/30 "2018-02-08T13:26:25Z")
> [@TriPolar](#): > > Hart did a few on parallel lines. In one strip he drags two sticks down the beach leaving parallel lines. Eventually he turns around and sees the lines converging in the distance where he started. He walks back and see they’re apart again, but converge in the other direction. When they converge, Jesus is dragging the stick . . . or something. CMC fnord! ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [February 9, 2018, 7:26pm UTC](https://boards.straightdope.com/t/parallel-lines-never-meeting/808333/31 "2018-02-09T19:26:59Z")
> [@DPRK](#): > > If we had a triangle on our mysterious (but known to be flat) surface, and two of the angles were right angles, what could the third angle possibly be? If anything like the Gauss-Bonnet theorem held, it would have to be 0 (or 2π), which seems problematic. Yes, but in order to have something like the Gauss-Bonnet theorem apply requires certain conditions. In fact I can thing of a simple example of a flat manifold where parallel lines intersect: Take the manifold with boundary defined by taking the set of points in the Euclidean plane {(a,b}), where x[sub]1[/sub]≤a≤x[sub]2[/sub], y[sub]1[/sub]≤b≤y[sub]2[/sub] and (x[sub]2[/sub]-x[sub]1[/sub])\>(y[sub]2[/sub]-y[sub]1[/sub]) for some Cartesian coordinate system. Now make the identification (x[sub]2[/sub],b) = (x[sub]1[/sub]+b,y[sub]2[/sub]) This gives you a flat manifold with boundary where the parallel lines do meet and you can simply remove the boundary to get a flat manifold. In this example the failure of the parallel postulate can be related to the failure of completeness. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [February 10, 2018, 2:58am UTC](https://boards.straightdope.com/t/parallel-lines-never-meeting/808333/32 "2018-02-10T02:58:45Z")
That is essentially your example of before, of a cone, right? Definitely, such conical singularities will modify the application of the Gauss–Bonnet theorem, and one can draw two parallel lines which come back around and intersect at some angle even though the rest of the surface is flat. The OP has not objected to this answer as far as I can interpret the comments. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [February 10, 2018, 4:36am UTC](https://boards.straightdope.com/t/parallel-lines-never-meeting/808333/33 "2018-02-10T04:36:30Z")
Note, by the way, that if your space is not a manifold, but, say, an orbifold, then its (appropriately defined) Euler characteristic may not be an integer at all. ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [February 10, 2018, 4:18pm UTC](https://boards.straightdope.com/t/parallel-lines-never-meeting/808333/34 "2018-02-10T16:18:23Z")
> [@DPRK](#): > > That is essentially your example of before, of a cone, right? Definitely, such conical singularities will modify the application of the Gauss–Bonnet theorem, and one can draw two parallel lines which come back around and intersect at some angle even though the rest of the surface is flat. The OP has not objected to this answer as far as I can interpret the comments. > [@DPRK](#): > > Note, by the way, that if your space is not a manifold, but, say, an orbifold, then its (appropriately defined) Euler characteristic may not be an integer at all. If you take the example manifold above and try to complete it (i.e. maximally extend all geodesics) you end up with a conical singularity, but the difference here is that parallel geodesics intersect on the manifold rather than at the singularity. The example I gave is a flat Riemannian manifold with a Euler characteristic of 0 (it is topologically equivalent to a cylinder), but I believe the Gauss-Bonnet theorem is modified by an additional term for the turning angle of the hole containing the conical singularity. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [February 10, 2018, 5:14pm UTC](https://boards.straightdope.com/t/parallel-lines-never-meeting/808333/35 "2018-02-10T17:14:52Z")
In a conical space, locally parallel lines never meet at the singularity. They meet somewhere beyond it. And that’s true even if you remove the neighborhood of the singularity from the space. ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [February 10, 2018, 6:30pm UTC](https://boards.straightdope.com/t/parallel-lines-never-meeting/808333/36 "2018-02-10T18:30:34Z")
> [@Chronos](#): > > In a conical space, locally parallel lines never meet at the singularity. They meet somewhere beyond it. And that’s true even if you remove the neighborhood of the singularity from the space. If you mean that when only a single diffeomorphism to a E[sup]n[/sup] is considered parallel lines don’t meet, then that is true but also trivial as parallel lines don’t meet in E[sup]n[/sup]. Even though that is not the same as saying they can’t meet at singularity as no neighbourhood containing the singularity/a hole can be diffeomorphic to E[sup]n[/sup]. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [February 11, 2018, 2:41am UTC](https://boards.straightdope.com/t/parallel-lines-never-meeting/808333/37 "2018-02-11T02:41:49Z")
As in, you take a neighborhood of the conical space which doesn’t contain the singularity, and hence is completely flat. In that flat neighborhood, you take two lines which both make the same angle with a line which crosses both of them, and which would therefore be parallel in the infinite flat space to which that neighborhood is homeomorphic. If we then step back again and look at the entire conical space, those two lines will eventually meet, iff the singularity lies between them. But they won’t meet at the singularity. You can also, of course, have two lines (or rays, at least, since it’s not clear how to extend them past the singularity) which do meet at the singularity. But those lines will nowhere resemble anything that might be called “parallel”. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [February 11, 2018, 4:50am UTC](https://boards.straightdope.com/t/parallel-lines-never-meeting/808333/38 "2018-02-11T04:50:15Z")
> [@Chronos](#): > > As in, you take a neighborhood of the conical space which doesn’t contain the singularity, and hence is completely flat. In that flat neighborhood, you take two lines which both make the same angle with a line which crosses both of them, and which would therefore be parallel in the infinite flat space to which that neighborhood is homeomorphic. If we then step back again and look at the entire conical space, those two lines will eventually meet, iff the singularity lies between them. But they won’t meet at the singularity. I think you have to be careful what you mean by “between” the two lines. The singular point could be very far away while you, an ant, walk along a narrow strip defined by two locally parallel lines which eventually intersect back where you started. All the while, you never encountered any singularities along the strip. ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [February 11, 2018, 11:24am UTC](https://boards.straightdope.com/t/parallel-lines-never-meeting/808333/39 "2018-02-11T11:24:30Z")
> [@Chronos](#): > > As in, you take a neighborhood of the conical space which doesn’t contain the singularity, and hence is completely flat. In that flat neighborhood, you take two lines which both make the same angle with a line which crosses both of them, and which would therefore be parallel in the infinite flat space to which that neighborhood is homeomorphic. If we then step back again and look at the entire conical space, those two lines will eventually meet, iff the singularity lies between them. But they won’t meet at the singularity. > > You can also, of course, have two lines (or rays, at least, since it’s not clear how to extend them past the singularity) which do meet at the singularity. But those lines will nowhere resemble anything that might be called “parallel”. I’m still not entirely clear what you mean as two different examples have been given flat Riemannian manifolds with conical singularities In the first parallel lines don’t meet, though the distance between them goes to zero at the singularity. In the second the condition for parallel lines to meet is that any neighbourhood containing both lines contains a hole due to the conical singularity, but this hole needn’t be in the strip between the two lines. ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [February 11, 2018, 11:37am UTC](https://boards.straightdope.com/t/parallel-lines-never-meeting/808333/40 "2018-02-11T11:37:23Z")
Just to be clear: by “lines” I mean line segments. [Previous page](https://boards.straightdope.com/t/parallel-lines-never-meeting/808333.md?page=1) [Next page](https://boards.straightdope.com/t/parallel-lines-never-meeting/808333.md?page=3)