parallel lines never meeting

[Dr.Strangelove]

I’m surprised no one’s brought up the real projective plane. Also called the extended Euclidean plane. Or have they in words I failed to recognize?

Totally flat. No singularities. All lines, even parallel ones, intersect.

[DPRK]

Asymptotically fat mentioned it (as a “flat möbius strip”) in Post #27, but it is good that you explicitly brought it up, lest we get too hung up on orientable surfaces.

ETA the entire real projective plane is obtained by gluing a disc along the boundary of your flat Möbius strip, in case anyone was wondering

[Chronos]

Quoth Asympotically Fat:

In the first parallel lines don’t meet, though the distance between them goes to zero at the singularity.

I don’t think this is true. If the distance between the lines approaches zero as you approach the singularity, then the lines aren’t parallel.

[Dr.Strangelove]

@DPRK

Asymptotically fat mentioned it (as a “flat möbius strip”) in Post #27, but it is good that you explicitly brought it up, lest we get too hung up on orientable surfaces.

Ah! That makes sense.

Though to be clear, I had been thinking of the infinite version, which I suppose still has the same construction if you take the distance of the strip as it goes to infinity, but (to me at least) is easier to imagine with a different construction: add a line at infinity, which looks like a circle (but with zero curvature so it’s really a line) that surrounds our whole space. All lines intersect the line at infinity, and parallel ones clearly intersect at the same point.

I (personally) like this better than the conical space example since there’s no finite distance you can travel before things start “getting weird”. Everything works normally in local space.

[Asympotically_fat]

@Chronos

I don’t think this is true. If the distance between the lines approaches zero as you approach the singularity, then the lines aren’t parallel.

Actually yes you are correct they meet beyond the singularity, depending on the metric you impose at the singularity.

[kaylasdad99]

@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.

And non-Euclidean is the LAST thing you want to be. Why? Because when Great Cthulhu returns he’s going to be in the mood to eat something new; that is, he’ll save the non-Euclidean folks for last.

You don’t want to be eaten last.

[DPRK]

Are you sure that you have read the novella carefully? Cthulhu & his friends ruled the Earth when it was young, so Euclidean space is old hat to them, and they have hung out in all sorts of abnormal dimensions as well; they have seen it all, really. As Thurston says, a time will come, but it is better not to think about it. Or think all about it, if you want—not like it makes much difference.