Wandering around Wikipedia, I came across the idea that if we violate the parallel postulate, we arrive at new, non-Euclidean geometries. Specifically, if you violate it in one direction, you get elliptic geometry, and in the other direction you get hyperbolic geometry.
It's a fascinating idea, but Wikipedia doesn't say a whole lot about it. I've spent a few weeks turning the idea over in my mind, and I now think I understand it. Basically I want to write down how I think it works, and have someone tell me whether I'm correct or not. However, I'm having trouble not making this into a 50-page essay that nobody will ever read!
As best as I can understand it, it's a question of space. Elliptic geometry doesn't have enough of it. Hyperbolic geometry has too much of it. Let me explain...
Euclidean geometry is the geometry of flat space. If you take a flat sheet of paper, cut wedges out of it, and glue the edges together, it forces the paper to curve. If you follow that curve far enough, it naturally closes into a complete sphere.
Sure enough, if two ships set sail from the north pole on different headings, initially the distance between the two ships grows linearly, just like Euclidean geometry would suggest. However, by the time they reach the equator, they are actually sailing parallel to each other, and after that they actually sail towards each other.
(Question: Is elliptic space finite in size? If you travel in a straight line for long enough, do you end up back where you started?)
Basically, as you travel outwards, elliptic geometry has "too little space", compared to what you would expect from Euclidean geometry.
Hyperbolic geometry is harder to think about; the Earth is spherical, but I'm not aware of any simple real-world shape that is hyperbolic. But, logically, if elliptic geometry is the geometry of missing space, hyperbolic geometry ought to be the geometry of too much space.
That is, as you travel outwards form a point, you find too much space around you. I don't know exactly "how much" extra space, but more than you would expect.
(Question: What's the formula for the circumference of a circle in elliptic geometry and in hyperbolic geometry?)
This suggests that if two ships set sail towards each other, provided they start off far enough apart and the angle between them is shallow enough, the "extra space" that keeps materialising as they travel onward potentially means they could actually miss each other - which would explain why you can have more than one parallel line. At some point, the angle becomes sharp enough that the ships' paths do cross, but at any shallower angle, they will miss. (And there are infinity such angles.)
Is any of this correct? Or an I barking up the wrong tree?