No JP, I used the triangle on a sphere as a definition that I found understandable but my question here is how two parallel lines can be seen to cross, and as I've seen it described 'merge'? As I see it there must be a space between those lines, and at no point will they share a same point (a slight pun, still, how I see it). And it's a axiom to me, as self evident as picking up those stones to me to count.

Ok, but it's a problem to use axioms that are "self evident," since our brains aren't really constructed to intuitively understand non-Euclidean geometry.

Here's one of the problems: what is a "straight line" on a sphere? A sphere is a curved surface, so nothing on it is straight, but how do we get close to a straight line?

The answer is that we figure out the rules for constructing straight lines in 2D and then figure out how to construct those on a sphere. One way of constructing a line would be to put your pencil on the paper and trace a path without picking the pencil up or having the path cross itself. But this isn't necessarily straight. I could draw a curve or a spiral if I wanted (Fig. A)

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The usual definition of parallel is that "lines" (whatever we define them as" don't ever touch except possibly at infinity. I could draw another curve that never touches the first curve. Are these parallel lines? Well, they never touch, but they're not lines. (Fig. B).

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To make the line straight, I can think of constructing it from a stretched rubber band on the plane. No matter how much I try to bend the rubber band, it snaps back into a straight line. The rubber band minimizes the distance between any two points along it's path in the plane, and is therefore known as a geodesic, which is a straight line on a 2D Euclidean plane.

Now I have a unique definition that captures what it means to be straight parallel lines:

1) The lines are constructed from the set of points that minimizes the distance between any two points on the line.

2) The lines don't intersect.

If you try to put a tight rubber band on a sphere, you'll find that it pops off as it tightens unless it's a great circle of the sphere. And if you put two such rubber bands on the sphere, the have to intersect. You

**cannot** apply both 1) and 2) simultaneously to the surface of a sphere. You either have "straightness" or "parallelness" but not both. Geodesics always intersect and non-intersecting lines are never geodesics.

From your description, I'm guessing you want to give up "straightness" and keep "parallelness." But then you can end up with weird cases like Fig. B above where you have wiggly contours on the sphere that don't ever touch, but clearly aren't intuitively parallel.

In terms of this, what do you mean by parallel lines on a sphere? Is it just non-intersecting?