A common misstatement of concepts from Riemannian geometry. First, note the evident falsehood of the statement: parallel lines are lines which do not intersect (or "cross"). It follows (as night day) that parallel lines do not cross (or "intersect") anywhere, including such mythical places as "infinity".

So why the saying?

For some mathematical purposes, it is convenient to add another degree of symmetry to the euclidian plane (i.e. the plane where we do elementary geometry). The Riemann sphere (go read about it first, or nothing will make sense) is a great way to do this. Every point on the plane is identified with a point on a sphere, but one point on the sphere has no matching point on the plane. Lines of the plane are identified with circles through that point on the sphere. Distances are of course hideously distorted (lines -- which have infinite length -- get transformed to circles -- which have a finite length!), but angles are kept.

Now, any two circles on the sphere can intersect on at most 2 points. Our "lines" are precisely all circles through our "special point", so they already intersect once. Two lines which intersect on the plane will intersect again on the sphere (as well as at the special point); two lines which are parallel on the plane will only intersect at the sphere's special point.

On the sphere, the special point is close to the images of all distant points of the plane. And every line passes through it. So it fulfills *some* of our intuitions of "infinity". Indeed, sometimes the special point is called ∞.

From the above, **ON THE RIEMANN SPHERE**, any two lines (parallel or not) meet at ∞. Of course, on the plane they do nothing of the sort -- ∞ is not a point of the plane!

But still, haven't we lost the idea of parallelism on the Riemann sphere?

Not really. Parallel lines on the plane intersect *once* on the Riemann sphere; non-parallel lines on the plane intersect *twice*. If we wished to define geometry directly on the Riemann sphere, we'd just change our axioms and definitions slightly -- parallel lines would be lines meeting *only* at the special point.

And remember: it's *called* "∞", but that's just a name. It's not infinity, it's not on the plane, and there *are* other ways (such as Gauss' hemisphere) to compactify the plane, where things happen differently!

It's just a convenient trick -- there's a lot to be learned from applying it in specific situations, but merely by definition we cannot learn anything new.