Re: two parallel lines will meet in the infinity

The behavior of parallel lines led to one of the most important developments in mathematics, the introduction of non-Euclidean geometry. In Euclidean geometry, a plane is like a tabletop or piece of paper, a flat object that extends forever in all directions. Using the axioms of Euclid, useful theorems can be proved like "the sum of angles in a triangle is equal to 180 degrees". However, this is only true on a plane. If you draw a triangle on a sphere (like a globe), you can measure the angles and show that they always sum to more than 180 degrees!

Parallel lines also behave differently on a plane and on a sphere. Two lines moving in the same direction on a plane will never meet at a finite set of coordinates. However, suppose that two people start at the Equator and head north. They are traveling in the same direction, but since they are on a sphere, they do meet! First they meet at the North Pole, and if they keep going long enough, they will meet at the South Pole as well.

Now the basic plane doesn't have a point called "infinity", it only consists of points describable by pairs of coordinates (x,y). However, some people (Einstein included) have found it advantageous to add a point at infinity. It works like this. We add a special point infinity with the property that all lines intersect the point infinity exactly once. There is only one point infinity, so no matter which way you move along a line, eventually you get to the special point infinity.

But now let's look at the consequence of this definition. Any two lines might intersect in the plane but they certainly intersect at the special point infinity that we have created. Even parallel lines intersect here. If you stand on railroad tracks running east west, you'll note that they seem to meet at infinity whether you look east or west. We've defined infinity so that parallel lines meet there no matter which way you go.

A sphere is not the most unusual geometry around. Hyperbolic geometries have the property that parallel lines will meet (converge) in one direction and will not meet (diverge) when moving in the other direction. Our modern view of gravity uses Einstein's General Theory of Relativity, where objects with mass actually bend space into these unusual geometries, so these non-Euclidean geometries are very important to physics.

I'd recommend the Encyclopaedia Britannica entries on Euclidean geometry, non-Euclidean geometry, and General Relativity to learn more about these topics. The NRICH online math club also has part of an article posted that answers this question.

Mark