Do parallel lines meet at infinity?

Question

Does gravity have an effect and the lines may not be close together, just parallel?

Answer 1

Only if you are doing projective geometry. But not in Euclidean geometry. There is no 'place' at infinity.

Answer 2

Consider an isosceles triangle (where 2 side are equal in length) - if you increases the length of the two equal sides the angle subtended by these lines decreases; as the length of the lines tend to infinity, so the angle tends to zero and the lines are therefore parallel. But as it is a triangle they must meet at infinity!

Equally, if your car could drive through an infinite field of snow in a perfectly straight line, the tyre tracks would never meet.

So I'm afraid the answer to this question is clearly both yes and no.


Added:

And if you're not convinced that the the lines are parallel at infinity, consider this:

1÷9 = 0.111111111111111111111111111111111111111111...
2÷9 = 0.222222222222222222222222222222222222222222...
3÷9 = 0.333333333333333333333333333333333333333333...
4÷9 = 0.444444444444444444444444444444444444444444...
5÷9 = 0.555555555555555555555555555555555555555555...
6÷9 = 0.666666666666666666666666666666666666666666...
7÷9 = 0.777777777777777777777777777777777777777777...
8÷9 = 0.888888888888888888888888888888888888888888...
9÷9 = 0.999999999999999999999999999999999999999999...

But 9÷9 is quite obviously =1 therefore
0.999999999999999999999999999999999999999999... = 1

So you might argue that at infinity there is still a small distance between the lines of the triangle, but the above equation shows that this is not the case.

Discuss ...

Answer 3

gravity can have no effect on a line which is just a mathematical construct joining two points by the shortest route. however in an open universe parallel lines will diverge, in a flat are always equidistant and in a closed will meet and cross over. this is due to a more precise definition of parallel ie. that two lines are parallel if a line tangential to one bisects the other at a tangent.
As a two dimensional example for each to help your visuallisation consider a saddle shape, imagine drawing a line across the saddle, now draw two lines tangential to it, you can see as they rise up over the sadle they diverge

consider a plat peice of paper, with two pencils stuch together now draw a couple of straight lines, obviously no matter how far you go the pencils are always seperated buy the same distance

imagine a globe with the north south lines drawn on, the lines bisect the equator at a right angle but this holds true of the lines at the tropics and at all other lines but the north south lines meet at the poles

Answer 4

It depends on the geometry.
For flat space such as a piece of paper they will never meet.
For hyperbolic space such as between 2 hills they will diverge.
For closed space such as the surface of a sphere they will eventually meet. For an example of this look at lines of lattitude which are straight parallel lines at the equator but meet at the poles.
The 'normal' laws of geometry make a hidden assumption of flat space, this is often called Euclidean geometry. In the real world non euclidean geometries are common. Another affect of this is that the interior angles of a triangle do not add up to 180 degrees.

Answer 5

Jay is correct. They don't meet at infinity.

In normal Euclidian geometry (the USUAL geometry taught in school), they don't meet at all.

In other geometries (Reimannian, etc.) there are some geometry rules where parallel lines will meet... but that isn't infinity either.

In ART, parallel lines can appear to meet (look down a long street and observe that the sides appear further apart where you stand than far off in the distance).

Answer 6

Lines are only are representation of an actual. however if two objects were traveling at the same velocity in a parallel direction they would never touch unless affected on by an outside force; however they would then no longer be parallel (two lines are parallel if they do not intersect)

Answer 7

You could use that as a definition of infinity:-

"Infinity is is the distance away at which parallel lines meet, but as parallel lines never meet you can never get there."

;-)

Answer 8

In theory there could be a race of beings that call themselves 'lines'. They might have a word 'parallel' - meaning male/female, or describing a region or type of 'person' etc. We would say that when coming together that these Parallel lines are meeting and it would be absolutely true.

If space/time distorts then if observing that distortion shows them to meet it isn't really happening - if you travelled through the distortion yourself you would not observe any distortion, surely.

Answer 9

I think the answer is basically relative to the original rule-base, that is, for example, should lines initially defined as parallel cross paths in a practical sense (observed or measured) due to a change in the time-space continuum, then they are no longer defined as parallel relative to the rules from which the definition originated from.

Answer 10

Parallel lines do not meet, and infinity is a continuation, not a destination, there is no definite end to infinity. Therefore they cannot meet at infinity.

Answer 11

according to Bolyai, they do! I was told by my teacher that space itself bends, when a huge mass is in it, and light bends in its gravitational field, and that's why, but that's not the explanation, if you're really interested i this subject, try Bolyai Janos's works, he was a Hungarian, Transylvanian genius, his father was also a mathematician too, who studied parallel lines, and gave his son the warning to stay out from these things, but Bolyai didn't, luckily, and developed a whole new geometry. Some people say that Gauss and he made these discoveries at the same time, so try to read some of his works too.