Parallel lines, am I thinking right?

Question

We established in a previous question that if you had two paralell lines, and if you followed them for enough that you would (north) eventually have the lines meet. My thought is that if you were on the equator that this would work. Exactly half of your line north, and half south of the equator, or your lines would cross,earlier on the shorter route and later on the longer one. Example if you were one mile from the pole and you set up two one half mile paralell lines they would not meet at the pole?

The question is have I lost my mind?

To eyeon, there is no question that the two parellel lines will neet, actually in two places on the globe and still be parellel lines. My thought is that they would meet half way around the globe and that their was nothing majical about the north pole as for as parallel lines meeting. My thinking is that the reason that they meet at the poles is because we start half way ( equator). If you take a equalateral triangle and place in on the globe you will find that it bends as it conforms to the shape of the globe. This is caused by the curature of space and it causes all kinds of uncommon things. This is what Uncle Albert came up with following his works on the theory of relativity based on the speed of light. This does not seem natural to us, but it is a fact that makes some very thought provoking things happen. I may not have this just right but I think that the curature of space is the fouth deminsion. Light is the constant time and space bows to it.

Answer 1

Some of your other answerers have been befuzzled by Euclid -- by Euclidean space, I mean. In Euclidean space, all the lines are straight, parallel lines never meet, and so on.

But you're talking about lines on the surface of a sphere -- and that is not a Euclidean surface. The rules are different there.

The easiest way to see this is to look at a globe. Suppose we start at the Equator, in the Gulf of Guinea, just south of Accra, Ghana, at zero degrees longitude along the Prime Meridian -- the one that goes north through London to the north pole.

The Prime Meridian is perpendicular to the Equator, forming a right angle (90 degrees).

Now look at the Galapagos Islands in the Pacific, south of Guatemala and west of Ecuador, where the 90th meridian meets the Equator. Those lines also form a right triangle, because all the meridians are perpendicular to the Equator.

Now follow these two meridians -- the Prime Meridian through London, and the 90th meridian through New Orleans -- up to the North Pole. At the North Pole, these two lines are 90 degrees apart; that is, they're perpendicular to each other, forming a right angle.

So now we have a triangle on the surface of the earth. Two of its vertices are on the Equator, at zero and ninety degrees longitude respectively, and the third vertex is at the North Pole. There are right angles at of the three vertices, so this particular triangle has a total of 270 degrees.

But according to Euclid, all (flat) triangles have exactly 180 degrees. Conclusion: the earth's surface geometry is Euclidean space, and Euclid's rules do not apply.

Notice, though, that each of the meridians is perpendicular to the Equator, and, according to Euclid, if two lines are each perpendicular to a third line, they're parallel to each other.

Okay, now let's go to your example where you set up two parallel lines a half mile from the North Pole. Instead of using that, suppose your lines are set up at 80 degrees north latitude (about 70 miles from the Pole).

Each of your two lines are perpendicular to the 80th parallel (so they're parallel lines). And if you run these lines north, they'll meet at the Pole.

So yes, on the surface of the earth, you do have parallel lines that meet. The reason is that the surface of the earth is non-Euclidean.

Oh -- one other thing. Notice that the Prime Meridian (or any other meridian for that matter) is perpendicular to both the Equator and, say, the Tropic of Cancer (one of earth's parallels). That means that the Equator and the Tropic of Cancer (or any other of the Parallels) are parallel to each other. But these parallels go around the earth and never meet. They do curve back on themselves, however. If you walked around the world along any parallel, you'd eventually end up back where you started.

So you haven't lost your mind; you're thinking along the right track.

Answer 2

Parallel lines do not meet...look up the definitions, please. If it is specified that two people maintain 100 feet between themselves as they trek northward, they will never meet. They are traveling pathways that, when projected onto a flat plane, will stay equidistant...they are parallel.

Understand this, if each walker started out walking due true north, one or both of them will not be walking due true north when they reach the closest distance from the North Pole. Prove it to yourself. Take a tape measure and measure two points on it one inch apart.

Now put that tape along the Equator on a globe of the Earth. Put one of the two points on a longitude...any one of the longitudes. Now pull that tape tight along the equator, keep it parallel to the equater, and move the one point up along that longitude to the North Pole.

See where that second point ends up when the first point (which is still on the preselected longitude) is at the Pole. I guarantee that second point on the tape is not on the North Pole. For that second point to also arrive at the North Pole it would have to also follow a longitude, in which case the tape between the two points would go slack as it gets closer to the pole because the two longitudes the two points started out on are getting closer. The two points are no longer keeping that one inch distance between them...they are no longer following parallel paths.

If each person follows a longitude, the north south imaginary lines on the surface of the Earth as a globe; then they will meet. All longitudes begin and end on the poles (both of them).

Answer 3

You may be confusing the issue by calling them "parallels". Two true parallel lines will never meet. And a "parallel" is the flat horizontal marks on maps (latitude). The word you are trying to use is "Meridian", which are the measurement stripes that run from north to south along the surface of the globe (longitude).

If you're somewhere on a one mile radius from the pole and you have two half-mile meridians run north, then no, they would not meet.

Answer 4

When you draw parallel lines on the ground, you need to follow them a certain distance before they can meet. I think you're doing so to prove the earth is round (sounds like an innovative idea)?? It doesn't matter where you draw your initial parallel lines, since there're no edges on a sphere. But you still need to follow the lines that same long distance.

So regarding your example (Example if you were one mile from the pole and you set up two one half mile paralell lines they would not meet at the pole?):
NO, they will not meet at the pole.

Answer 5

If the lines are not great circles, they do not have to meet. Parallels of latitude are not great circles and never meet. Meridians of longitude are great circles and do meet, at the poles.

A great circle is any circle drawn on a sphere that divides it exactly in half, like the equator does.

This is assuming we are talking about spherical geometry and not Euclidean geometry.

Answer 6

your thinking is incorrect on both counts because this is not the definition of parallel LINES. lines are, by nature, straight, and the "lines" you are defining are on the surface of the earth and are therefore not straight and cannot be parrallel. this is especially true if we note the definition of parallel as follows:

two lines or line segments or line rays which propegate equidistant from each other for so long as they continue to exist in each direction.

since the paths you note on the earth surface are neither straight, nor continually equidistant, they are not defined as parallel and can therefore cross each other. that's why you think you're crazy

Answer 7

And never the twain shall meet.

Answer 8

if the lines are truly parallel, then they never intersect

Answer 9

there was a guy, his last name was kelly, from new york that could really double talk your a s s off!