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Name: kaos
Status: N/A
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Date: Around 1995

Do parallel lines meet at infinity?

Could be. Nobody has ever managed to go to infinity to find out. Actually, the question of what "infinity" means for spaces of more than one dimensions is a little tricky. You could have one infinity, which is often done with the complex plane, or you could imagine infinities in all directions - an uncountable number of them in fact. If there is only one infinity, then parallel lines must of course meet there, since there is nowhere else to go. But if there are an infinite number of infinities, parallel lines could go to different ones, I suppose. Although, since they are headed in the same direction, you might think they would meet in the same place. In any case, it is something that is often said, but I am not sure there is that much meaning to it.


In the context that you are probably using to think about geometry, the answer is that parallel lines do NOT meet. I say this since almost everyone interprets geometry as Euclidean Geometry. The true statements in Euclidean Geometry are those which can be derived from the definitions and postulates assembled by the Greek mathematician, Euclid, (400 BC). In this context, parallel lines cannot meet. In fact, one definition of the word parallel is "lines that do not meet" (in a plane). As Ross Perot might say, "end of discussion." However, I think you mean something more, e.g., do two common perpendicular lines to a single line in a plane ever meet? While the answer is, no, in Euclidean Geometry, there are definitions and postulates that have been studied for centuries in which the answer is "yes" instead. Projective geometry is a case in point. You should be able to find informa- tion in your library about this. There is a way to visualize this situation so that a Euclidean plane is contained in a projective plane.


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