Parallel Lines: Euclidean and Non-Euclidean GeometryDate: 4/25/96 at 12:25:57 From: Anonymous Subject: Parallel lines If two lines are parallel, do they intersect? Thanks. Dennis Date: 4/26/96 at 8:15:29 From: Doctor Steven Subject: Re: Parallel lines Well, it really depends on what kind of geometry you are looking at. In planar geometry (geometry on a flat surface, also called Euclidean geometry), no. The definition of parallel lines is that they extend infinitely long in both direction without intersection. The fifth postulate of Euclid says that if a straight line falling on two straight lines makes the angles on one side of the line less than 180 degrees, then the two straight lines if extended will intersect at some point on the side of the other straight line where there angles added to less than 180 degrees. The fifth postulate was a source of contention for many mathematicians in ancient times and even in relatively modern times. The thought was that it could be proven from the other four postulates that Euclid gave: 1) From any two points a straight line can be drawn. 2) Any straight line can be extended indefinitely. 3) A circle can be drawn of any radius at any point. 4) All right angles are equal to one another. Many mathematicians spent their time trying to prove this postulate using the others; if they could it meant that the fifth postualte should not be a postulate but rather a theorem. Probably Euclid would like to have proved it, so parallel lines could be proved to intersect a falling straight line with interior angles equal to 180 degrees. Unfortunately for him he could neither prove or disprove the fact, so he added it as a postulate. Now other mathematicians also thought it should be provable from the other four postulates and so they tried - for hundreds of years! Finally two mathematicians decided that if the 5th postulate was provable by the other four they should be able to "replace" the 5th postulate with a contrary one, and they would get a contradiction somewhere. Well they took out the 5th postulate and added a new contrary one and they got crazy statements like: Every triangle must have strictly greater than 180 degrees as the sum of the interior angles. But no contradictions. They had invented a new system of geometry, of course they called this Non-Euclidean geometry. This new geometry they created perfectly described what we call spherical geometry. After their discovery other mathematicians decided to see what they could get by adding different contrary postulate in the 5th postulates place. In a Non-Euclidean geometry such as spherical geometry, two lines can be parallel and still intersect. To see this think of the globe and two lines of longitude. Look at the lines of longitude, say 30 degrees W and 40 degrees W, note that at the equator these lines are parallel, no look at either of the poles and you will see that they intersect. Such geometries as this are called non-Euclidean, and there are many. To learn more on some of these you can read up on G. Bolyai, Lobachevsky, Riemann, and Felix Klein. -Doctor Steven, The Math Forum (with some help from Dr. Patrick) Check out our web site! http://mathforum.org/dr.math/ |
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