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Question:
Parallel lines. I was playing with a lock in geometry class on day and my teacher asked me if the outside line and the inside line of the lock are parallel. said that it was, but he told me that it could not be, but he never gave me a reasonable explanation. I think that the inside and outside lines could in fact be parallel. Could you please give me an explanation as to why they are not considered parallel? Everything I have looked up about them never says anything about the lines not being able to curve like that in an arch.



Replies:
I do not know exactly what you mean by the "outside" and "inside" lines on a lock, but I can tell you that curves can be considered to be parallel. Before talking about these curves, let us review what two parallel lines are: two lines in a plane that do not intersect. These lines are straight and this definition is for Euclidean geometry (the kind we study most in high school). In non-Euclidean geometry, we can talk about "curved lines" called geodesics, in curved spaces. In two-dimensions, consider a curved "plane" like the surface of a sphere. A great circle has the circumference of the sphere and divides the sphere into two equal hemispheres. Even though the earth is not a perfect sphere, you can consider the equator to be a great circle. A segment on a great circle is called a geodesic. Two geodesics are considered parallel if they do not intersect anywhere on the curved surface on which they lie.

That may be a lot more information than you wanted, but without seeing your lock, I think there is a pretty good chance that you and your teacher may not have clarified whether you were speaking about Euclidean or non-Euclidean geometry. From a non-Euclidean perspective, your two curves may indeed have been parallel.

Dr. Eric Hagedorn



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