Einstein, Euclid, and Parallel Lines
October 28, 2006
Parallel lines eventually meet. All you have to do to prove it is to look down the railroad tracks toward the horizon. You can see clearly that the rails touch.
A contributor informed me, without much tact, that my scientific method was lacking in rigor. "Sheepandgoats, you moron! It’s perspective! Walk down to where they seem to meet and you will see they are as far apart as ever."
So I walked down to where they seemed to meet and, sure enough, they were just as far apart as ever. Okay, so apparently they remain separated by an unchanging distance. Parallel lines never meet.
Or do they?
With mathematical lines you can accomplish exactly what perspective suggests.
Imagine a straight horizontal line. Mathematical lines, you will remember, are endless. They extend forever. Now picture a line perpendicular, a vertical line. Of course, the two lines will intersect. Call that point of intersection point I. Now travel two feet up that perpendicular line, two feet above I, and choose another point. Call it point P. We call it P for pivot. Pretend that you can pivot the entire line around that point, as if that vertical line was a compass needle pivoting around the center.
As you pivot the vertical line, what happens to point I? It moves farther and farther down the horizontal line, doesn’t it? As you continue to pivot your vertical line, so that it is more and more starting to resemble a horizontal line, 2 feet above your original line, point I really zooms out there. And it seems like, when you finally get your “vertical” line parallel (from the pivot point P of view) point I must “jump the track.” It must leave that horizontal line. It must disappear. Otherwise, your two lines can never really become parallel.
But where is that point of jumping the tracks? Can you identify it? Pick a point at random. Call it point J. That is the last point the two lines have in common. After that the two lines are separate. They never touch, as we’re told parallel lines never do.
But, geometry also teaches us that between any two points, it is always possible to draw a straight line connecting them. So take a point one foot further than J on the original line. Call it point F. And you can draw a straight line from point P to point F! So point J is not the last common point after all! You can quickly see that there never will be a last common point.
So parallel lines do indeed intersect, at infinity!
Obviously, then, mathematics does not really describe reality. Or does it?
Well….if you build on your new parallel lines derivation, you come up with a different, oddball, non-Euclidian geometry. And it turns out, that geometry does have application in reality, because reality is decidedly oddball, as we know from trying to wrap our heads around relativity. And, what’s even worse: quantum mechanics.
When science is experimenting in the lab, it is easy to explore and deduct. You just mix chemicals, take cover, and see what happens. You taste or feel or weigh the results. But you can’t do that on the cosmic scale….it’s too far away. And you can’t do it on the subatomic scale….it’s too tiny to insert your fat fingers. Therefore scientists use mathematics to go where their instruments cannot, utilizing the fact that math correlates highly with the way things are.
A total eclipse of 1919 furnished proof of Albert Einstein’s theory of special relativity, first published in 1905. If Einstein was right, an object of huge mass (the sun) would bend light (from the stars behind it) and the angle of the bend could be recorded by scientists. If Einstein was not right, there would be no bending of light, and that too, could be verified by scientists. Einstein was right.
But the frizzy haired physicist wasn’t on pins and needles the night before the big test. He wasn’t sweating it. He knew his theory would hold.
The math worked.
If you try the parallel lines trick on your pals, (amaze your friends!!) some, depending on who your friends are, will grasp it right away. Others will argue with you forever. And some will get mad. I suspect the third group feels threatened. Indeed, we may need to live forever to figure this out.
“God wrote the universe and the language that he used was mathematics.” Galileo
That there are lines that are perpendicular doesn't mean that there doesn't exist lines that are straight. Light that travels from a sun to a planet, without interruptions of other planets or moons, can travel in a straight line from point A to point B.
Posted by: Robin | April 28, 2009 at 07:07 AM
Agreed. I think. Not sure what you are getting at.
Posted by: tom sheepandgoats | April 28, 2009 at 09:18 AM
You are confused again, I think. Just because you can't find the last point of intersection for a non-parallel line, does not mean that parallel lines don't exist.
Indeed, there are an infinite number of non-parallel lines as you turn your line around the pivot, P. However, a horizontal line through P exists and will never touch your line.
This is exactly like points on a number line - you can never name the largest value that is less than 2 but that doesn't mean that 2 doesn't exist.
PS (there are indeed other geometries in which parallel lines intersect - but your example is just a misunderstanding of Euclid.)
Posted by: Beth Gallis | December 21, 2010 at 07:29 AM
Okay. perhaps. what is the proper understanding of it?
You are remarking on this statement, perhaps:
"Well….if you build on your new parallel lines derivation, you come up with a different, oddball, non-Euclidian geometry."
That could have been a leap on my part, I admit. Sure is an odd quirk, though.
YIKES! Your return link takes me to Philadelphia Central High Home of the Lancers AP Calculus Math! I'm not at all sure I want to mess with them!
Thanks for the comment. Feel free to offer any followup you like.
Posted by: tom sheepandgoats | December 21, 2010 at 06:28 PM
I have been thinking about this and I think you are wrong in your claim that because the limit of L as line tends to parallel is infinity the lines meet at infinity.
Think of it as like this. As the lines tend to parallel the point l tends to infinity, in the same way as x tends to zero 1/x tends to infinity. That doesn't mean that 1/x = infinity.
1/x is undefined and the meeting point of parallel lines is also undefined.
Posted by: Andy | December 01, 2012 at 05:50 AM
Okay. Still, is that not acknowledgement that parallel lines do meet, even if nailing down the precise point is beyond us?
Posted by: tom sheepandgoats | December 01, 2012 at 12:02 PM