Euclidean Geometry

If we took a survey asking whether we live in a universe that follows the principles of Euclidean geometry or non-Euclidean geometry, I expect almost everyone who understood the question would choose Euclidean geometry, since that is what is taught in school. Those who studied the history of geometry may recall that there were lots of very puzzled mathematicians in centuries past who tried to prove by contradiction that the parallel lines never meet (or something like that) and wound up inventing a bizarre but self-consistent system of geometry. The actual history differs slightly from this, but the essential point is that everyone knows Euclid’s version was the right one and the others are just mathematical quirks that only have abstract applications in the real world, just like imaginary numbers.

About a year ago I was studying different coordinate systems to see which one would be the most useful for storing the contours of a randomly generated planet (spherical coordinates were best: basically latitude, longitude and altitude) and stumbled across a surprisingly real application for non-Euclidean geometry. It seems that if you constrain yourself to the surface of a sphere Euclidean geometry does not hold true. For instance, the sum of the angles in a triangle is always supposed to equal 180 degrees, but on a sphere it is always greater than 180 degrees. An extreme example should make this clear: the meridian lines all meet at the poles and cross the equator at right angles, so if you start at the north pole and choose meridians that are at right angles to each other then these two lines and the equator will form a triangle with three right angles. Of course, if you look at it in the full three dimensions, you see that these “lines” are actually curves, so you don’t really have a triangle at all. Euclidean principles have not been broken and all is well with the universe. Or is it?

As you just saw, it was the bending of our surface, and therefore lines, that resulted in the non-Euclidean weirdness. It turns out that Euclidean geometry only works in flat planes or space. However, in his famous theory of general relativity Einstein found that gravity is actually the warping of space-time, so every massive object in the universe is bending space into a ridiculously complicated, definitely not quite flat shape. This leaves us with the starling conclusion that Einstein overthrew not only Newton, but even Euclid. A little research shows that this is indeed a valid conclusion.

That’s not to say we should throw out Euclidean geometry or even that the schools are wrong to teach nothing else. Since most objects we know of are not massive enough to make a serious dent in space, Einstein’s discovery does not invalidate the practical, everyday applications of geometry, any more than it invalidates the everyday applications of Newtonian physics. The only place you would need to worry about the difference is near a black hole, in which case I’d say you have much more pressing issues to worry about (no pun intended).