Parallel Lines
Dec. 14th, 2010 07:58 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
purplecatI promised ![[livejournal.com profile]](https://www.dreamwidth.org/img/external/lj-userinfo.gif)
lukadreaming that I'd try to explain about parallel lines in non-euclidean space. Firstly anything I may have said on the subject in Birmingham should be discounted. Never ask a mathematician questions outside of their field while simultaneously making them watch direct-to-video films.
So we will assume that a straight line is the shortest distance between two points. Now if you happen to be on a frill (or, you know, a globe) then you have to go over the bumpy bits to get between the two points so the shortest distance may not be as "straight" as you might think. For instance great circles are the "straight lines" of a globe and the reason why long distance flights often appear to be taking a longer route than necessary.
Euclid attempted to set out the rules for how geometry worked. His aim was to have as few assumptions as possible and derive everything else from those assumptions by reasoning. He managed to get his assumptions down to five, e.g. you can draw a straight line between any two points but he was never really happy with the fifth of these. This postulate, as Euclid stated it, says that if you draw three lines which cross in at least two places and the sum of the two angles facing each other where they cross is less than 180 degrees then in fact all three lines will cross each other (and form a triangle). That's a bit complicated (compared to, you know, you can draw a line between any two points) and you can see why he wasn't happy with it. The fifth postulate has turned out to be equivalent to all sorts of facts and I'm going to pick one which states that if you pick a straight line and a point (which isn't on the straight line), then there is only one way you can draw a straight line through this point which doesn't cross the first line. There is one and only one line parallel to another through any given point.
However, on a frilly surface all bets are off.
I've attempted to create some straight lines (defined as shortest distance between two points) on the hyperbolic plane I've crocheted up. The book insists you get a straight line by folding the crochet piece in the easiest way but, it must be said, I'm not terribly convinced by the resulting lines. However I have maths to back me up, even if those aren't straight lines on my hyperbolic plane I'm hoping you get the idea.
You can see that because of their curviness neither of the two lines that cross on the right will ever cross the third line on the left. Thus contradicting Euclid's fifth postulate, but only if you assume that to get between two points you have to go up and down all the frills.
One thing mathematicians like to do is to play about with rules and assumptions. Sometimes they do this for the love of it, and sometimes they do it to try and find out where they hit a patent absurdity so they can work back from there and find out what assumptions are wrong. So people investigated geometries where the fifth postulate didn't hold and they found that a lot of geometry still worked in these circumstances and, in fact, told us useful things about, for instance working with big distances on a globe (great circles (and thus efficient aeroplane routes) are an application of non-euclidean geometries). There is some evidence* that the effect of gravity on space and time means that the most efficient routes through the universe may not be the ones we would intuitively think of as "straight", hence the phrase "space-time curvature" and so non-euclidean geometry will have applications for long-distance space travel if we ever develop it.
*I think this may actually be established fact but I'm not a physicist. *looks hopefully at physicists on the flist*.
lukadreaming that I'd try to explain about parallel lines in non-euclidean space. Firstly anything I may have said on the subject in Birmingham should be discounted. Never ask a mathematician questions outside of their field while simultaneously making them watch direct-to-video films.So we will assume that a straight line is the shortest distance between two points. Now if you happen to be on a frill (or, you know, a globe) then you have to go over the bumpy bits to get between the two points so the shortest distance may not be as "straight" as you might think. For instance great circles are the "straight lines" of a globe and the reason why long distance flights often appear to be taking a longer route than necessary.
Euclid attempted to set out the rules for how geometry worked. His aim was to have as few assumptions as possible and derive everything else from those assumptions by reasoning. He managed to get his assumptions down to five, e.g. you can draw a straight line between any two points but he was never really happy with the fifth of these. This postulate, as Euclid stated it, says that if you draw three lines which cross in at least two places and the sum of the two angles facing each other where they cross is less than 180 degrees then in fact all three lines will cross each other (and form a triangle). That's a bit complicated (compared to, you know, you can draw a line between any two points) and you can see why he wasn't happy with it. The fifth postulate has turned out to be equivalent to all sorts of facts and I'm going to pick one which states that if you pick a straight line and a point (which isn't on the straight line), then there is only one way you can draw a straight line through this point which doesn't cross the first line. There is one and only one line parallel to another through any given point.
However, on a frilly surface all bets are off.
I've attempted to create some straight lines (defined as shortest distance between two points) on the hyperbolic plane I've crocheted up. The book insists you get a straight line by folding the crochet piece in the easiest way but, it must be said, I'm not terribly convinced by the resulting lines. However I have maths to back me up, even if those aren't straight lines on my hyperbolic plane I'm hoping you get the idea.
You can see that because of their curviness neither of the two lines that cross on the right will ever cross the third line on the left. Thus contradicting Euclid's fifth postulate, but only if you assume that to get between two points you have to go up and down all the frills.
One thing mathematicians like to do is to play about with rules and assumptions. Sometimes they do this for the love of it, and sometimes they do it to try and find out where they hit a patent absurdity so they can work back from there and find out what assumptions are wrong. So people investigated geometries where the fifth postulate didn't hold and they found that a lot of geometry still worked in these circumstances and, in fact, told us useful things about, for instance working with big distances on a globe (great circles (and thus efficient aeroplane routes) are an application of non-euclidean geometries). There is some evidence* that the effect of gravity on space and time means that the most efficient routes through the universe may not be the ones we would intuitively think of as "straight", hence the phrase "space-time curvature" and so non-euclidean geometry will have applications for long-distance space travel if we ever develop it.
*I think this may actually be established fact but I'm not a physicist. *looks hopefully at physicists on the flist*.