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14 December 2010 @ 07:58 pm
Parallel Lines  
I promised 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*.

This entry was originally posted at http://purplecat.dreamwidth.org/28732.html.
 
 
 
philmophlegm: Sumatraphilmophlegm on December 14th, 2010 08:43 pm (UTC)
I think the phrase "on a frilly surface, all bets are off" deserves wider use. I plan on using it in conversation myself at some point in the future.
louisedennislouisedennis on December 14th, 2010 09:18 pm (UTC)
I await a report of the result with much anticipation!
rodlox: going to hugrodlox on December 15th, 2010 09:29 am (UTC)
at the very least, we need it on more icons.

i volunteer to help.
lukadreaminglukadreaming on December 15th, 2010 09:00 am (UTC)
Thank you! I understood that. I think *g*.

I don't remember frilly surfaces from maths O Level (actually, I remember nothing from maths O Level, in which I scraped a grade C *g*).

And I second philmophlegm's call for a wider use of "on a frilly surface, all bets are off." We're going for our work Christmas lunch today. I fully intend to wow people with it (and I did try to explain your crochet book to a colleague the other day!)
louisedennislouisedennis on December 15th, 2010 10:54 am (UTC)
Well I do not believe that "frilly surface" is the technical term for them, but I thought it more self-explanatory than hyperbolic plane *g*

Have fun at your Christmas lunch!!
rodloxrodlox on December 15th, 2010 09:29 am (UTC)
if nothing else, you are working hard to keep us sane on the day when the stars are right...

seriously, I enjoyed your post. thank you for explaining it to us.
(and the picture helped)
louisedennislouisedennis on December 15th, 2010 10:55 am (UTC)
Thanks! I'm having fun with the crotcheting, although I lot of counting is involved which makes it harder than anticpated to crotchet and do other things!