purplecat

I'm wondering if a recap is needed here. In short, a non-euclidean surface is one in which the most efficient way to get from point a to point b is not necessarily a straight line (e.g. you have to go over some kind of bump or frill). It turns out that lots of geometry continues to work if you treat these most efficient routes like we do straight lines in normal geometry.

The bumpiness of the surface we are working with is called its curvature. If you have positive curvature you end up with a ball shape (like the Earth), if you have negative curvature you have "saddle" shapes - shapes where the surface is curving up in one direction and down in the other. These give you frilly surfaces.

On a normal flat surface, if you draw a circle, say, with a particular radius then you expect this circle to have the same area wherever you draw it. On the frilly surfaces I've been crocheting up so far this isn't the case, basically the amount of frilliness has varied as the surface got larger. Below the cut is a surface with uniform negative curvature which has the same amount of frilliness everywhere (It must be said I've not checked this, I just believed the spreadsheet I used which told me how much "bigger" to get on each crochet row).

( Crotchet Model Beneath the Cut )

This is probably my last non-euclidean crochet model. There are a few more in the book I've been using (Crocheting Adventures with Hyperbolic Planes by Daina Tamina) but they aren't nearly so pretty - though I'm a little tempted to have a go at a Klein bottle. That said, various people have requested hats and other things, so I may well post more crotchet pictures in future but probably less intellectual ones.

This entry was originally posted at http://purplecat.dreamwidth.org/35287.html.

( Pseudospheres )

Pseudospheres are basically just the hyperbolic planes I've crocheted up before only this time in a spiral. Historically they are the first shapes that were hypothesized as examples of non-euclidean surfaces with negative curvature.

Practically speaking it's difficult to know when to stop crocheting up a pseudosphere...

This entry was originally posted at http://purplecat.dreamwidth.org/29537.html.

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.

( Crochet picture under the cut )

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.

This is basically the same as my last crochet effort, except a bit frillier, but I was experimenting with the knit-purl version of crotcheting.

( Under the Cut )

This entry was originally posted at http://purplecat.dreamwidth.org/27876.html.

Mum bought me Crocheting Adventures with Hyperbolic Planes for my birthday. I have just finished chapter 1 (go me!!!)

( Photo of my very first attempt at crocheting under the cut )

The curvature of a surface is a measure of, well, how curvy it is. Most curved surfaces have positive curvature, turning them into spheres or ellipsoids and the like. If a surface has negative curvature you end up with, well, a frill, which is what I crocheted above by the cunning device of adding in an extra sixth stitch for every 5 on the row below (hence the ratio of 5:6).

This entry was originally posted at http://purplecat.dreamwidth.org/26700.html.