Uniform Negative Curvature
Feb. 17th, 2011 12:22 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
purplecatI'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).
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.
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).
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.
(no subject)
Date: 2011-02-17 04:44 pm (UTC)BTW, are you taking requests for hats? *g*
(no subject)
Date: 2011-02-17 05:55 pm (UTC)I was also thinking I might make myself a non-euclidean poncho....
(no subject)
Date: 2011-02-17 05:58 pm (UTC)Could I possibly have a black one, like Luka's, but with a blue or a green (or both) thingy on the side?
Am happy to repay with fic or a donation to charity. *g*
A non-euclidean poncho would be frilly beyond belief!
(no subject)
Date: 2011-02-18 11:04 am (UTC)I will ponder upon suitable payment...
(no subject)
Date: 2011-02-18 11:27 am (UTC)Black acrylic is good, as wool mix can be a bit scratch for hats.
(no subject)
Date: 2011-02-18 12:52 pm (UTC)100% acrylic *phew!*
(no subject)
Date: 2011-02-18 01:36 pm (UTC)(no subject)
Date: 2011-02-18 11:13 am (UTC)this or this
(no subject)
Date: 2011-02-18 11:29 am (UTC)Either of those look good.
(no subject)
Date: 2011-02-18 01:24 pm (UTC)(no subject)
Date: 2011-02-18 01:37 pm (UTC)