8 ms·
Can planes be tied in knots in higher dimensions the way lines can in 3D?
- OJFord 11y agoSurely a plane can be tied in a knot in R^3 (like a sheet of paper, a blanket) - and therefore trivially so in higher dimensions?
- AgentME 11y agoA sheet of paper or a blanket would only be a plane segment.
- grandalf 11y agoA string is just a line segment
- Taniwha 11y agoI can imagine an infinitely long line with a know in the middle in 3d ... I guess the interesting question is "can you have an infinite plane with a knot in the middle in 5 dimensions?6 dimensions?"
- ustolemyname 11y agoYes, but the ends could be extended to infinity in arbitrary directions, so we can tie a knot in a line, not just a segment. You can't do that to the edges of a blanket knot (two sides run into the knot).
- baddox 11y agoA mathematical knot is like a loop of string. There are no ends. Of course, you could cut any mathematical knot wherever you like and make the two ends go to infinity if you'd like.
- ustolemyname 11y agoThank you, I appreciate the correct definition.
- OJFord 11y agoI understand a bit better after that, thanks. But a knot in a plane segment doesn't need to fold the ends in - I can grab a blanket in the middle, and tie a knot there, then extend edges infinitely still?
- GFK_of_xmaspast 11y agoA knot is an embedding of S1 in R3, i.e. a closed loop and not a line segment.
- cynicalkane 11y agoAny such knot could be topologically deformed into a knot made of arbitrarily thin plane segment, which would end up being the same as a line knot. So knots made out of finite plane segments are uninteresting.
- OJFord 11y agoI should have been more clear though, I was imagining an infinite 'blanket', which I grab in the 'middle' (not an end) and tie a knot there. Or does that not count as (mathematical) knotting?
- GFK_of_xmaspast 11y agoA sheet of paper or a blanket is equivalent to D2, the disc, which is not a plane.
- gohrt 11y agoFor it to be a mathematical "knot", the edges have to be sewn together into form a knotted sphere. The trivial way to knot a blanket doesn't yield a trivial way to connect all the edges without untying the knot.
- grandalf 11y agothis seems to hinge on a peculiar definition of a knot.
- throwaway2048 11y agoits a mathematical knot, which is essentially definable as a loop of "string" that has crossings that cannot be removed (ie the loop returned to a normal doughnut shape/torus) via any sort of transformation that doesn't involve cutting it. the trefoil in the article is an example of such a thing. Most knots we encounter in every day life are not a knot in a mathematical sense, because they can be untied, or don't form a loop.
- anyfoo 11y agoI'm not familiar with the subject, so I don't quite get that last sentence. It seems to me like most knots I encounter in everyday life is similar to this one: http://www.nationbydesign.com/simple-knot.jpg http://www.nationbydesign.com/simple-knot.jpg This, to me, seems like a knot that cannot be removed via any sort of transformation that doesn't involve cutting it, assuming that the piece of string has no ends and continues forever. Is my understanding of a knot in the mathematical sense wrong? Or of the involved transformations? If so, can you elaborate?
- jakobegger 11y agoI guess most strings in the real world don't go on forever... And if you have two lose ends, you can untie the knot trivially. I think that is what your parent comment was getting at.
- Someone 11y agoYour understanding is fairly correct, but it is mathematically easier/nicer to tie the ends together rather than let them run to infinity. That guarantees that you describe exactly how the ends would 'continue forever', and fits better with topology. See https://en.wikipedia.org/wiki/Knot_theory https://en.wikipedia.org/wiki/Knot_theory
- glitcher 11y agoYes https://en.wikipedia.org/wiki/File:Airplane_art_1.jpg https://en.wikipedia.org/wiki/File:Airplane_art_1.jpg
- cbd1984 11y agoBut that's not important right now.
- johnhattan 11y agoSurely you can't be serious.
- livatlantis 11y agoI am, and don't call me Shirley.
- jdblair 11y agoI'm so happy someone beat me to this joke.
- huuu 11y agoWell I was expecting an article about air traffic control :o
- zodiakzz 11y agoNot sure why this is the top comment since this isn't Reddit bla bla. I've had my previous account hellbanned because of upvoting a comment like this. (I can only assume, since I hadn't posted anything in a long time).
- pc86 11y agoI don't think accounts get hellbanned based on voting behavior unless it suggests they are part of a voting ring.
- cbd1984 11y agoBut that's not important right now.
- a_c 11y agoNeed someone's insight: Isn't klein bottle a 3D knot?
- jordigh 11y agoKnots are typically seen as embeddings (non self-intersecting image) of S^n, a closed orientable surface. For example, S^1 (the circle) in R^3 (3-dimensional space) is what we usually think of as a knot. The simplest knot, trefoil knot, is just a circle embedded into the ambient space in a way that cannot but "untangled" without self-intersection. The Klein bottle, being non-orientable, is not usually considered a knot. But the important thing is that a knot is all about the ambient space. If you are allowed 4-dimensional motions, then you can always untangle any embedding of S^1.
- Pxtl 11y agoNo, a Klein bottle is a 4d moebius strip.
- catnaroek 11y agoA Möbius strip has a boundary. What is the boundary of a Klein bottle?
- startling 11y agoThe surface.
- catnaroek 11y agoThe surface itself is the Klein bottle, so I would expect its boundary to be some 1-dimensional object, in the same way the boundary of a disk is the circle surrounding it.
- Manishearth 11y agoA Mobius strip is a twisted non-orientable 2D surface that exists in 3 dimensions, with a 1D edge. A Klein bottle is a twisted non-orientable 3D volume that exists in 4 dimensions, with a 2D "edge" surface.
- ingenter 11y agoAs a matter of fact, yes, planes can be tied in knots. Here's a page from "Knotted Surfaces and Their Diagrams" by J. Scott Carter: http://i.imgur.com/2rIKJ8E.png http://i.imgur.com/2rIKJ8E.png
- amelius 11y ago> Ordinary knots (that you can tie with string) can only exist in exactly 3 dimensions. Or a higher dimensional space.
- ingenter 11y agoIn a 4-dimensional space, 1-dimensional knots are trivially untied, because you can pass strings through strings by "lifting" the string into fourth dimension (strings can "pass through" each other).
- amelius 11y agoAre there any (practical or theoretical) applications for theories of knots in higher dimensional spaces?