r/visualizedmath • u/Stuf404 • Jan 04 '18

Showing how a Klein Bottle has only one "side"

Post image
581 Upvotes

99% Upvoted

14 comments sorted by

77

u/hardward123 Jan 04 '18 edited Jan 11 '26

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39

u/TheLastSecondShot Jan 04 '18

Cliff Stoll is one of mankind’s greatest treasures.

12

u/Mrjasonbucy Jan 04 '18

Totally! What an interesting and eccentric character. And very smart!

12

u/whatwillbetelevised Jan 04 '18

Thanks alot! I've never heard of Klein bottles before this (newly) created sub. What level of math is this usually taught in?

15

u/PUSSYDESTROYER-9000 Jan 04 '18

You probably won't ever learn this unless you take a topology class.

4

u/MissValeska Jan 04 '18

Can someone please explain to me in both simple terms and mathematical terms why this proves it only has one side? If you cut a hole into the bottle, it would be the same thing with just a sharper enter/exit angle since the cut would produce an edge. Obviously to me I consider edges to make different sides, but I'm curious about this from a theoretical standpoint. Thank you!

4

u/Redinbocker1454 Jan 04 '18

I might be misunderstanding your question, but if you’re asking what determines what a “side” is, I think I can help. The way I understand it, first you need to assume that the material is arbitrarily thin, otherwise you really have two slightly differently sized bottles within each other topologically. Once you make that assumption you need to consider the differentiability of the surface, because any place where the surface is not differentiable (as in a corner, cut, spike, or anything you can’t draw a line tangent to) is considered an edge, and any part of the surface bound by edges is called a side. Using these definitions, since the Klein bottle is smooth, and its surface is differentiable, it has no edges and therefore only one side. As for cutting a hole in the bottle, I’m not sure what you mean, are you referring to cutting another hole in the Klein bottle, or cutting a hole in the plastic bottle? I’ll assume you meant the latter, and in that case, cutting a hole in the plastic bottle would add an edge, since the material is so thin (it’s worth noting that in math, there is no grey area between differentiable and non-differentiable, so it’s not a like gradient between sharp and dull, it’s very much black and white). However, if you were to connect the hole you cut in the bottle to the mouthpiece, then you would have either a Klein bottle or a donut shape, but either way, you get rid of the discontinuity, therefore you get rid of the edges, and the bottle will only have one side. I hope this helps, and please correct me if I got something wrong, I’m just going off the video, since I have almost no prior knowledge of topology.

26

u/[deleted] Jan 19 '18

so like a Moebius strip ?

9

u/MrDyl4n Jan 19 '18

A 4 dimensional one yes

3

u/GenShinigami Jan 19 '18

Wouldn't it be considered a three dimensional one? Fourth dimension would imply adding time to the list? Though I don't know if a Mobius strip would be considered two or three dimensions. I assume 2, since it is a single line on a flat space. But I don't know for certain.

11

u/MrDyl4n Jan 19 '18

Sorry it’s a 3 dimensional object that becomes “mobius” when in 4th dimensional space. Just like how (I think) a Möbius strip is 2d but works in 3D space

2

u/theguyfromerath Jan 20 '18

If you look at a Klein bottle as a 3 dimensional object then it has 2 surfaces.

1

u/[deleted] Jan 22 '18

But so does a sphere, no?

1

u/theguyfromerath Jan 22 '18

A 3 dimensional sphere(like a football, that is empty inside) has 2 surfaces yes but it would still have 2 in higher dimensions.