Klein Bottles

>>7715081
Klein bottles aren't 4D though. They're 3D.

This guy sells them.
https://www.youtube.com/watch?v=-k3mVnRlQLU

>>7715083
I think what OP means is an ideal klein bottle where the neck doesn't intersect the base in 4D.

>>7715285

this is relevant to my interests. why does in intersect in 3D but not in 4D. please explain to someone who knows nothing about topology.

>>7715306
what? you don't need topology to understand klein bottles you goon
>why does in intersect in 3D but not in 4D.
try projecting a mobius strip into 2D. you'll see that it intersects. same logic applies here.

>>7715306

A typical 3-dimensional object has all its points defined by x, y, and z. Now add a 4th dimension, w.

A typical 3 dimensional object lies "flat" in the 3 dimensional space, meaning that all values of w would be the same.

Now imagine some values of w were modified, so that the object were curved or bent into the 4th dimension.

That's how the neck could seemingly pass through the side of the bottle. It's actually passing "over" it in the 4th dimension.

A klein bottle is a closed surface, i.e. a 2d object!!

But it is non-orientable, which implies it cannot be embedded in R^3 without self-intersection. However, it can be embedded in R^4 since we can just move the intersecting areas apart in the 4th dimension. Makes sense?

>>7715382
>>7715365
>>7715285
What is the relation between a manifold, and the embedding spaces dimension that allows non intersecting embedding?

Mobius strip is 2D, must be embedded in 3D or higher
Klien bottle is also 2D, must be embedded in 4D or higher

?

File: vortex_in_vortex_out.gif (36KB, 398x376px) Image search: [Google]

36KB, 398x376px

inb4

>>7715732
a manifold is a collection of points describing a surface

the number of dimensions necessary to define the surface doesn't matter

>>7716656
What the fuck are you talking about

>>7715081
A three-dimensional volume which loops on all sides, but flips each connection through a mirror image.

Imagine a room with six doors - four on the walls, and two hatches on the floor and ceiling. If you walk through one door, you come out the door on the opposite side, but you've been flipped into your mirror image: your left hand has become your right hand, etc, and so everything appears reversed to you.

Just like a Klein bottle is a 2-D surface connected this way on all four sides, and a Mobius strip is looped twistily in just one direction.

>>7716656
No, a manifold is a topological space with the property that for each point in the space, there exists an open set containing that point which is homeomorphic to some open set of k-dimensional Euclidean space, where k is fixed.

>>7715081
What happens when you cut a klein bottle in half longitudinally?

>>7717167
https://www.youtube.com/watch?v=I3ZlhxaT_Ko

>>7715081
http://dl.acm.org/citation.cfm?id=73859

>>7717167
You get a Mobius strip. A Klein bottle is a pair of Mobius strips with the edges glued together into a tube.

If you cut a physical 3-d embedded Klein bottle in half, the edge traces out the 2-d embedding of a Mobius strip.

>>7715732
It's not at all obvious how the dimension of the manifold relates to the minimal dimension of a Euclidean space in which it can be embedded (I don't know if there is even an answer to that)
However, a theorem of Whitney states that an n-dimensional manifold can always be embedded into R^{2n} (therefore that minimal dimension is always lesser than or equal to 2n), and this is the best possible bound as there are examples of n-manifolds that cannot be embedded in R^{2n-1} for every n.

>>7717432
Sorry, not for every n*

>>7715081
Useless nonsense.

Stop thinking about this mathematical shit that has nothing to do with the real world. At least take it to another forum for bullshit like this.

>>7717432
thanks

>>7715083
what in the FUCK

>>7717432
For anyone who finds this as interesting as I first did way back when, there's an absolutely wonderful, surprisingly simple proof of a more elementary version of this theorem. Amazingly, it relies most strongly on some very unexpected mathematical objects for the topic: Graphs.

The proof involves graph switching, Seidel adjacency matrices, spectral properties of the Seidel adjacency matrix with regard to switching, and (finally) the connection (via two-graphs) between all of these rudimentary graph theory concepts and some special lie groups that are heavily related to sets of equiangular lines.

>>7717439

>/sci/ - Science and MATH

Did you not read the title

It seems you're the one who doesn't belong here.