More 4-Dimension Objects: The Klein Bottle

Submitted by: MacGuffin 4 years ago Science

This is a Klein Bottle: a 4-D object that is "non-orientable." It is to 4-D what a Mobius Strip is to 3-D.
There are 32 comments:
Male 347
@MacGuffin: I have written software that allows one to visualise true 4-d objects in 3-d using 3-d glasses. I have spent a good amount of time over the last 30 years working on it, and I have gained a solid understanding of 4-d geometry in the process.
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Female 2,602

>>>@MacGuffin: I don`t know why you keep insisting that Klein bottles are 4D objects when they are clearly NOT.

...

Hell, you can *buy* Klein bottles!
http://www.kleinbottle.com/ <<<


Well, if you can *buy* 3D projections of the real thing on the internet and everything, that *must* prove that the 4D object that said 3D projection is a projection of can`t exist.

Read this, then realise how mistaken you are. To quote:

"While the Möbius strip can be embedded in three-dimensional Euclidean space R3, the Klein bottle cannot. It can be embedded in R4, however."

As I said, this *is* a 3D projection of such an embedding in R4 - that`s how the object is shown to pass through itself. That only happens in 4D.
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Female 2,602

[quote]@MacGuffin--a 4-d Klein Surface would not show an intersection and would rotate without any flattening.

This is a matter of some expertise for me. [/quote]
Again, this is a *3D projection* of a 4D Klein Bottle. Nobody is saying that it is the 4D object itself. If it was, we`d be unable to perceive it, since we live in 3D.

I guess it`s not that much of an area of expertise for you, or you`d be able to understand that simple concept.
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Male 621
@MacGuffin: I don`t know why you keep insisting that Klein bottles are 4D objects when they are clearly NOT.

Seriously, no part of a Klein bottle needs to project into the 4th dimension for it to be a Klein bottle, therefore it is, by definition, *not* a 4D object.

Hell, you can *buy* Klein bottles!
http://www.kleinbottle.com/

Since we cannot build 4D objects, doesn`t that prove to you that they`re *not* 4D?!?

(Or do you have some strange definition of 4D that most of us don`t use?)
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Male 6,227
Yeah, I uhhh... have *no* idea what`s going on here.

~looks for this thread`s exit~
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Male 347
@MacGuffin--a 4-d Klein Surface would not show an intersection and would rotate without any flattening.

This is a matter of some expertise for me.
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Male 2,592
Up, down, left, right, forward, backwards... in, out.
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Male 2,592
When I imagine a 4th dimension to space, I`ve imagined myself sprinting in one direction, looking as if I`ve stayed in the same position in 3d space but becoming ever smaller as if I`ve ran a 1/4 mile away.
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Female 2,602

[quote]What you are seeing is a 3-d Klein Bottle being rotated in 4-space and being projected into 2-space. The tell-tale is when the object appears to flatten itself during rotation.[/quote]
That`s incorrect. It`s a 3D projection of a 4D Klein Bottle. If you look at the video carefully you`ll see that the bottle never completely "flattens". Only parts of it appear to as those parts briefly sit at right angles to two or more of the three dimensions that make up 3D space. Those parts which appear flat at any one time, are actually sitting at 90 degrees perpendicular to at least two of the axes that make up 3D space at those points. We get the same view at those moments as a 2D being would get of a flat square seen edge-on.
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Female 2,602

[quote]This is not a 4-d representation. [/quote]
Nobody said that it was. It`s a 3D projection of a 4D object.

[quote]The Klein Bottle is a surface that only appears to not intersect itself in 4-space. [/quote]
It doesn`t just appear not to self-intersect. It really doesn`t intersect itself in 4D. Just as the squares that comprise the 2D projection of a 3D cube I showed below really are clearly separated once you can see the object they represent in full 3D.

...
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Female 2,602

[quote]MacGuffin: I disagree. That only works, again, because our brains work in a `three dimensional world`, but the cube representing a 3D object on a 2D plane isn`t `3D` either, and so if you have someone who doesn`t understand the third dimension, that is not going to be sufficient to teach it.[/quote]
Whether you "disagree" or not isn`t relevant. I wasn`t trying to convince you of anything, I was merely stating a mathematical fact that exists whether you`re aware of it and recognise it or not. It`s not a matter of opinion.

You can project a 3D object onto a 2D surface, and you can project a 4D object into 3D space. Whether you understand/agree with these undisputed concepts or not is immaterial. You might as well disagree with gravity.
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Male 228
BZZZT! but thanks for playing. Now go away.
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Male 347
This is not a 4-d representation. The Klein Bottle is a surface that only appears to not intersect itself in 4-space. What you are seeing is a 3-d Klein Bottle being rotated in 4-space and being projected into 2-space. The tell-tale is when the object appears to flatten itself during rotation.
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Male 338
So many people confusing a concept with a nice glass trinket for keeping wine badly
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Male 2,688
Someone has a failed interpretation of what 4D means... This is a 3D object that doubles back the envelopes itself on one end... As previously stated, it`s nothing more than a 3D version of a Möbius Strip.
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Male 2,737
Looks like a failed attempt at a bong.
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Male 15,832
I still count the hole in the side as an edge.
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Male 1,647
MacGuffin: I disagree. That only works, again, because our brains work in a `three dimensional world`, but the cube representing a 3D object on a 2D plane isn`t `3D` either, and so if you have someone who doesn`t understand the third dimension, that is not going to be sufficient to teach it.
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Male 621
Ugh. No. Just... no.

A Klein bottle is *not* 4D, it`s a 3D object with a single contiguous surface and no edges. Basically, it`s the 3D equivalent of a Möbius strip.

They`re quite interesting in and of themselves, you don`t need to make up stuff about them being 4D somehow.

Wikipedia: Klein bottle
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Male 1,243
where can we get the 4d glasses?
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Male 628
Most certainly a very awesome shape, although not very interesting if you don`t know much about maths.
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Male 14,775
They`re a biatch to use. You get an air-lock that prevents the wine from pouring.
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Male 1,497
You can accually buy one. It`s glass and 3d. Looks good on my mantle. Here. Acme Klein
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Female 2,602

It gets interesting when you consider that the Universe itself is also an object that (from our 3D perspective at least) has only one surface and no known boundary.
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Female 7,838
FFS - I am honestly not an idiot- but I think visually and this gives me a headache!!
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Female 2,602

A Klein Bottle has some remarkable properties. Like a Mobius strip, it only has one edge. But unlike a Mobius strip it has no boundary, and doesn`t intersect itself (although 3D projections of it appear to, just as the squares that make up a cube appear to overlap when projected onto 2D).
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Female 2,602

>>>Forgive my physics stupidity, but how can they represent a 4D model in a 3D engine? How can they represent a 4D object in our 3D perception?<<<


In the same way that you can represent a 3-dimensional object like a cube in 2D by projecting it onto a 2D surface:



You can also project a 4D object like a Klein bottle or the tesseract that I posted about the other week onto three dimensional space. When you do, you see the 3D `shadow` of that 4D object. That`s what you`re seeing in this video: the shadow of a Klein bottle in 3D as it rotates in a fourth spatial dimension.
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Male 1,647

Moments later, after doing some googling, I found some.. stuff. http://eusebeia.dyndns.org/4d/cubinder

However, while I get the `idea`, I still think it`s ridiculous to try and present a 4D object in only 3 dimensions... I`m all for breaching new territory in science, but seriously, it`s kind of like trying to show color to a blind person. The blind person is never going to see what you want them to see, just as we are not going to be able to see 4D in our 3D world, currently.
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Male 15,510
The 3D shadow of a 4D object
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Male 646
BORING

It looks like a pooty french horn
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Male 1,647

Forgive my physics stupidity, but how can they represent a 4D model in a 3D engine? How can they represent a 4D object in our 3D perception?
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Female 2,602
Link: More 4-Dimension Objects: The Klein Bottle [Rate Link] - This is a Klein Bottle: a 4-D object that is `non-orientable.` It is to 4-D what a Mobius Strip is to 3-D.
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