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

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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.

[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.

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?)

~looks for this thread`s exit~

This is a matter of some expertise for me.

[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.

[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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[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.

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

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.

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).

>>>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.

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.

It looks like a pooty french horn

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?

- 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.