Science a GoGo's Home Page Forums General Discussion General Science Discussion Forum 11th dimension?

Kline bottles are physically and logically impossible in three spatial dimensions: Try again.

dvk wrote:
"I am not talkin English"

Why would anyone expect anything else from you?

Weren't you going to go to some other URL and share your profound gift for inarticulate sentences?

DA Morgan: wrong again... a Klein bottle can locally be described using three coordinates, hence as a manifold it is three-dimensional.

This answers the question about the dimension of a Klein bottle.

But it is non-orientable and cannot be covered by a single three-dimensional map. And it cannot be embedded in three spatial dimensions.

You should try to read about the Klein botttle in the wikipedia or some other place.

To make a Moebius strip you start with a two-dimensional sheet (say, a square piece of paper). You then glue two ends together with a 180 degree twist - the Moebius strip is still locally described using two numbers.

To make a Klein bottle, you start with a Moebius strip (or a sheet of paper again) and glue the remaining sides - now you need three local coordinates.

Here is an analogy: a string is an extended object; you need only one coordinate to specify a point on a string. But a string does not really move in one dimension, but in two or more....

A mathematician named Klein
Thought the M?bius band was divine.
Said he: "If you glue
The edges of two,
You'll get a weird bottle like mine.

I wonder? Is there a real world object that has the shape of the Klein bottle?

Oh, by the way have a look at this:

http://www.kleinbottle.com/index.htm

Neat, het!

Kasper Olsen, Ph.D.'s inability to quote correctly notwithstanding I will repeat my original statement ... Klein bottles can not be constructed in anything less than four physical dimensions.

1 dimentional objects cannot exist
2 dimentional objects cannot exist
Do you agree with the above statements?
I will continue if the answer is yes

Based on what you wrote I would say, without hesitation, you are incorrect.

You see the problem is that strings and branes, thought by many highly qualified mathematicians and physicists to exist or at least to be possible solutions ... are precisely what you are asking me to say do not exist.

That I can not do.

Rob said: "Please name one shape that cannot be described using 3 dimentions and 3 dimentions only." And you, DA Morgan, said: "A Kline bottle.

Want to try again? ;-)"

and later: "Klein bottles can not be constructed in anything less than four physical dimensions."

Your last statement is correct - at least if you make precise what you mean by "constructed".

But your former statement is wrong. The Klein manifold is three-dimensional in a precise mathematical sense. Roughly it can locally be covered by the real space R^3 - in this sense it is three-dimensional.

The question about embedding the Klein bottle in R^n is another thing. Let's talk about embedding a manifold M, of dimension m, in the space R^n.
How large must n be, in terms of the dimension m of M? The so-called Whitney embedding theorem states that n = 2m is enough. For example the real projective plane of dimension 2 requires n = 4 for an embedding and the Klein bottle of dimension 3 requires n = 6, but can actually be embedded with n = 4.

You are confusing m with n above.