>Tim Martin wrote:
>>
>> AIUI, a Klein bottle is a 4 dimensional object. What they have on that
site
>> is a 3-dimesional projection of a Klein bottle (Think of trying to draw a
>> mobius strip on 2D paper). It's as close as you can get in a 3D world,
and
>> it is still a damn fine mug.
>>
>> Tim
> Not at all, a Klein bottle can exist in 3D space (as the mig
>demonstrates). What is impossible is to have a Klein bottle that doesn't
>cross itself in 3D space, you need at least 4D for that. As for drawing
>a moebius strip on 2D paper, you can't because a moebius strip is a
>surface (eg it has 2 dimensions) but the klein bottle is also a surface
>and not a volume, that's where your analogy fails...


I quote from http://mathworld.wolfram.com/kleinbottle.html :
"A closed nonorientable surface of Euler characteristic 0 (Dodson and Parker
1997, p. 125) that has no inside or outside. It can be constructed by gluing
both pairs of opposite edges of a rectangle together giving one pair a
half-twist, but can be physically realized only in 4-D, since it must pass
            ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
through itself without the presence of a hole."

The model of the Klein bottle is an immersion of the 4D object in 3D space,
and has to intersect itself just as a 2D projection of a mobius strip has to
intersect itself. As such the 3D Klein bottle isn't a real Klein bottle any
more than a drawing of a mobius strip is a mobius strip.

The issue of volume is a somewhat misleading one, since AIUI topologists
usually deal with infinitely thin sheets and the volumes they enclose,
rather than with solid objects. A surface can be 2D but still require 3
dimensions to exist, as the mobius strip demonstrates, and similarly it can
require 4 dimensions to exist, as in the case of the Klein bottle. I could
be wrong though, I have no formal topological training.

Tim
--
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Website: http://www.alphafish.f9.co.uk/

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