"mimus" <tinmimus99@[EMAIL PROTECTED]
> wrote in message
news:uYCdncq7zKAt1bXVnZ2dnUVZ_rHinZ2d@[EMAIL PROTECTED]
> On Mon, 12 May 2008 23:25:47 +1000, Peter Webb wrote:
>
>> "mimus" <tinmimus99@[EMAIL PROTECTED]
> wrote in message
>> news:WZGdnYk6ysIsorXVnZ2dnUVZ_qHinZ2d@[EMAIL PROTECTED]
>>
>>> On Mon, 12 May 2008 18:36:20 +1000, Peter Webb wrote:
>>>
>>>> "mimus" <tinmimus99@[EMAIL PROTECTED]
> wrote in message
>>>> news:8uCdnZuH0sBCzIDVnZ2dnUVZ_hKdnZ2d@[EMAIL PROTECTED]
>>>>
>>>>> On Sat, 03 May 2008 21:33:59 -0500, Tim Weaver wrote:
>>>>>
>>>>>> mimus wrote:
>>>>>>
>>>>>>> At least, I swept up at least twice as much glass as could
possibly
>>>>>>> have
>>>>>>> been in the original.
>>>>>>>
>>>>>>> And that's only possible if you do an infinite decomposition of
the
>>>>>>> object, as exemplified by the Tarski-Banach ball.
>>>>>>>
>>>>>>> Maybe this is a sign I should mop my kitchen-floor.
>>>>>>>
>>>>>>> Will I need an infinite mop?
>>>>>>
>>>>>> Yes, if you have a mobius shaped floor.
>>>>>
>>>>> Mobius strips are usually finite.
>>>>>
>>>>> Just unbounded.
>>>>
>>>> Technical note: Mobius strips have a single boundary.
>>>
>>> ok fine.
>>>
>>> (As soon as I read that, my head tried to encompass an unbounded
Moebius
>>> strip and couldn't do it.)
>>
>> Well ...
>>
>> Imagine the width of the Mobius strip was infinite. You would then have
a
>> surface that was unbounded in both directions, finite in one direction
>> and
>> infinite in the other - sort of like a cylinder, but its not. Nor is it
a
>> Klein bottle or cross-cap. It can't be embedded in R^3 as it self
>> intersects. But it is a reasonable interpretation of an "unbounded
Mobius
>> strip", whatever its real name is (if it has one).
>
> <squint>
>
> A single-sided infinite plane or saddle or a single-sided sphere, is
> what it looks to me like what we're lookin' at, yes it does. Yes.
>
No. A single sided infinite plane is infinite in both directions, a single
sided sphere is finite in both directions. So my "unbounded Mobius strip"
is
not isomarphic to either of these.
> I think Klein bottles cheat with that penetration business-- tearing is
a
> no-no in algebraic topology,
No tearing is neccessary. You simply lift the "neck" of the bottle into
the
fourth dimension, move it "over" the outside surface, then reconnect it.
The problem you are having is that a Klein bottle, despite being a
perfectly
reasonable 2D surface, can't be embedded into R^3. (And nor can my
infinite
Mobius strip). However, it is only your parochial desire that 2D surfaces
should be embeddable in R^3, there is no reason that they should be - they
are both embeddable in R^4 without any problem, and there is no reason to
limit the consideration of 2D surfaces to those which can be constructed
in
3D, even if you think you live in 3D world.
> even though that's how you make a Moebius
> strip, and also in a sense how they work up the matricial representation
> of one, swapping connection-points or vertices in the matrix
representing
> an ordinary strip or tube.
>
> http://www.kleinbottle.com
>
> --
> tinmimus99@[EMAIL PROTECTED]
>
> smeeter 11 or maybe 12
>
> mp 10
>
> mhm 29x13
>
> Given a manifold M with a submanifold N, N can be knotted in M
> if there exists an embedding of N in M which is not isotopic to N.
> Traditional knots form the case where N = S1 and M = S3.
>
> < Deep wisdom from on high (Wikipedia)
>
|
