On Tue, 13 May 2008 11:29:17 +1000, Peter Webb wrote:
>
> "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.
<sullenly>
It might be if it was both infinite and unbounded.
>> 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.
Maybe that's how the d00ds at Acme Klein Bottle are doing it.
But it still looks like they're cheatin'.
--
tinmimus99@[EMAIL PROTECTED]
11 or maybe 12
mp 10
mhm 29x13
Time was by ill luck arrested hereabouts on a Thursday evening,
and so the maid is out indefinitely.
< _Jurgen_
|
