Re: request

Michele Cerulli <cer...@dm.unipi.it> wrote:

> In article <200...@bunrab.ronnet.moc>, 
> col...@yahoo.com says...
>>
>>
>>On Tue, 25 May 2004, Michele Cerulli wrote:
>>> It is a possible figure, it's odditiy is that it has only one
>>> surface. However, if you want an "impossible" moebius like figure,
>>> you can try with 
> the
>>> Klein Bottle.
>>
>>It's impossible in 3 dimensions, but if you will allow use of time
>>as the fourth it is very easy: Take two rings sitting on top of each
>>other. Lift the top one and turn it over and set it back on the
>>fisrt. The moving ring sweeps out a surface something like this:
> 
> In fact it is "impossible" in the sense that we cannot represent
> using the standard representations of drawing, painting, or
> sculpturing, but as soon as we allow other kind of representations,
> we can represent it. 
> 
>>Using [the edges of] two coins you can bet your mathematician
>>friends you can make a klein bottle and maybe win said two coins :-)
> 
> In fact a mathematichan would represent it simply as:
> 
> 
> .----->>-----.
>|............|
>|............|
> V............V
>|............|
>|............|
> '-----<<-----'     mc
> 
> 
> With an equivalence relationship on the sides of the rectangle that
> considers as coincident the the two sides with a "V" wich are
> "glued" together as they, while the two sides ">>" and "<<" are
> "glued" together reversing one of them, so that the two aros "<<"
> coincide exactly with the two arrows ">>"....I guess you can find a
> clearer explanation on any introductory book of topology, or 
> equivalent web sites". 
> 
> Michele Cerulli

The Klein-Bottle-Variations posted are costructs from the following:

.--------->>---------.
|....................|
|....................|
|....................|
|.....,---<<--.......|
|.....|       |......|
|.....V       ^......|
V.....V       ^......V
|.....|       |......|
|.....'--->>--'......|
|....................|
|....................|
|....................|
'---------<<---------'     

which has the same property of being non-orientable, but possible to 
build. (like you can actually build a Moebius "Strip")

-- 
Dirk-L�der Kreie                the email address is correct *as is*
http://www.nord-com.net/dkreie/            dlk...@deelkar.ath.cx

  DLK  <|\            
        | \          
~ ~~_*-_|__\__~~  ~~   @~^"=~^"=~^"=~^"=~^"=~^"=~^"=~^"=~^"=~^"=~^"@
 ~~ \________/  ~~ ~   )  ho wo kakete kokorobosoge ni yuku fune   )
  ~~   ~~`�  ~~ ~~     (  no ichiro kanashi mo uraraka nareba...   (
~~    ~~  ~~   ~~ ~~   @._,=._,=._,=._,=._,=._,=._,=._,=._,=._,=._,@