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From: Michele Cerulli <cerulli@dm.unipi.it>
Newsgroups: alt.ascii-art
Subject: Re: request
Date: Wed, 26 May 2004 11:41:26 +0200
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In article <20040525210714.G2072-100000@bunrab.ronnet.moc>, 
colonel_hack@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