Conference rusure::math

    I won't attempt to draw one but it's easy enough to make one.
    
    Take an A4 sheet of paper. Guillotine off a strip from the long side of
    the sheet. A 2cm width strip should be OK. Now if you bend the strip
    upwards at both ends so that the two short sides meet you don't get
    anything unusual (a very squat tin can with the top and bottom cut out)
    BUT if you give one end a half twist before joining the short sides you
    have a Mobius strip.
    
    (Now onto Klein bottles...I've always wondered whether milk kept in a
     Klein bottle would go off. Can the bacteria get in? :-))

    You can draw a Klein bottle fairly easily... sweep a figure 8
    around a circular contour while rotating it so it makes 1/2
    a turn in all.

    This is not the usual picture that's shown, but is actually more
    elegant and symmetrical.

    - Jim

I tried to draw a Moebius strip using typed characters but it's very hard.
Later on I will try doing it in PostScript, which I think will be clearer. 
Once you try the twisted strip all will become clear. Make several and try 
the effect of cutting them across and along the length of the strip. Then 
experiment with a full twist, then 1.5 twists instead of 1/2 - that version
has even more intriguing properties. 

I used to own a model of a Klein's bottle, made for me by a Chem major who 
had gotten pretty good at blowing glass. He started with an ordinary test
tube and some 1/8" tubing. It was quite neat. Yes, it held fluids, and you
could put a cork in it. Sadly, I can't recall what happened to it. I think
I have seen a picture of one very like it made by someone's ray-tracing
program. 

If you split a Klein bottle along its plane of symettry, you get one left-
and one right-handed 1/2-twist Moebius strip. Conversely, if you have two
opposed strips you can *in principle* join them to make a bottle. It takes 
quite a bit of planning to get the "strips" in the right shape to join
easily. 

    re .1 : ... or use the border you tear off your paycheck.

There is another way of making one that is quite neat: make a figure - 8
loop of a piece of stiff wire and double it back on itself so that it
doesn't touch itself. Now you have a smaller, round, double loop which is
to be the edge of the strip. Then dip the whole thing in a container of
soap-bubble mix (you can buy it in any toy store). Voila! A Moebius strip
bubble is formed. It may be necessary to leave a little handle at the place
where the wire loop is closed, so your fingers won't break the bubble. 

re .3
    
>Yes, it held fluids, and you could put a cork in it.
    
    I'm confused (which is my own fault since I was the one who mentioned
    Klein bottles) . . . where does the cork get put? Assuming I'm visualising
    the same thing, I would guess the cork goes in the bottom.
    
    I can't see that the bottle in .2, which is presumably topologically
    equivalent to mine, is very useful for holding fluids and how do you
    cork it? Granted .2's bottle has elegant symmetry and shows the
    connection with the Mobius strip more easily.
    
    One final thing...I thought the Klein bottle needed to be embedded in
    4-space which is why the bacteria have a hell of a time getting in???

What my chemist friend did was take a test tube and melt holes in the
bottom and side. Then he ran an S-shaped tube through the side hole and
sealed first the bottom, then (with some real glassblowing artistry) merged
the two circular ends at the top, and sealed the side hole. The
cross-section looked like: 
	___	___
	| |     | |
	| |     \_|___
	| \_________  \
	|	  | \ |
	|	  | | |
	|	  | | |
	|	  | | |
	|	  | | |
	\	  / | |
	 \__   __/  | |
	    | |____/  |
	    \________/

The cork goes in the top. Unless you turn it upside down.

    re .7
    
    Nice picture. Thanks.
    
    Yes, this is what I was visualising except that mine was upside down
    i.e. imagine taking a wine bottle and softening the glass in the neck
    so that it can be drawn out, bent around and passed through a hole cut
    in the side of the bottle and then joined to another hole cut in the
    bottom of the bottle.
    
    I am still wondering how this is equivalent to the bottle in .2 but I
    will try first to run the program in 1468 to display it.
    
    (re 1468.2: The C compiler seems to do some magic and accept U*IX
     style path names but, yes, you do need to define the logical name
     X11.)

>    This is not the usual picture that's shown, but is actually more
>    elegant and symmetrical.

	More elegant, yes;  more symmetrical, no.

	The "bottle" has an obvious plane of symmetry.

	The "dancing figure of eight" admits one non-trivial symmetry.   Look
at the locus of the centrepoint of the figure of eight.   This is a circle.
There is a unique axis passing through the centre of this circle IN THE PLANE
OF THE CIRCLE which permits a rotation of pi which preserves the shape.

Andrew.

PS: Very pretty picture.

I have made a PostScript file, using MAPLE, of a Moebius strip that is 
kinda pretty. You can get it and the MAPLE commands that generate it from 

civage::disk$user6:[lynn.maple]moebius.*

for a while, at least.

Here is a cute paradox concerning Moebius strips. 

Pick a point on the edge and cut with scissors directly across the strip.
You will then have a single piece with two sides and one edge. If instead
you had from the same point cut a semicircle to a point nearby on the same
(only one!) edge, you would produce two pieces: the original strip with a
semicircular chunk cut out, and the chunk. In each case you have cut from
one point on the one edge to another point on the same edge, but the
results are quite different. 

At what point along the edge does the behavior change? 

    re .11            
    
    As far as I can see the resolution of the 'paradox' is that the
    result of cutting from a point A on the edge to another point B on
    the edge does not depend on A and/or B but instead on the path chosen
    between them.
    
    For all distinct A and B, it is possible to cut from A to B so that the
    Mobius strip reverts to being a 'sheet' and it is possible to cut from
    A to B so that the Mobius strip remains, with a sheet having been cut
    from it. The difference between the two cases presented in .11 is that we
    instinctively choose a certain path depending on the relative position
    of A and B. Thus the difference in behaviour is that of the human and
    not of the Mobius strip.
    
    More detail (hopefully accurate - if in doubt get out your paper and
    scissors)...
    
    Cut from A to B by cutting across the strip 2/3 of its width, then
    while you are not within a distance of 1/3 of the width of the strip
    from B, cut parallel to the edge (which direction?), then cut directly
    to B (the remaining 1/3 of the width). Call this an S-cut. This cut
    produces just the one piece (destroying the Mobius strip) even when A
    and B are 'close'.
    
    Cut from A to B by cutting across the strip 1/3 of its width, then
    while you are not within a distance of 1/3 of the width of the strip
    from B, cut parallel to the edge, then cut directly to B. Call this a
    U-cut. This cut merely shaves the Mobius strip - producing two pieces -
    even when A and B are 'opposite'.

Another way of dealing with the 'paradox' is to think of the edge of the 
strip as orientable, i.e. assign a direction to a point on the edge and see 
what happens to it. 

Suppose we assign 'left' to the nearest edge as we start cutting. If we
follow the arrow along the edge to the 'opposite' side of the strip we
find it also points to the left. So if we cut straight across (and keep
the scissors upright!) we will have cut from one left-pointing edge to
another, but if we cut to the nearby edge, we will have to turn the paper
(or scissors) so we end the cut at a *right* edge. Rule: if the scissors
cuts to an opposite- oriented edge, we get two pieces; if both edges are
oriented the same, we get one piece. 

There are several ways of getting confused about the orientation: by 
turning the paper over, or turning the scissors over, or turning the paper 
around ... but if we mark the paper carefully, and are careful not to 
change orientation in mid-cut, we will find that the rule holds true and
that the half-twist in the strip can be used to produce either one or two 
pieces, and with as many loops as we wish.

!p          
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I tried a number of combinations.  First, I typed it on my greyscale screen.
The picture hardly shows up.

Next, I printed it with:

	$ print foo.bar/param=data=regis

This produced two neat figures, except the triangular one overlapped the
lower left portion of the rectangular one.

I got different results with

	$ print foo.bar/param=(page_orient=landscape,data=regis)

but still overlap (but different amount).

Did you intend an overlap ?

I tried a number of combinations.  First, I typed it on my greyscale screen.
The picture hardly shows up.

Next, I printed it with:

	$ print foo.bar/param=data=regis

This produced two neat figures, except the triangular one overlapped the
lower left portion of the rectangular one.

I got different results with

	$ print foo.bar/param=(page_orient=landscape,data=regis)

but still overlap (but different amount).

Did you intend an overlap ?

No Eric, it is not intentionnal and in fact I had never printed it.
it is OK when I type it on my screen, so I don't know why it is not
when printed!
the two following notes are the file separated.
JcD_______________________mm_/��\_mm______________________________________

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