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This week I'm going to give a talk on fiber bundles, and I found myself with an unexpected problem. Since I'm not using slides, I'll need to draw a Möbius band on the blackboard. Usually what I do is simply draw a rectangle with some arrows to indicate the identification, but I think I could do better.

Is there a method that can help me to draw a good Möbius band on the blackboard? I'm looking for something that looks good and is easy to draw.

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    $\begingroup$ I would show one made of paper, and I would practice drawing from good images online. $\endgroup$ – Sue VanHattum Nov 9 '15 at 19:34
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    $\begingroup$ If fact, it is so easy to make paper ones everyone in the class should be encouraged to make a copy. $\endgroup$ – Joseph Malkevitch Nov 9 '15 at 20:22
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    $\begingroup$ By talk do you mean a class you are teaching, or a research seminar, or something else? Paper ones will not be entirely appropriate in some contexts. $\endgroup$ – Jessica B Nov 10 '15 at 7:04
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    $\begingroup$ @JosephMalkevitch Very good idea! In my class we actually made our copies, and that was in high school. As a very beautiful and meaningful mathematical concept I wear a Möbius strip as my wedding ring, but as an object it is unfortunately too small for demonstration in front from a whole class. $\endgroup$ – Pavel Nov 10 '15 at 17:11
  • $\begingroup$ Also look at: davidparker.com/janine/mathpage/topology.html#MOBIUS $\endgroup$ – Joseph Malkevitch Nov 11 '15 at 18:44
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Draw the bottom three-quarters of an oval: mobius step 1

Flesh that out to make the bottom half of the strip: mobius step 2

Connect one of the open ends at the top to the bottom on the other side: mobius step 3

Now draw a straight line across the top: mobius step 4

Finally fill in the last edge at the back: mobius step 5

An advantage to this approach is that it highlights the fact that the mobius strip starts out as an ordinary strip, but has a twist in one spot.

Note I figured out these steps by making various physical mobius strips with very thin width relative to their length, and manipulating them so that they were as loop-like as possible. One of the key features is the straight line at the top. That is due to how a curve in paper looks from the side -- same as at the sides of the picture.

Finally, I echo everyone else's sentiments: you should bring strips of paper so that everyone can make a real physical mobius strip of their own. There's nothing like seeing the physical real object to support the imagination!

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    $\begingroup$ I was wondering something similar, but I also thought it might be too broad. This question matheducators.stackexchange.com/questions/7864/… was on a similar theme, but didn't have a "how can I actually do it by hand" answer (yet) $\endgroup$ – DavidButlerUofA Nov 10 '15 at 1:49
  • $\begingroup$ I'm using yours. It's amazing. $\endgroup$ – Newman Nov 10 '15 at 11:04
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    $\begingroup$ @Newman If you are using this one, you might want to accept his answer by clicking the checkmark at the upper lefthand corner of the post. $\endgroup$ – Steven Gubkin Nov 10 '15 at 15:19
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    $\begingroup$ @BenjaminDickman I've put an answer to that other question on quadric surfaces, if you're interested matheducators.stackexchange.com/questions/7864/… $\endgroup$ – DavidButlerUofA Nov 13 '15 at 4:50
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    $\begingroup$ UPDATE: I used this drawing in my talk. It went great. This drawing is specially good to show that there isn't a natural homeomorphism between the fibers and the typical fiber, but two such homeomorphisms , and they are related by "reflect in the midpoint". $\endgroup$ – Newman Nov 15 '15 at 17:20
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I searched google images and found many nice renditions. Here one that you may prefer from this link

Valid XHTML
(source: umich.edu) .

I especially like that it shows the width of the paper and doesn't draw the strip as a line or piece of string. Having the students make them is always a hit.

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  • $\begingroup$ I really like this one--per chance are there other figures you draw nicely? $\endgroup$ – Addem Nov 10 '15 at 4:17
  • $\begingroup$ @Addem I wish I could draw anything. I got this from a link as explained in the beginning of the post. $\endgroup$ – Amy B Nov 10 '15 at 12:29
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Since I use Seifert's algorithm to construct surfaces a lot, I tend to go for the flat approach:

flat Mobius band

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Instead of drawing on the blackboard, use a 3D model and pass it around when you're finished using it for demonstration.

If the class is small enough, you can bring in a strip of paper for each student to make his/her own.

The demonstration will be much more powerful this way and the lesson more likely to stick.

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  • $\begingroup$ This allows people to actually vote up this particular answer as an independent idea instead of just always having it in the comments. $\endgroup$ – WBT Nov 10 '15 at 16:34
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A Möbius band can be constructed by cutting a cylinder and then rejoining its ends after a twist. So draw it the same way.

Draw a cylinder. This should be easy enough to do. Cut it by rubbing out parts of it. And then link it back with a twist.

This is in the same spirit as David's answer, with the added benefit that your audience can follow its construction with their imagination.

However, the use of blackboard may present a problem, as erasing tends to make an area less readable. Perhaps a different colour for the twist would help.

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