# What is a good method for drawing a Möbius band on the blackboard?

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.

• I would show one made of paper, and I would practice drawing from good images online. Commented Nov 9, 2015 at 19:34
• If fact, it is so easy to make paper ones everyone in the class should be encouraged to make a copy. Commented Nov 9, 2015 at 20:22
• 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. Commented Nov 10, 2015 at 7:04
• @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. Commented Nov 10, 2015 at 17:11
• Also look at: davidparker.com/janine/mathpage/topology.html#MOBIUS Commented Nov 11, 2015 at 18:44

Draw the bottom three-quarters of an oval:

Flesh that out to make the bottom half of the strip:

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

Now draw a straight line across the top:

Finally fill in the last edge at the back:

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!

• 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) Commented Nov 10, 2015 at 1:49
• I'm using yours. It's amazing. Commented Nov 10, 2015 at 11:04
• @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. Commented Nov 10, 2015 at 15:19
• @BenjaminDickman I've put an answer to that other question on quadric surfaces, if you're interested matheducators.stackexchange.com/questions/7864/… Commented Nov 13, 2015 at 4:50
• 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". Commented Nov 15, 2015 at 17:20

I searched google images and found many nice renditions. Here one that you may prefer from this link

(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.

• I really like this one--per chance are there other figures you draw nicely? Commented Nov 10, 2015 at 4:17
• @Addem I wish I could draw anything. I got this from a link as explained in the beginning of the post. Commented Nov 10, 2015 at 12:29

Since I use Seifert's algorithm to construct surfaces a lot, I tend to go for the flat approach:

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.

• This allows people to actually vote up this particular answer as an independent idea instead of just always having it in the comments.
– WBT
Commented Nov 10, 2015 at 16:34

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.