## I Learn to Fold Evenly

The article “Why do Automobile Sunshades Fold Oddly?” by Curtis Feist and Ramin Naimi won the George Pólya award for expository writing in The College of Mathematics Journal for the year 2010.  You can access the article here.  Dr. Feist is a professor at Southern Oregon University. Two years I attended a talk he gave on the subject.

Three Felt Ribbons

Afterward  I started to play around with his ideas.  The result was that I made the three felt ribbons on the left and used them to explore extensions to the main assertion of the Feist-Naimi(F-N) paper.  Spoiler alert! – The F-N article is constructed so that the reader  finds out the answer to the title question only at the end.  So if you want to experience the full pleasure of reading it, do so now.

The idea of the paper is that you can think of folding the automobile sunshade as winding a ribbon with the ends joined together (like the center ribbon above) around a cylinder without any twists.  The surface of the ribbon should fit smoothly on the cylinder.  Note that the right hand object above is a möbius strip.  Since it can be thought of as the surface of a cylinder cut and twisted 180 degrees, it clearly will  not fold around a regular cylinder with no twists.  So one-sided ribbons like the möbius strip will not be discussed here any further.

After I had seen Dr. Feist’s lecture, I went back to my office and created these paper models among others.

Three Paper Models

The right one is a ribbon with the ends joined normally.    The middle one was twisted 360 degrees before the ends were taped together and the left ribbon has a 1080 degree twist.  I first established the truth of the article – that the normal ribbon needs to be formed into three loops to have no twists.  Here it is.  If you trace around the ribbon you can see that the inside surface would lie flat around a cylinder.

Normal Ribbon with Three Loops

The F-N paper also stated that loops could be added in pairs.  Here is the ribbon folded into five loops.

Normal Ribbon with Five Loops

So far, so good.  Now I was interested in how many loops a ribbon with a 360 degree twist could form.  I discovered this by trying a paper model (middle of the paper models) but if you go back to the F-N article you will see that the boundary edges of the twisted ribbon have a linking number of one so the number of crossings is odd and thus the number of loops must be even and indeed here is the twisted ribbon with two loops.

Twisted Ribbon with Two Loops

So now four loops and six loops are possible.

Twisted Ribbon with Four Loops

Twisted Ribbon with Six Loops

If you twist the ribbon 720 degrees, you get a linking number of two and thus an odd number of loops and if you twist a ribbon around three times you get a linking number of three and an even number of loops.  Here are pictures of a triply twisted ribbon with its loops.

Triply Twisted Ribbon with four Loops

Triply Twisted Ribbon with Six Loops

Triply Twisted Ribbon with Six Loops – End View

Triply Twisted Ribbon with Eight Loops

Half of the fun of this project was confirming and extending the results of the F-N paper.  The other half was the sensual pleasure of making and manipulating the felt models. The following pictures show the felt objects from various angles and with various foldings.  Since I am basically topologically challenged I get renewed enjoyment every time I try to form the felt ribbons into the proper number of loops.

Felt Ribbon with Three Loops

Felt Ribbon with Three Loops – End View

Twisted Felt Ribbon with Four Loops

Twisted Felt Ribbon with Four Loops – End View

I originally wrote this article last summer but didn’t publish it.  Since then I have made more models.  I think I have gotten better at it – at least more colorful.

Felt Ribbons

I leave the models on a table in my living room and challenge curious visitors to wrap them around their arms with no twists with a given number of loops.

This crafty approach to topology is very pleasurable.  See this TED talk for a more fully formed example.