Flying Circus of Physics newsletter

Flexagons

Say what you want, even that I am a complete nerd, but I am now going to reveal one of the high points of my otherwise angst-filled years in high school. One day I discovered the Scientific American article by Martin Gardner about hexaflexagons. If you have never seen any type of flexagon, these paper devices will sound dull, but I must warn you that they are addictive. Indeed, many people fell in love with the flexagons soon after their discovery (including Richard Feynman), and then after Gardner’s article appeared, the addiction went worldwide.

The discovery

The first flexagon was discovered in 1939 by graduate student Arthur Stone who happened to play with the paper strip he had cut from his American notebook paper in order to fit the paper into his narrower British binder. Stone found that by folding the strip on itself in 10 equilateral triangles and then fastening the ends to each other, he could form a hexagon with six triangular “petals.” More important, when he flexed (pinched together) the adjacent triangles at one corner while pushing the opposite corner toward the center, the hexagon opened up and then folded outward and down to show six new petals --- a few face. When he flexed it again, it again revealed a third face of six new petals. In Gardner’s words, “the hexagon would open out again like a budding flower.” The term hexaflexagon refers to the flexing capability of the hexagon shape.

By the next day, Stone discovered that by similarly folding a strip with 19 equilateral triangles, he had an even more complex hexaflexagon. Repeated flexing the device revealed that it had six faces, each with its own set of petals. Three of the faces were easy to find, but three required planning, as if they were hiding in secret. Here is where Feynman and others got involved, to develop strategies (such as “the Tuckerman traverse”) to flex through all six faces.

Video 1

I am about to give you links to videos of hexaflexagons, but they each flow very rapidly. So, let’s start with some static diagrams of how to label and then fold the six-face device, known as a hexahexflexagon.

http://britton.disted.camosun.bc.ca/hexahexa/hexahexa.html

Now get ready for a presentation at nearly the speed of light. Vi Hart will first demonstrate making a trihexaflexagon (three faces) and then a hexahexaflexagon.

http://www.youtube.com/watch?v=VIVIegSt81k

Video 2

(If you have limited time, skip this video and scroll down to my favorite, the Mexican hexaflexagon.) In Vi Hart’s part 2, we see how Feynman and others figured out how to flex their way through all the faces of a hexahexaflexagon and also explained why three of the faces are more common the other three faces. Vi Hart also explains how the article about flexagons got Martin Gardner a permanent column at Scientific American, where he inspired many millions of readers, including me.

One of the letters Gardner received in response to his flexagon article described how a man in New Jersey, United States, was flexing rapidly when his tie got caught in his hexahexaflexagon. To the astonishment of his coworkers, he disappeared into the center of the flexagon. Although the coworkers searched frantically through the faces of the flexagon, they could not find him. However, strangely enough, months later when someone was “idly flexing” a hexahexaflexagon in Glasgow, Scotland, the missing man was “disgorged” from the flexagon, totally bewildered.

Here is Vi Hart’s second video:

http://www.youtube.com/watch?v=paQ10POrZh8

You can follow Gardner’s suggestion of cutting up a photograph and gluing the pieces to a hexahexaflexagon’s petals. You can position them such that certain faces recreate the original photograph when they show up. Instead you can position them such that they never recreate it, much to the frustration of person flexing the flexagon. If you want more flexagon designs, you can start at this next link. The decahexaflexagon has 10 faces which can appear in 82 distinct ways.

http://www.mathematische-basteleien.de/flexagons.htm

Mexican hexaflexagon

Hexaflexagons were a high point of my high school life. So was the Mexican food of my home state of Texas. Vi Hart has managed to link these two high points in this next video. What joy she has brought me. (Again, say what you want.)

http://www.youtube.com/watch?v=GTwrVAbV56o

Cheers, Jearl Walker

09.05.2014