Intro... How to make... How to flex... Theory Net resources References Feedback

Intro...
If you've not come across the joys of flexagons yet, now would be a good time to start smiling. Flexagons were discovered in the 1930's by a bright university maths student, who was idly playing with an offcut strip of paper. He made some triangles, and wow, what's this - it's a trihexaflexagon! Well, he probably didn't say that exactly - but he did promptly invent some more complicated versions. Possibly a bit on the brainy side, I suspect.

So....... a flexagon is a strip of paper folded up, which you can "flex" to reveal hidden faces. There are two types - hexaflexagons which look like hexagons (six sides), and tetraflexagons which look like squares (four sides).

Once you start flexing, it's easy to become compulsive - you see a flexagon lying around, and unconsciously pick it up to give it a couple of extra flexes.

Trying to figure out how they work is good practice for your brain ;). They're also a lot of fun for making with your children on a wet afternoon.

There is quite a lot of stuff on the 'net already, and I've listed some pages at the end of this document, along with some references. My contribution to all this is that I've made some files which you can download and print out - thus saving you the bother of making templates and trying to draw accurate triangles. Most of the stuff already on the net deals with hexahexaflexagons (6 faces to find) - but I've also included some other versions. OK let's go......

How to make...
I've made template files in a format called 'Windows Metafile' or .wmf for short; these are suitable for Microsoft Windows. I've also converted the template files to Postscript (or .eps) format which may be more suitable if you're using another operating system; or you may be able to send the postscript files directly to a laser printer. Both these should print in good quality. If neither of these is suitable, try the GIF format version - the printed quality should be pefectly OK. I suspect the .wmf format will be a good choice for many people.

Download the zipped template files by clicking
***here*** (.wmf format - size 200kB) or
***here*** (.eps format - size 180kB) or
***here*** (.gif format - size 120kB)

In the zipped files are:

You can use the instructions from these web pages to make your flexagons, or if you want a better quality of printout, the instructions are available as zipped up .wmf (not .eps) files ***here*** (size 1.1MB)

Once you've downloaded the files, unzip them, for instance using WinZip. The readme.txt file tells you the name of the template files (eg flex4.wmf has the template for a tetrahexaflexagon).

For the .wmf files:
Open up your favourite publishing/dtp program (eg MS Publisher, Serif PagePlus) or word processor (eg MS Word, MS Works) or anything else that you can insert a .wmf file into. Start a new document of US letter size or UK A4 size, orientated so it is wider than it is high, and insert one of the .wmf files into it. You may want to make your page margins smaller. The picture should come out the right size, but if you want to make it larger or smaller, that's ok (just remember that the square shape I included as a reference should always be printed out as a square!). Print out the page(s).

For the .eps files:
If you cannot send the files directly to a postscript/laser printer, or insert them into a publishing program or word processor as above, you could load them into a postscript viewer (eg the free Ghostscript, available for many platforms) and print from there.

For the .gif files:
Simply open up the relevant .gif template file (eg flex3.gif) in your web browser and print the page. This should work on letter or A4 size paper, but you may need to reduce the page margins first. Alternatively if you use an image viewer such as Lview Pro (MS Windows) you can resize the image if necessary, and then print.

Now (all formats) follow the instructions as given below to make your flexagon (print them out first if you like).

The instructions are available online as follows:

How to flex...
Having made your flexagon, you can now start flexing.

First, pinch together two triangles at a corner of the hexagon. Push the opposite corner down and in towards the centre (center if you're in the US ;) ). You can now open out the middle to reveal a new face.

If you can't get it to work, try the next corner.

If you really want to be impressed, check out my Java 3-D Simulation of a flexing flexagon.

You can paint or number the faces as you find them. Make sure you can find them all - if you can't, click here.

The faces don't always appear in the same orientation - sometimes they point different ways. You can decorate the faces to show this - say with a picture that only comes together once in a while, or with geometric shapes.

Most of all, I hope you have fun with them...

Theory

I've done a bit of thinking about how and why these things work. If you're curious and want to exercise your brain, click here.

(Note: the theory pages are not included in the zipped download on this page).

Net resources

Wikipedia's Flexagon entry (with further links)

flexagon.net Colourful site with lots of templates and links.

H McIntosh's papers Excellent papers about the theory, many by him. These appeared on the web after my webpage, and are well worth checking.

References

Crampin, J. ``On Note 2449.'' Math. Gazette 41, 55-56, 1957.

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 205-207, 1989.

Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 62-84, 1979.

Gardner, M. ``Hexaflexagons.'' Ch. 1 in The Scientific American Book of Mathematical Puzzles & Diversions. New York: Simon and Schuster, 1959.

Gardner, Martin. Mathematical Puzzles and Diversions. Penguin Books, or Chicago Press, 1965.

Gardner, M. Ch. 2 in The Second Scientific American Book of Mathematical Puzzles & Diversions: A New Selection. New York: Simon and Schuster, pp. 24-31, 1961.

Johnson, Donovan A. Mathmagic with Flexagons. Activity Resources Company, 1974.

Maunsell, F. G. ``The Flexagon and the Hexaflexagon.'' Math. Gazette 38, 213-214, 1954.

Oakley, C. O. and Wisner, R. J. ``Flexagons.'' Amer. Math. Monthly 64, 143-154, 1957.

Wheeler, R. F. ``The Flexagon Family.'' Math. Gaz. 42, 1-6, 1958.

Hexahexaflexagrams, Mathematics Teacher, 44

A Six-Sided Hexagon, School Science and Mathematics, 52

T.B. McLean, "V-Flexing the Hexahexaflexagon", Am.Math.Monthly 86, 457-466, 1979.

Feedback

Hope you enjoy this stuff

Please feel free to comment - I'd welcome a quick email to tell me if it worked for you and, if you had any difficulty, what I can do about it in future...

David King last update 19Nov01