Sunday, March 27, 2011

The Amazing Trihexaflexagon!!

A combination of math and art, this annual 6th grade project intrigues everybody. We began making them when my 22 year old son was in 5th grade, and for an enrichment class he researched math puzzles and games. His interest in them waned many years ago, but I've been intrigued ever since, and so have the kids. First we break down the word: tri = 3, so it has 3 sides, or 'faces'. Then hex = 6, and it has 6 edges. Then of course flex = bend, so we have a 3-faced, 6-sided, flexible polygon! Then, an added bonus, each face has an additional 'rotation'. What you see above is the first side, and its rotation.
Above is side 2 and its rotation, and below you see it being flexed, or rotated. The final step would be to reach into the center and 'bloom' it like a flower.
And below is side 3 and its rotation, still being colored (by one of the Sarahs).
It all starts with a sheet of bristol or tagboard, 3-1/2" x 22", measured into triangles. Accurate measurement is critical, so it is a good exercise for these kids. Then they learn to score the paper for it to fold cleanly. It's easy to do with good instruction, but hard for me to explain here on the blog. Here's a video of one:
The story of the origin of the flexagon is really interesting and fun. If you type "flexagon" into google, or even "trihexaflexagon", you'll find endless information on how to make them, and much more. (The trihexaflexagon is only one of many flexagons, but easy enough for kids to put together. We also make a tetratetraflexagon,which is even easier to make, but a little less dynamic a result.)
I've taught workshops on making flexagons at my NY state art teachers' conference, and may teach one again next year, so if you are in NY and are interested, I'll see you in Westchester in November! It's much easier to learn hands-on than from printed instructions, believe me!!
If you've read this far, you're in luck, because..Ta Da!! Here's a scan of my instructions that I used for my workshop! Feel free to copy and use them:


  1. WOW - I wonder if 5th graders could do you know where I could find step-by-step directions?

  2. Geez, blogger is going NUTS tonight. First it kept putting in miles of extra spaces, which I kept deleting and kept showing up, and then finally the spaces disappeared, and I couldn't space my paragraphs AT ALL. So finally I think it looks OK, with centering photos and adding in stars that I didn't really want. Anyone else have a problem with blogger tonight?

    It's late now, so I'll have to post the flying pigs tomorrow...

  3. Nancie Kay, you know your students better than me. They need to be able to measure, ACCURATELY. An 1/8" off will make the thing not go together properly. I demo on an old-fashioned overhead projector, and I use a clear acrylic ruler with centimeters. My centimeters correlate to the students' inches. I go around the room and check after EVERY step.

    As for where to find directions, there's stuff all over the internet, but some of it is very hard to follow unless you are a mathematician and know what the heck they are talking about. I have printed instructions I've used for my workshops, and tomorrow I'll try to figure out a way to get them to my readers. So check back!

  4. I've been waiting for this post since you mentioned it earlier!! Thank you so much! Can't wait to try it : )

  5. Phyl,
    I don't know where my comment disappeared to? I commented yesterday saying thank you for the directions. I can't wait to try these with my students!

  6. I am having issues :(. I've tried making one as an exemplar first, to see if I can get the hang of it.

    I cut, folded and assembled correctly... but I just can not get the hang of how to flex it. I've looked on so many sites, including yours of course, and they all say to push two triangles in the center etc., but mine doesn't... "bloom"?

    I figured out my own way to get it to move... but I can NOT figure out how to get a third blank side to decorate?! I wish you could see what I'm doing... but any advice?

  7. Oh Hannah, I would like to help you! (This is why I teach it as a hands-on workshop. Come to NYSATA next November!) I'm betting you are not assembled right. Make sure that when it is assembled the sides are all "weaved" in and out of each other, in other words make sure when you fold that the end gets fed under the beginning. It's a frequent beginner error. If you look at your assembled flexagon and there's an opening all the way across, it's wrong. I'm not sure if this makes sense, but hopefully...

    Then, when you fold it, pinch together two triangles like you are pinching your nose. Make sure they are two triangles that are NOT attached to each other at the fold where you pinch. Then push in the other side so it looks like the one in the photo that is squeezed together, and reach into the top to open.

    A crazy idea - you are in western NY right? I will be in Rochester on Saturday for "ArtAwake", an event sponsored by U of R, where my son is a senior. As I've mentioned before, it is a day filled with musical performances, art displays, interactive arts, food, and more, and my son is one of the chief organizers. it begins at 3pm and runs I think to 1am. Check it out here:
    Anyhow, the crazy idea would be to actually see you there. Email me if you decide to check it out, and I'll tell you how to find me!

  8. I did it! I did it! Phyl... you don't understand... I watched at least 5 videos on youtube, even had my husband try last night! I tried every possible combination but I finally got it! I'm sure I had it right before.... but I think I was nervous to pull down before because it was just a LITTLE snug from my folding- like... a 1/16" maybe overlapped a bit and I was afraid of ripping it... but I did it! I did it! YAY! I think what really made me double check my folding was making sure I "weaved" and didn't have an open space like a pocket.

    Ya hoo!! And I was TOTALLY planning on going to Rochester this weekend, believe it or not... I even wrote it down in my planner the first time you mentioned it... but then the Sabres had to go make the playoffs... and I'm now actually going to Philly this weekend to see some family, but really to go root for the Sabres.... I don't think you get how crazy Buffalo fans get when ANY team makes the playoffs... I hope you have a super time though, which I know you will. I can't wait to see some pictures!

  9. So glad you made it work!!!!!! I always tell the kids there's a "fudge factor" - the imperfection of my paper cutter - so once you've "got it", I tell the kids that the more they flex it, and kind of press it down each time until it softens up and eases into place, the easier it will flex after a while. The kids agree flexing them is addicting.

    Have fun in Philly! I'll definitely be posting pics from ArtAwake.

  10. I saw this and thought of you! :

  11. My question is how do you illustrate it?? Do you illustrate it with the triflexahexagon taped, or do you do it with it open in a long strip?? Thank you so much for sharing all that you do!!

  12. Mrs. Cohan, you DEFINITELY need to illustrate it with it assembled. It makes absolutely no sense when it's in a long strip. I recommend that you draw, flex, correct and re-draw, so that the sides connect in either rotation, if that makes any sense. It's often a good idea to begin with a ruler, connecting the hexagon's corners into a 6 pointed star, and then using that as a framework for a more complex design. Does that make sense?

    We color with felt tip markers (both fat and skinny), but NO SHARPIES - they soak through in all the wrong places, or with colored pencils. All black is done LAST so that other colors don't pick up and smudge the black.

    I'm assuming you found this post from my more recent post, where I mentioned the specific design challenges I gave my students this year.

    Let me know if I can be of any more help.

  13. Hi Phyl. Do have another link to your instructional video that you provided above. The Google Video link does not appear to work. Cheers! Chris in Sechelt, BC

    1. Chris, you are right,so sorry the link no longer works but unfortunately it wasn't my video. I don't have any videos, but if you google 'trihexaflexagon' you'll find all sorts of stuff and I bet you can find lots of videos. Thanks for visiting!


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