geometry and paperfolding · Paper Toy

Flexagon 2020

I’m ushering in the new decade with a new family of flexagons.

The first flexagons originated from the fiddley hands of Ph.D. mathematics student Arthur H Stone in 1939. What he discovered was ways to fold paper so that it could flex to reveal hidden faces.

Martin Gardner popularized flexagons in the 1950’s, and Vy Hart made them totally adorable with her videos, which were made during this past decade. There are likely an uncountable number of flexagon configurations just waiting to be discovered. Ann Schwartz , who I met this past summer at MoMath’s paper-folding conference, and whose folded discoveries include a 12-angle flexagon, has told me that she thinks that this one that I’ve made is something new.

My flexagon has a great deal in common with Octaflexagons and Tetraflexagons in that all of these are have square faces embedded in them, and the octaflexes, like mine, are full of isosceles triangles.

Some of the differences between my flexagon and the others is that mine has pockets and fins. It’s also constructed from a different shape than other flexagons, which generally depend on strips on paper. This flexagon starts with a square.

I created these graphically partitioned squares with the idea in mind that I wanted the various surfaces of my flexagon to be recognizable distinct.

Like it’s easy to see that the surfaces above are completely different from the owl-like face below.

Static photos are not the best way to view flexagons. Videos are much better. Here’s the video.

I’m saying that my flexagon is part of a family of flexagons because I’ve realized that if I make slightly different decisions in the constructions of these flexagons that different variations, which have their own distinct characteristics, emerge. There are at least three more variations in this family. I’m looking forward to sharing everything about them in this coming year.

I’ve done a bit of production-making with these. Just made 20 of them. Most of what I’ve made are spoken for but I have 9 that I’m selling on Etsy. Why nine? I finally ran out of my stash 11″ x 17″ Strathmore 25% cotton writing paper that these are printed on. These 9 flexagons that’s I’m selling will be the last of the ones that are made in 2019, and are signed and dated.

These have been great to have all over my desk, but now they need a new home. Etsy.

12 thoughts on “Flexagon 2020

  1. Thanks for posting. I am trying to bring some of these ideas to my K-5th class but I find it hard. Anyway I enjoy your detailed posts and this blog has been a great discovery in 2019. Have a good and creative 2020! Thanks!


    1. You teach K-5th?!?! That’s quite a spread! I have some number line posts that I will be getting to soon that might be just the kind of thing that is just right for the range of classes you are teaching. Stay tuned! Thanks for your note. Best wishes for the new year!

      Liked by 1 person

    1. Hi Denise,
      Exactly! That Infinity Card is a flexagon, and quite a wonderful one. The fact that it is based on two squares that are split then rotated is something that I love about this one. I’ve used it a number of times with classes, putting information on the surfaces, and it’s always surprising how much info it fits. Thanks so much for commenting Denise. I am so grateful for all that the amazing info you gather in one place. Your presence online is such a gift. My best wishes for the new year!


  2. Dear Ms Beardell Krieg
    I truly love reading your posts. Your creative ideas are a true inspiration.
    I have been making flexagons for a few years, and teaching them to our Paper Group, but we have never tieid anything as elaborate as yours. I LOVE it.
    Our group has been also been playing with the Chinese Thread Book, and your posts about that gave us lots of new ideas. I especially like the version of the Twist Box that is all folding, and no cutting, but I found it hard to find any printed instructions for that so I had to make my own!
    Many thanks for sharing your wonderful work and I hope 2020 is another very productive and fulfilling year for you.
    Cheers from Tricia in Australia


    1. Hi Tricia in Australia, thank you for your note. I can’t tell you how much I appreciate your thoughtfulness in sending this to me. Glad you’ve been making flexagons with your Paper Group. They are quite wonderful. It got me thinking about how I learned them. At first I couldn’t remember because I was trying to remember WHO taught me about them. Finally I remembered, more than twenty years ago I first saw them in them in a book, Complete Origami by Eric Kenneway, while I looking for projects to make with an after-school book arts class for 3rd and 4th graders. Flexagons are such great fun. The one I’ve designed as actually easier to fold and assemble than most. I will be writing up, or at least making a video of its construction in the coming month.
      I’m so impressed you made up your own directions for a no-cut twist box!
      I would love to see, or at least hear more about the ways you’ve been playing with the Chinese Thread Books. To me, they are all about finding all the different ways of thinking about the pieces.
      best wishes for the New Year,


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