geometry and paperfolding · holiday project

Simplest Flexagon

Folded things, flexagon not included

This evening I’m teaching a class that Center for Book Arts is called Seasonal Folding. As I’m sitting here preparing my materials, I’m also reading over the class description, with the thought in my head “what was I thinking????”

Actually I know what I was thinking. I was thinking that this would be a three-session class, for a total of about 7.5 hours. What happened next was we decided to do it as a one-session class, for three hours. What didn’t happen was a change in the class description about what we’d be covering. There’s no way we’ll get to everything that’s in the write up.

I am so totally against anything that remotely resembles bait and switch that I’m working out how to deal with this. Fortunately, I’ve written about most of what we are doing, and have video links that I can point people towards. This clever little flexagon structure isn’t something that I’ve done anything with on these pages, so, crazy as it may be, part of my prep for this evening is making a short video of how to create this.

It’s really an incredibly sweet structure. Some of my friends, most notably Ed Hutchins and Mark Kaercher discovered this before I did. Ed has a small collection of these, with different designs on them, collected from various Cracker Jacks boxes from many years ago. Mark was introduced to it by Chuck Stoffle, who, I think stumbled upon it through a Crayonal site, which calls it the Past, Present and Future Me Never-ending card. It was Mark’s version that caught my eye when he posted it on twitter.

The directions on the Crayola site has too many folds for my liking, so, naturally, I had to make my own version of how to make it.

If you can decipher this illustration, which is entirely possible, you won’t have to watch the video, but, even though, at this moment I haven’t made the video yet, I promise it will be short. After all, I have a whole workshop to ready for in 90 minutes.

The video:

As it turns out, I didn’t finish the video in time for the class. Didn’t seem to matter. It was a great group. With just four minutes left, they were up for one last project, which is saying something, since the class was three hours long.

Here’s what else we made:

We made paper springs, scaled origami pockets, starbursts, one-cut fold-and-cut five pointed star, spiraling paper ornament, mobius strips with cuts, equilateral triangles, snowflakes, and puffy pentagon box. What a night!

Flexagons · folding

Video Tutorial for My Flexagon

About a month ago I zoom-taught a group of about 200 people how to make this very sweet fidgety flexagon that I had created. The Museum of Mathematics hosted the event.

As I want to show this to anyone who wants to learn it, I (finally) made a tutorial video of my flexagon.

Start with any square, or the cool graphic above, which is my work. I love this version with the MoMath logo on it.

Here’s the video:

A big shout out to MoMath, who, once again, were a pleasure to work with!

If you make one of these, send photos! Have fun.

Flexagons · hexagon · How-to

Hexagon-Flexagon: Post #3, Instructions

How to Make a Hexagon-Flexagon
How to Make a Hexagon-Flexagon

After posting  about Hexagaon-Flexagons on November 10 and November 16, I started working on making a template for teaching this tricky paper invention. Even though this  structure is well covered on the net, what I want to add to the mix is something to make the folding easier.

When I’ve taught this structure the part that people have the hardest time with is creating precise folds. I made the instruction sheet above because it provides a way to create score lines so that the folding is easier.

Ruler and paperclip
Ruler and paperclip to use to score lines onto paper

Scoring is PRESSING lines onto paper, so to help facilitate folding.  If a score line is firmly pressed into paper, it will fold easily on the score line. A pen that has no ink in it, or a paper clip, both make good scoring tools, as they will make a thin line. Bookbinders use bonefolders for scoring, but for this template I prefer a paper clip because it makes a thinner score line.


Here’s a picture of my daughter’s hand scoring  my template. The template is on a firm but not hard surface: the surface ideally should have a bit of ‘give” so that it can be pressed into. (A stack of newspapers or a catalogue works well.)  Place the ruler on the indicated line, holding it securing in place, then run the edge of the paper clip along the edge of the ruler. Press firmly, but not too hard, as you don’t want to rip through the paper. If you look closely at the image above you can see the the score lines that have already been pressed into the paper.

Cick on this image for to print or to enlarge

That’s about all I have to say for now. Hopefully the instruction sheet above clear enough to follow. If you want to print out template without the instructions, this is the a link to my non-annotated hexagon-flexagon .

And for some more inspiration, here’s a hexagon-flexagon decorated by Michele Gannon:

By Michele Gannon

Now, here it is again, flexing….

The Flexing of a Flexagon
The Flexing of a Flexagon

….to next reveal the design that was formerly tucked inside:

This next picture is that same design, after the image has been flexed. Be sure to notice how the dragonfly images change direction.

addendum: Here are a couple of links to another person’s take on the hexagon-flexagon: and

Happy flexing!

Addendum 3/12/2017: just came across another great flexagon resource!

Flexagons · math · Paper Toy

Hexagon-Flexagons: Post 2, Fractions

Except for the last picture, all of the images in this post exist on one hexagon-flexagon. In my last post I showed hexagon-flexagons that were about making designs with pencil, paper, and gouache. When I was making the design for this post I was thinking more about math.

Hesxagon-Flexagon divided into three rhombuses

Okay, now here I go,showcasing my hexagon-flexagon as way to illustrate fractions, rather than to be a kaleidoscopic toy.
The hexagon-flexagon above has been illustrated to be understood as thirds,showing that three thirds create a whole.
Now, this concept might be better understood if the parts were actually labeled.

Flexing the structure to reveal another design

I worried that labeling the parts could cause a problem, because, after flexing the structure, the design changes. I did a mock up to see what would happen, and this is what I got:

Fractioned parts, labeled

Not bad. I like this way of working with the hexagon-flexagon.

Hexagon-Flexagon, two trapezoid showing dividing the structure in half

One thing is really clear to me, and that is that I would love to work with a graphic designer who would add words and numbers that looks snazzier than my handwriting. Oh, and another thing that is clear is that labeling the hexagon as two halves doesn’t work well after it’s been flexed. (I’m not providing a picture here of how the broken up halves looks).

Six equilateral triangles dividing the hexagon-flexagon into sixths

This side of the hexagon-flexagon shows that six parts equal a whole. Ideally, I would label each of the blue triangles with their own “1/6′ fraction, then write equations all around the outside edges (such as 1/6 + 1/6 = 2/6 =1/3)

Flexed version of the previous image

I think that this is going to be a long-term project, perfecting these images with better text graphics. I like how the geometric patterns work out here. It’s just the labeling that I can’t get right.

So, everyone has heard students question why they have to learn math, particularly algebra and trig, as they don’t foresee ever using it. I have at least one good answer to that age-old question. The reason to master math is that it keeps a person’s options open. I recently spoke to the someone who was helping her 30ish-year single parent daughter with Algebra because it is required for a nursing degree. Today I spoke a woman who has gone back to school to get a degree is Public Health. She is struggling with her required economics course. My son needs a to complete two semesters of Calculus towards his Biology degree, which will qualify him to go on to Chiropractic studies. So, there you have it: learning math a keeps options open for the future.

Now here’s another way of decorating a hexagon-flexagon, and hey, it’s even seasonally correct, as it can easily pass for a six-sided snowflake.

Hexagon-Flexagon by Michele Gannon

Math and art together: always a great idea.