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.


Pleated Star and Template

Creating this radial pleating occupied much of this past week.

Would love to see other people make something like this, so am sharing the process and the template.

This began by seeing a folding designed by Anna Rudanksi in Paul Jackson’s Complete Pleats book, page 199…

…and Tomoko Fuse’s infinite folds, Origami Art Works by a Modern Master, page 84.

I tried out a few different ways to set up the folding. The PDF below shows the final pattern I made then folded. The paper is 11″ x 17.” For more about how to make all these creases, take a look at my video https://youtu.be/92EZeFjha5c which shows how to do this sort of folding.

When I was testing out the folding pattern, I was using a 24 lb paper. I think I would like something slightly heavier if I was making the whole thing out of paper. What I actually used is something that is a synthetic paper, which can get wet, because the finished piece is going to hang on an outside door.

Scoring then folding

I made 10 of these pieces, first scoring then folding.

I did the folding over several days.

Next step was to group them together in pairs, overlapping by 25%. Used linen thread to sew them together, since I didn’t trust this synthetic paper with adhesives. Also did a turn in on the lower (inside) the edges. Not really sure if this was really necessary. For the turn in, I used the method that Joe Nakashima shows in his Origami Fireworks video between 7’10” and 8’20.”

Here’s me with my husband, hoping that eight units would be enough. Nope, needed two more.

Finished piece is 25 inches cross. Rays are every 12 degrees. One step is left, which is to mount it on board so that it can hang up.

I’m quite happy with the way this piece turned out.

Not everyone in my household shares my enthusiasm.

Addendum: If you liked this post and want to consider supporting a children’s Lunch, Learn and Play program in my rural community, please visit this link to discover some math/art that you or some children you know might find inspiring. It’s a collection of math/art cards to color, which I designed and which have a story and the link to the math that they are built with. https://etsy.me/3uJkJuk


Folding, Teaching, Zooming

Fascinated by Folding: maybe that’s what I should called the next set of classes I’ll be teaching at Center for Book Arts, via zoom, over 4 weeks starting afternoons at the end of April or for 4 weeks in the evenings starting in the beginning of May.

Alas, these classes are called Flat Foldable Pleats, and Edge Release Explorations. I don’t think anyone will even know what that means. I’m hoping that the picture will sell the class,

I’ve been taking this deep dive into pleating and edge-release folds, which is a whole different thing than symmetrical pop-ups, which I also love. After playing with unusual foldings, like miura folds, and examining Paul Jackson’s books for years, I started playing around with the idea of teaching these lesser known structures.

My last couple of submissions to the Bridges Organizations Math Art shows, like the piece above, have reflected my interest these pleated folds.

They are fun, challenging, and always surprising. Some of the folds, like the hexagon bellows there with the compass leaning against it, are a bear to fold.

Duck and Fold

Other folds, like simple one above, in which an edges of the paper are released by cuts from the folds they might have been bound to, create gorgeous architectural effects, which become even more delightful with some thoughtful photography.

In fact, many mornings this past winter I would get up to do early morning folding just so that I could photograph the constructions in the early morning sunlight. It was a satisfying way to start the day, especially during those stressful days from early November until mid January.

Another thing I’d like to mention is that it appears that I’m getting the hang of teaching on zoom. I feel like I’m figuring out how to create the feeling of connection that I like so much about in person teaching. One big discovery for me is that it’s great to ask people to unmute themselves for our whole class. Don’t know why more people don’t do this. This leaves the way open for people to interject comments, ask questions at critical moments, and lets me know if my pace needs to be adjusted.

I’m finding, too, that some people who take workshops have figured out how to adjust their cameras so that I can see them AND their workspace. Seeing people’s work and as well as their expressions as they work is such a pleasure.

A year ago I had no idea that I’d be able to do this kind of teaching from a little production studio in my home that I could not have even imagined being there. The twists and turns of life never cease to surprise me.

folding · Process

Simple Folds that are Hard

A folding Map Booklet

Folding a square diagonally is not in most people’s skill set.

There are so many applications of diagonals, not just in paper folding but also in math, that it’s a worthwhile concept to chase down and cozy up to, but it needs to be done carefully.

Having this extra time home these days, I find myself wanting to write about these big/small details that teaching has taught me, that feel important to share.

Paper folded in half the “regular” way

If I ask a student to fold a piece of paper in half, they fold the paper in half. Generally I ask students to refine their intentionality with the paper so that their paper is folded evenly. The older they are the more they love this lesson. The whole experience of folding paper is half is a pretty happy place. Until it isn’t.

What shape is the paper in when you fold a square in half? Think about it before you raise your hand, and don’t change your answer because of what someone else says, and I’m going to call on everybody who has their hand up.

I see every hand go up. Rapid fire around the room, rectangle, rectangle, rectangle. Then someone say triangle. Just one person. That person is so brave. I ask that student and another student to come to the front and prove their response to me and to the class. They are both correct. We celebrate.

Folding a square point to point, from the top down
Folding a square point to point, from the top down

The fact that making a diagonal fold fries brain circuits was driven home to me one day when I asked a class of third graders to make three folds with a square piece of paper. I didn’t anticipate any problem as I already had already done quite a few folds with this group.

The lesson for this day was to do one diagonal fold, unfold it, flip it over, fold vertically, open it, and then fold horizontally.

I could not have been clearer in my instructions. I had drawn out the directions, which were projected on to the classroom smartboard. The papers I handed out were printed so that they could compare their progress easily with my sample.

I have rarely so quickly and completely experienced the chaos that followed. The students were totally confused; there were renegade folds on nearly every paper. Students struggled with paper orientation. They fearfully charged ahead in complete darkness. My perfect simple lesson was simply a disaster.

This was a class of 25 students. I would be facing four more classes who needed to learn this folding sequence.

The problem wasn’t the diagonal fold. The problem was the sequence of folding. Did I lose you for a moment when you read this:

“The lesson for this day was to do one diagonal fold, unfold it, fold vertically, open it, and then fold horizontally.”

Turns out that making the diagonal fold in the first step of this lesson disoriented the class in a way from which they could not recover. Immediately, students were felt unsafe, afraid of doing something wrong, and I could not gather them back to me.

The fix turned out to made perfect sense. I showed the diagonal fold last instead of first.

origami base for world map

Students folded papers in half (the “regular” way), opened the paper, folded it in half in the other direction, opened it, flipped it over and made one diagonal fold. Before any confusion took hold of their brains, their folding was done.

World map popping up
World map popping up

I want to write more about the diagonal, how to make it safe, and why it’s important, but that’s another story.

Today’s story is simply about how something that is simple can go haywire.