Hexaflexagons in the Summertime

July 21, 2019July 22, 2019 Paula Beardell Krieg2 Comments

This past Wednesday was my third three-hour meeting with a group of teenagers I am doing projects with this summer. I’m getting to know these young people a bit more, which is the best part of what I do.

It means a lot to me to be able bring projects to them which they enjoy, that are dynamic, and that might teach them something new. I missed writing about last week’s project…oh well.

I’ve been planning three things per week. My thought is to start with with something really short but really cool. This week I started with this amazing puzzle I saw on a post by Mike Lawler :

What you’ll see if you watch this video is a square piece of paper that has a square cut out of the center which a CD must fit through. It looks impossible. It’s quite mind blowing. Take a look.

The main event the afternoon was making hexaflexagons, which I’ve written about numerous times. Basically, they are a tricky foldable structure that, as they flex, transforms the patterns that are applied on them.

For instance,

these are two views of the same sides of the a flexagon. The way that the paper folds rotates the sides to create an illusion that you are looking at something entirely different.

It was great fun to watch these teens discover the different transformations of their designs.

These cats were a surprise. Mostly people were doing purely geometric designs. I had no idea how these cat motifs would work out. Just loved how they paired up!

The fellow that did this one, with the black square and the blue and red circles within, has surprised me during each class. He leaves me wondering if he’s going to participate at all and then I look over after awhile and see that he’s done something stunning.

Here;’s a PDF for 11″ x17″ paper. You need just one of these strips of 10 equilateral triangles to make a hexaflexagon

We made the flexagons using a template I created. What is needed is a paper strip which folds into 10 equilateral triangles, so this template I made can be used to make four separate hexagon-flexagons.

For some reason I kept messing up showing the group how to fold. One of the older teenagers, who’s position is counselor, really understood the folding well, so I took a video of her explaining how it goes:

If you still haven’t seen enough, well, I took just a couple more fun photos of the work of this talented group.

So cool!

There was one last project we did during the last half hour together. It was doing some origami, but I had them each cut out a separate rectangular piece of paper. Each rectangle was a different size but proportionally the same. I have a thing about scaling: I want all kids to know how to do it.

After the paper was cut, I walked them through the steps of making an origami toy boat, not because I wanted a toy boat, but because I wanted to stack the different sizes and see what happened. Each person had a different size paper

This is what happened.

It stands on it’s own, and looks kind of like a ziggurat , or maybe it looks like a big hat.

I think we decided it looked like a hat.

After having spent most of the afternoon making flexagons with this group I came home and checked my twitter feed. Coincidentally, seems that my friends had been all atwitter about flexagons, starting with this from Vincent: https://twitter.com/panlepan/status/835988773875892224

Within the thread was a link to Dave Richeson’s template and instructions for what he calls a Cube Tri-Hexaflexagon, but it’s what I’ve been calling the hexaflexagon. I made one of these immediately. It’s a great template.

Ok. It’s nearly time to start planning my next project with this group. Looking forward to it!

Flexagons · Paper Toy

Hexagon-Flexagon: Post #4, Student Work

November 28, 2011November 28, 2011 Paula Beardell Krieg2 Comments

I’m ending this series of posts on hexagon-flexagon with images of work done by students in Michele Gannon’s art classes in the Adirondacks. I taught the students how to do the paper-folding. Michelle’s considerable artistic influence supported the students’ creative output. Each of images that I am showcasing here is followed by the same image in different configuration on the same structure. Notice how the patterns change.

This lovely design was created with Prismacolor pencils. Some, but not all, of the Prismacolors show up well on black paper.

This next set shows a really simple design that works really well, as the look of it changes dramatically when the image is flexed.

Ta-Dah!

Since each hexagon-flexagon has three distinct surfaces to design and decorate I brought in a variety of novel tools to help motivate the students to keep working. In the photos below the students used my Chinese Dragon paper punch. Paper punches are a great motivational tool, and the dragon is the most popular of all that I have.

By the way, the students that made these images were about eleven years old.

I like the way the relationship between the dragons changes when the flexagon flexes.

Another tool that makes a big hit with students is Crayon Gel Markers. These markers take about 15 seconds to fully show up on the paper, thus adding a bit a magic to the process of decorating.

I also use some of the Crayon colored pencils. Only some of them work well on black paper. I used to be able to find Crayona FX (?) pencils that worked on black paper, but I haven’t seen them around for awhile.

I have to say that that only reason that I felt comfortable teaching this structure to these students is that I knew I would be working with small groups. I don’t think that I had more than 10 or 12 students in each class. The fact is, the folding for this structure is so specific, that I doubt I could successfully teach it to large classes. But I think it would be a great project to teach to home schoolers, or in an after school art class.

The hexagon below has a completely different look when it’s flexed.

The middle circle totally changes, and that white line becomes a whole different thing, too.

These were so much fun to teach. It’s a real pleasure to watch the expressions on the students’ faces when they see their own design be transformed when the hexagon flexes. That’ s my favorite part of the process, too.

Intro to hexagon-flexagons

Fractions Flexagaon

How to Make a Hexagon-Flexagon

Flexagons · Paper Toy

Forest of Flexagons

April 28, 2011August 8, 2011 Paula Beardell Krieg4 Comments

I don’t often have the chance to make flexagons with a class. But recently, Mrs. Schroeder, who works with an artistic group of fifth graders, asked me to plan a project with these students that could showcase their creative energies while still requiring that they use research skills.

Mrs. Schroeder chose Adirondack trees as the object of research. Each student was assigned their own individual tree. They were then asked to research such things as the tree’s habitat, uses of its wood, characteristics (the look of the bark, shapes of the leaves, etc) and its latin name .

Making flexagons to house these research facts was my idea. Here’s my instructional PDF for Flexagon Squared.

For the uninitiated, a Flexagon is a paper structure whose surfaces rotate in unexpected ways when the folds in the paper are flexed.

There are many shapes and styles of flexagons. The ones pictured here are based on a square.

Ok, I know it doesn’t look like a square, but imagine unfolding the lime green paper….

…now, here it is, half-way unfolded….

…and all the way unfolded. There are still surfaces on this one flexagon that remain unseen.

The French teacher at this school saw this project and was tres impressed by the work of these energetic and talented students. In addition to the fine work done by the fifth graders, I told her that the secret to the success of this project was Mrs. Schroeder. She kept the class on task, and raised the bar high for research, spelling, and presentation. Mrs. S. remained consistently positive with and respectful to the students while staying clear in her intentions that they produce. It was really great observing how responsive these young people were to meeting Mrs. Schroeder’s standards.

My part in this project mostly consisted of providing instruction on how to construct the flexagon, and providing colorful papers and compelling materials to support motivation and inspiration.