Piling Kaleidocycles

Piling Kaleidocycles

They did it! This group of fifth grades did this hand-lettering kaleidocycle class project! I described the details of this project a few posts back so check out that post for more details. Here’s the general gist: After introducing the project, which is a 3D paper construction with rotating faces that will be graced with references to the Bill of Rights, students were given pages of letter fonts to choose from.


Using the windows of the library as light boxes, students traced out letters to created one phrase each that described one of the first ten constitutional amendments, aka The Bill of Rights.

At the window

Every single student was highly engaged. Really.

Within two class sessions the students produced something that I could take home and scan into my computer . I won’t lie..scanning and cleaning up their work took time. Above you can compare what they gave me, on the left, to what I ended up with on the right. Some pages required much more work than others. The middle example above was so easy to work with that next time I will encourage students to just give me outlines. The most time-consuming letters to work with were those that were colored in and touching other letters. I moved things around a bit, like in the top example you can see I centered the word “OF.”

I brought home their work, scanned them into my graphics program, cleaned them up and laid them into a kaleidocylce template. Brought them back for students to cut around the perimeter and score.

Scoring the paper

Scoring the paper

Students made score lines so that the paper would fold easily and accurately. Scoring is generally done with bone folders but we used glitter pens to score the lines. They worked great, and kids were excited to be using the gel pens.

Assembling the Kaleidocylce

Then came the folding and gluing. I didn’t take many pictures of this process as I was, like, really really occupied helping move this process along.

One of four faces of the Kaleidocycle

One of four faces of the Kaleidocycle

This project turned out so well. Not everyone had a chance to finish up and decorate, but the wonderful school librarian will be able to help with the few than still need finishing.

Kaleidocycle yellow borders

Kaleidocycle yellow borders

Students enjoyed individualizing their own kaleidocycles.

Another side

Another side

I tried to get them to use completely different color schemes on each face, so that the differences between the four rotation of faces were dramatic. Students didn’t much listen to my suggestions.

Here’s one of their kaleidocycles in action:

I consider this project a great success. I got to talk to the students about design, about hand lettering, and they got to work with some cool geometry. I’d even go so far as to say that they are also much more familiar with the Bill of Rights , as they were constantly asking each other, which one do you have, which one is yours, and talking about their own.  I have to say that at first the students were confused about what I was asking them to do, after which the librarian told me that doing a group project was pretty much out of their experience, so the concept was hard to grasp at first.

One thing that made this possible was that this was a small class, just 12 students. I often work with 60 to 70 students in a grade level: I wouldn’t do this project with a big group. OH, but it was so delightful doing this with a small group.

Tower of Kaleidocycles

Tower of Kaleidocycles

Do I get to pick a favorite project of my teaching season? Yes? This is it.

Kaleidocycles and Tetrahedrons

February 19, 2017


I looked up the definition of a tetrahedron today, I figured out how to spell kaleidocycle a few hours ago. Just saying.

kaleidocycle-colors-1 Paula Krieg

Pattern for one kind of Kaleidocycle

Sometimes an exploration pursues me. It’s always a gift to be preyed upon by ideas, but if my desk is already full and messy, and I think I can’t bear adding one more layer I pretend to kind of ignore the newcomer. No, this strategy doesn’t work.

I didn’t know that tetrahedrons were following me around. Like I said, just this morning I finally looked up the definition (a solid having four plane triangular faces; a triangular pyramid).

The image above is where this all started. This is not such a startling set of pictures until you know that the image shows the same, unchanged structure viewed from front and back. It’s on the facebook page of someone whose name is written in an alphabet I don’t understand. This is the link to the page on facebook https://www.facebook.com/artsmathematics/videos/718044448365422/ . Take a look if you can. It’s such an amazing bit of transformation, which I have yet to figure out how to do. What’s going on here is that this structure to made up of connected 3D shapes that rotate together to reveal different surfaces. It’s very tricky and fun to see the shapes turn, revealing new surfaces.

The next piece of this story is that a teacher just a bit south of me in Upstate NY posted some directions on how to build a certain  geometric shape, and he asked, via twitter, if anyone would be able to test drive his tutorial. It looked simple enough to me, so I thought I’d try it out the following Saturday morning. I thought it would take about 2o minutes. Ha ha.



Looking back, I think if this teacher, Mr.Kaercher, had done a tutorial on a simple tetrahedron it might have gone more quickly and I might have finished up knowing what a tetrahedron was. But, no, Mr. K provided directions for a tensegrity tetrahedron, and since I didn’t have much of a clue about the definition of either term, I didn’t really have much of an idea of what I was doing.

Tensegrity Tetrahedron

Tensegrity Tetrahedron

Even so, after a megillah of failures, I got it done and was quite pleased with myself.

In the meantime I was still thinking about those images from that facebook page.

I showed the FB clip to book artist Ed Hutchins. He told me that what I was looking at was a type of kaleidocycle.

Oh, and Ed just happened to have a hot-off-the-presses copy of what is probably the world’s most amazing example of a hexagonal kaleidocycle, designed by Simon Arizpe. (This is a fully funded Kickstarter Project, which you can view to see the book in motion.)

This structure tells a story as it rotates. Since these rotating sides can turn forwards or backwards, the sequence of the story is determined by the direction the viewer rotates the kaleidocycle. The way that I choose to turn it, it begins with a bear peeking out at a stream…


…the bear opens his mouth, a salmon jumps out…

The Wild by Simone Arizpe

The Wild by Simone Arizpe

The Wild by Simon Arizpe

The Wild by Simon Arizpe

… and then the salmon jumps into the river. There’s one more frame, but I’m not going to be a spoiler and show it to you.

So what does this have to do with tetrahedrons? I’m getting there.

As it turns out, the last couple of times I’ve gone lurking at the Lawler family math page, they’ve been looking at, yes, tetrahedrons.

This shape that the Lawler’s were considering was beginning to look familiar to me. Part of the reason for this was that, ever since Ed had given me the gift of the term kaleidocycle I had been Googling around then assembling kaleidocycles.

Kaleidocycle, unhinged

Kaleidocycle, unhinged

Here’s one of my first attempts. Notice that I forgot to attach the ends together before I closed things up. This turned out to be a good thing, because, wait! these shapes appear to be repeated echoes of the shape that the Lawler family was exploring.

Just to pile it on, it certainly helped that just yesterday a package came in the mail, all the way from France, from Simon Gregg. In the package was, can you guess?… a tetrahedron.

Tetrahedron from Simon Gregg

Tetrahedron from Simon Gregg

That Saturday a few week ago that I tried, time after time, to create my tensegrity tetrahedron, I had been posting my failures publicly on twitter. I imagine that Simon thought that it might be merciful to send me some bamboo, as the straws that I was using would sometimes collapse. Included with the bamboo rods, Simon also gave me a collapsible tetrahedron, held together by stretchy cord.

With all of these pieces floating around me it,  I finally made the connection that units of kaleidocycles are series of tetrahedrons. 



So cool.

Now to reward you for making it all the way to the end of this post, here is a pattern for a kaleidocycle that you can make yourself.


click the image to get the printable PDF

Just cut it out, use it alone or attach it to the one near the top of this post, but, in either case, do  make sure you attach ends to make it circular. Here’s a pleasant little video to show you how it’s put together.

I still intend to figure out how to make the kaleidocycle that I saw on FB. When I do sit down and try it out, at least, now, I feel like I’m starting with some helpful understandings.

I have no big attachment to figuring it out for myself, so if you are inspired to decipher it, please let me in on its secrets!


That’s it for now. Thanks for staying with me through these meanderings.

Addendum 2/20/2017

truncated tensegrity tertrahedron

truncated tensegrity tertrahedron

Used bamboo sticks with bobby pins in the ends to make another one of the Mark Kaercher project.  The bamboo worked out great! If I was to make this again with straws, I think I’d try to first put stirrers, like what Starbucks provides to stir coffee, inside the straws. But love the bamboo!




Although I intend the title of this post to refer to what I’ve been messing around with this weekend, I’m not really sure it means anything. What’s been happening in my studio is that I’ve wanted to mix up some interesting lines with some interesting folds.

After a person with the handle of GHS Maths posted a group of images made by rotating the graph of a trig equation I got it into my head to see what one of them would look like on a hexagon-flexagon.

If you don’t know what a hexagon-flexagon is, you haven’t watched enough Vy Hart videos. In 2012 Vy offered her own utterly delightful interpretations of what she thought people should know about this piece of paper wizardry in Hexaflexagon, (6 MILLION views!), Hexaflexagon 2, and the sequel Hexaflexagon Safety Guide .

 A Hexagon-Flexagon has three distinct sides, which results in six distinct designs: I’ve written about these here, here, and showcased student work here. I haven’t thought about these in a while, but it seemed to me that the image at the top of this post, and others that I had been working with lately, might be interesting to put on a hexaflexagon. 


I had  ideas for all sorts of images but I became so enchanted by what the variations of the image above that in the end this is what I went with.


My computer did not like this idea at all. I spent half my weekend redoing what I lost when my program crashed, half my weekend watching that blue swirly thing going around, and half my weekend coddling my computer so it wouldn’t crash. I know that I’ve listed three halves, so if that bothers you, here’s what I did with  fourth half: I was able to actually make an image that became a hexaflexagon.

This is what a hexagon-flexagon looks like on two sides of copy paper. I always want things to fit onto standard copy paper, so I had made this template: hexagonflexagontemplatecolor


It’s a bit tricky to follow, but it actually works really well. I love being able to print these up on my little printer.

This is a dynamic structure, that is not easily appreciated in still photos. I am going to either get my son to make a video of me working the structure or will post the more appropriate stills that I can come up with. Tomorrow. Edited into this post. See you then.

Update: I made a quick video! My first one! Based on the image in this PDF which is printed on both sides of the paper.

UPdate #2

For a hexaflexagon template that has a snowflake, a Christmas wreath and a Star of David, visit Chalkdust magazine at http://chalkdustmagazine.com/blog/how-to-make-christmas-special/. You’ll find a link there, and now here too, for Martin Gardner’s famous article on Hexaflexagons.

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.


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.

Related posts:

Intro to hexagon-flexagons

Fractions Flexagaon

How to Make a Hexagon-Flexagon

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