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!



4 Responses to “Kaleidocycles and Tetrahedrons”

  1. Steve Morris Says:

    I am thoroughly baffled! Not by you, but by the three-dimensional world.


  2. goldenoj Says:

    Awesome sharing of personal learning. Thanks, Paula!

    Liked by 1 person

  3. […] February 19th, Paula Beardell Krieg published a post called Kaleidocycles and Tetrahedrons and there it was again. The post included information on a fabulous pop-up version by Simon […]


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