Book, peeking out of its box by Paula Beardell Krieg

Book, peeking out of its box

When Miriam Schaer was assembling her teaching collection to send to Telavi University in the Republic of Georgia, I very much wanted to contribute, but nothing I had on hand seemed right. In the nick of time, some thoughts came colliding together. Polygon Fractal book by Paula Beardell Krieg

This structure started out with an exploration of a shape which I wrote about a few weeks ago after watching family math video made by the Lawlers.

Inside Outside Book by Paula Beardell KriegThe book  opens in an accordion-like fashion, but front and back are structurally different.

Polygon Fractal book by Paula Beardell KriegThe colorful pages rotate open to create these double layered corners. The polygon fractals on the pages here are harvested from Dan Anderson’s openprocessing page then toyed with in Photoshop.

To see the fractals in their full radiant radial symmetry one must rotate the book. There are six completely different images to be seen. But it gets more interesting, because there is a whole other side to see.

The folds of those double layered corners completely reverse to form a cube!

You can’t imagine how excited I was when I saw this cube emerge from the folds!

This folded structure totally suggested that, whatever I use on it, that it be about the dual nature of….something….a suitcase (no, too obvious), a politician’s statements (ugh, too boring)…actually wanted to use images that didn’t imply any hierarchy, hiding, agendas, or judgement about contrasting inner and outer manifestations.

It was this thinking, about duality but equality of visuals, which led me to using Dan’s code along with the polygon fractals that it creates. So perfect. Code and images are perfectly linked, simply completely different ways of seeing the same thing. You know, like Blonde Brunette Redhead 

Now, I do have a lingering unresolved issue with this book. I’m not thrilled with the paper that I’ve used. It’s 32lb Finch Fine Color Copy paper. It takes color beautifully, folds well, but I’m thinking that the folds might be more prone to tearing than is comfortable. Not sure what else to use…am open for suggestions. Miriam’s copy has been shipped, but I’m still happy to check out different papers to use.

I can’t help but wonder if people will be able to figure the transformation of these  pages without seeing this post or reading the brief explanation I’ve provided on the back page of the book? Dunno.

Oh, and here’s my favorite variation:

Hanging a tea light from a pencil so I can see the inside and outside at the same time.


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 . 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!



Paper folding in the morning

Paper folding in the morning

Yesterday I watched a video that showed the Lawler family looking at shapes.

One of Mike’s sons said he liked the top shape in the image above. You can’t see from the photo, but it’s a full sphere. The image above is only half of the sphere, the other half looks no different than the half that is showing. I’ve played around with constructing foldable versions of shapes that look something like the one above, and I thought I’d be able to make a foldable version of what was on Mike’s screen.

Template for folding and cutting

Template for folding and cutting

I’m not showing all the steps that led to this map of folding and cutting because what I’m most interested in showing here are the wonderful visuals I got to experience along the way of creating the final structure.

Folding and Cuttin

Folding and Cutting

Silly as it may seem, one of my first realizations was that the indoor, nighttime lighting in my workspace was just all wrong for photographing what I was about to fold. Morning light would be best. So I went to bed.

Assembling in the morning light

Assembling in the morning light

Of course I forgot to recharge the battery of my phone camera before I went to bed, so I didn’t get the earliest light.


Hmm, I don’t really want to say much more about this process. I’m just going to post pictures now. (Haven’t had my coffee yet.)






Ok. Time for coffee. Am heading to Rochester today to bring my daughter back to college. Will be thinking about all shapes that this structure made. (Which reminds me of a question someone once asked me, “What, do you just sit around thinking about folding paper?” Well, yeah.)

Zhen Xian Bzo in progress

Zhen Xian Bzo in progress

Last spring, when I saw Elissa Campbell’s posts about the Chinese Thread Books that she was making, they reminded me that this was structure that I had been intending to explore. Elissa expressed, in her posts, that she just couldn’t stop making them. I didn’t understand what she meant by that at the time, but, wow, I get it now.

img_0513Before finishing my posts on how to make this structure I thought I’d take a step away from making the models and put together some of these thread books, experimenting with different papers. It was much harder to make a completed Zhen Xian Bao than I could have ever imagined, but not for the reasons that are obvious.

img_0537My problem was that at each and every step of the constructions I had to stop and admire the way everything looked. The geometry of the  each piece, as well as how they looked together, just blew me away.

Not only that, but, also, I got to use papers that I have never been able to find uses for, like this delicate paper embossed with a spiderweb motif. It’s like I was holding on to it for all these years just so I could use it now. Not only did I like the way it looked, but the spider web pattern seemed to be completely appropriate for a thread book.

img_0528At each step, shapes and patterns emerged. I would have to just pause and look at everything over and over again. This is no way to work if you want to get something finished.

img_0547Even the pieces that I was cutting away to discard caught my attention. It was a bit ridiculous.

img_0555I also got to use this nearly sheer blue, fiber filled Japanese paper that I had stored in my flat files for many year. I love flat files.

img_0554It felt painful to attach the boxes together in the proper way. I wanted to be able to look at all pieces of the thread book at once.

img_0541I took way too many pictures.


One of the hardest decisions I had to make was what to use for the covering material. Traditionally the Zhen Xian Bao is covered with indigo cloth. I happened to have, tucked away in my flat files, a large sheet of sturdy,handmade indigo paper. It wrapped nicely around my boxes. Have I mentioned that I love flat files?


When I was finished I decided I needed to make another thread book immediately. This one was smaller because I had some beautiful orange paper that was 21 inches wide, so I started with 7 inch squares so as not to waste paper.


In fact, I just kept raiding stashes of gorgeous papers that I have collected over the years but could never find suitable projects to use them in. Part of the adventure of making these books is seeing how all these lovely papers work together.


My favorite papers to use were thin, strong, textured papers. All my models that I’ve made in my previous posts were made from smooth, machine-made, standard weight (about 24 lb) papers. I liked using them for demonstrations and practice, but enjoy that more unusual papers for finished pieces.

img_0602As soon as I finished the second Thread Book I wanted to make another. I was thinking of using paper bags from the grocery store, to see how they worked. Actually, I want to try out so many different papers and combinations of papers. This feels like such an adventure. But I stopped here, as I actually have to do other things in my life.

img_0605My final decision about these books is still not made…what to use to keep them closed? I have many ribbons, but they all seem to slick.  For now, I have them secured with this pale lavender ribbon that is water stained in a way that seems to go with the wrappers, but I’m not convinced it’s perfect.  In any case, enough of this for now. I’ll be turning my attention back to the final bits of construction soon. The big tray part of the Zhen Xian Bao is what’s up next. And it’s awesome.

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