an open source architectural beadwork project from Kate McKinnon and a worldwide team of innovators

The CGB Pre-Releases: The Kaleidocycle

Whew! It was fun, publishing our Mirror Tetras version of the Kaleidocycle to the pre-order list. Did you pre-order any of our upcoming books, and if so, get yours? If not, please email us at: and let us know.  If you ordered, you are definitely on our list, but you may need to add the above email address to your Contacts list, or update your email address in our database before we can reach you effectively.

Our email of last week featured our Primary Cycle, a wonder done in black and white beads, with yellow and green and blue and red calling out the Hinges and Edge Beads.

We have kits for this Cycle for $60 in the CGB shop, if you’d like to make it easy.
Order yours here!

Dots Face CGB Primary Cycle web

There are a lot of ways to build a cycle, and as far as we can tell, most people who have made one have followed in the footsteps of Susannah Thomson, who beaded one in Flat Peyote Triangles and called it a Hexaflexagon.
(Which it would have been, if a hexaflexagon grew up to be dimensional. As it turned out, it was a Kaleidocycle, done in size 10 Delica beads.)

Cath Thomas had done one earlier and shown it to me, but it was in full lozenge shapes rather than the easily-accessible Triangle and it was more of a beautiful anomaly to me, a one-off done out of curiosity and love. She crafted it after an origami folding cycle. It didn’t occur to me to try to re-engineer it in triangles, but looking at the paper model of it now, I can’t think why not. We weren’t ready, perhaps.

When Susannah made hers, our whole team was mesmerized by the idea of being able to do it in peyote Triangles; anyone could make those. When we began investigating the mechanism, though, immediately we wanted to make a hinged version.
We didn’t start with the Nets.

After all, we reasoned, the Origami Folding Net was meant for paper folding. If we were engineers, using building materials, we’d know the opposing-hinge tetra chain as a Bricard Linkage, after the mathematician Raoul Bricard, who wrote about them at the turn of the last century. Studying the linkages convinced us that the best engineering build would be from Tetrahedra, which is where the Cycle ends up anyway. The question is really this: do you want them sharply tailored and hinged, or softly folded?

I experimented with using Pyramids instead of Tetras, and also what I call Lab-Grown Tetras, which are just Pyramids with triangular bottoms. It will all be in the book. Anyway: Bricard Linkages! We had our teeth in the science of the thing now.

Most people know Kaleidocycles because of a popular craft book published in the 1980s, celebrating the work of M.C. Escher. Wallace Walker, a graphic artist, had tumbled onto the linkage in origami, and named it a “Kaleidocycle.” He teamed with mathematician Doris Schattschneider to translate a stack of Escher patterns onto origami folding nets for the Bricard Linkage, and a book was born.

Despite the fact that Bricard Linkages were already well known in mathematics and engineering (but presumably not to Wallace Walker) he actually got a patent for the Kaleidocycle (also presumably because he gave it a new name and classed it as a toy?)

Too bad for Bricard, but I don’t think anyone really noticed. Math and engineering continue to pursue the linkages, though, and you can see them in architecture, virus chains, and crystals.  Here is a paper from Cambridge, 2015, just an example.

As it happens, Pat Verrier and I are writing our own science paper about this so we can officially connect the engineering, the math, the computational origami, the paper folding, the architecture, the beadwork and the topology. Few of these groups seem to be recognizing the other, but we plan to fix it.

Screen Shot 2017-04-23 at 4.12.14 PM.png

The Cycle form is so compelling (and Escher is so timeless) that the book from the ’80s endures,  and was even picked up by Taschen (above).

How did we get started beading them? My story, and I’m sticking to it, is that Susannah Thomson saw a paper piece made from that book, and I saw what she made, and I published it on the Contemporary Geometric Beadwork Facebook page.

As we didn’t want to release a pattern without studying the form, a lot of bead groups excitedly formed to make it on their own. Hundreds, if not thousands, of folding net cycles were born, along with a smaller wave of Tetra-based models from our extended team. There was a lot to know about bead size, type, and hinging.

We were pretty sure that hinged Mirror Tetrahedra were the way to go, and we took the time to study the problem. Along the way, we made a few Folding Nets, of course. My favorite is the one below by Dustin and Kim, which I love not only for its beauty but also and especially because it shows the mirror Triangle faces in three neat groups of two instead of splitting the third set in the confusing way the paper folding patterns usually do. In this one, you can SEE the Tetrahedra in flat-pack

This particular Net Cycle is also pleasing because the Join Beads are all 15° rocailles, which make for the best hinges EVER.

Screen Shot 2017-04-06 at 8.49.58 AM

Above, a glorious Kaleidocycle from an Origami Folding Net, with 15° rocaille Hinges, by Dustin Wedekind and Kim Van Antwerp.

CGB Rasta Machine Poppies face web.png

This machine, and its many implications, sent our whole team down a year-long rabbit hole. We made hinges and cycles and Tetras and nets and more Tetras and more hinges and more Triangles and we asked people to send in their Cycles from all over the world so we could study and photograph them. I still have a good handful of them, and am so grateful to everyone who has them in on loan.

Franklin Martin cycle web

Above, a cycle from an origami folding net by Franklin Martin, Jr., and below, an early Mirror Tetra cycle from Kim Van Antwerp.

Screen Shot 2016-03-22 at 6.44.15 PM

You can fairly easily see the difference in construction. Each are beautiful, and we show both techniques in the full book chapter (in addition to a few other neat ways to do it.)

The PDF that we sent out to pre-orders this past week is an edit proof, so don’t worry if you see wonky page numbers or blue text or anything else. It’s all part of the magical journey, and you are counting on us to know what we’re up to.

Remember, if you pre-ordered, the only way you can make sure our emails get through to you is to add the email address of to your Contacts list. Email is an imperfect science, as are blog posts.

More soon! So much more.

About katemckinnon

Kate McKinnon has devoted herself to the study of how things are done, and how they could be done better. Find her at or on some city street, walking fast, smiling at strangers.

7 comments on “The CGB Pre-Releases: The Kaleidocycle

  1. rene lindstrom
    June 16, 2017

    I LUV your kaleidocycle!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
    It would be very helpfull if you would provide blank graphs for the triangles with several on a page. I’ve hunted on the internet. Some aren’t dark enough and others have too many rows and have to enlarge.
    THANK YOU!!!!!!!!!! Rene

  2. Sydney Duenas
    May 5, 2017

    I made my own kaleidocycle after discovering your blog a couple months ago, and have been itching to make a second one.

    I used the same size cylinder beads for the hinges, as seen in these photos:

    View this post on Instagram

    Hey look, a wild kiwi fruit appears

    A post shared by Sydney Dueñas 🌺 (@lostaloha) on

    From what I’m gathering, you think it’s better to use size 15 rocailles for the hinges? I haven’t pre-ordered the book yet, so I don’t have your pattern. I kind of went rogue and just made triangles and joined them together.

    Thanks so much for all the research and information you put into your work and this blog! It’s reignited my passion for beadweaving 🙂

    • katemckinnon
      May 5, 2017

      Hi! Thanks for writing in, your comments are great to hear.
      Yes, we just love how easily the 15 rocailles move as hinge beads, whether in a Folding Net or as actual door-hinge-style hinges.

  3. Cath
    April 24, 2017

    I tried with triangles, but it didn’t work well to have “material” at the joins, and also not well to draw images. So I abandoned… because there were other designs that wanted to be created through my hands.

    I love what happened after. We can ask ourselves many times why we didn’t think of “this or that” before. I think that when the Universe wants something it will push people to play with shapes and ideas, but people do not always listen (me. for example :D), and others do not always listen, or are simply too busy with other things… until finally the Universe finds a person who gets it.

    The whole CGB adventure shows beautifully that it takes the effort of more than one inquiring mind and plenty of little hand to finally achieve something that really works.

  4. Kit Hardy
    April 24, 2017

    Thanks Kate X

    Sent from my iPad

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