folding · Paper Toy

Kaleidocycle, revisited

12 face Kaleidocycle

I take no pleasure in this. It’s not like I want to spend my time working on a kaleidocycle template, but I can’t get away from it while I have this thought in my head.

I, along with everyone else in the world, has made the templates for kaleidocycles a certain way: The attaching edges are always either triangles or rhombuses, which I have always found to be awkward to glue. A couple of nights ago I decided to try out making them using double tabs. It worked so much better. I don’t want to forget this, so I made a video (see end of post.)

Reworking Kaleidocycle template

Here are the PDFs you can print out on medium weight paper, something in the area of 150 gsm of 65 lb.

Full Color designs Kaleidocycle PDF

Black and white Kaleidocycle, make your own designs PDF

IF you don’t have any idea what I’m taking about because the word kaleidocycle is not in your radar, well, that’s all for the better. Just watch sit back and enjoy the video below.

Kaleidocycle with Double Tabs Video

How-to · Paper Toy

Paper Rods: Something from Almost Nothing

At this time of year I’m usually working in a summer programs, trying out new projects with kids without the time constraints of being in classrooms. The projects that kids connect to the most become part of what I do with my arts-in-ed sessions in the schools. Turns out that just because there’s no summer programs during this 2020 season, and there is not much chance I will have arts-in-ed work, that doesn’t mean I’ve stopped thinking about new projects. There are a few that I’m particularly eager to share, which is what this and some future posts will be about.


This exploration started with seeing a project posted by Chuck Stoffle in which Chuck made paper rods (he calls them paper supports) by rolling newspaper around a skewer and securing the roll with tape. I liked what he made so much that I had to try it out, but could I make them without using tape?

I started thinking about how, when glossy catalogs get wet, their pages stick together and thought that maybe this could be a tapeless way to make the rods.  Chuck’s method of using tape has the advantage of being able to use the rods immediately, whereas my tapeless method requires overnight drying time, but, hey, I’ve got time.

Here’s how it goes,

Make a 1-1/2 inch fold on the long edge of a page, then fold that in half, and repeat two more times, then start rolling


I start with one of the catalogs that are always showing up in my mailbox, looking for one with glossy pages (uh, they all have glossy pages), but also is not too thin or too thick, and also is colorful on the edges.. Turns out that the Lands End catalog gave me the results I liked the best, which is fortunate as they show up at my house frequently.

Here’s the work flow:  take out the staples, cut each page in half along the center line, then fold up a 1-1/2″ flap on the one of the long edges. Next fold the flap in half, then fold that in half again, and finally fold that last flap in half a fourth time. This last fold is quite tiny. Then start rolling.

Here’s a video of how it looks:

After the paper is rolled up, give it a shower right under a water faucet.

Choosing pages thoughtfully results in rods that are quite lovely.

Now this is where I really miss having groups of kids to play with. What I would like to do is to just hand the rods over to kids and watch what they do with them.

Fortunately my friend Mark Kaercher is a person who is like a group of kids. After we talked about this over Zoom he made a bunch, and figured out that he could use sections of pipe cleaners as connectors.

I really like the way that the pipe cleaners worked to connect the rods!

One of the challenges I made for myself was to connect only three rods together, tripod-like, then see how many more I could add just using gravity.

Or what about building something over a tomato?

I, uh, think a group of kids would have done something more interesting than what I came up with using the tomato.


What about purely linear arrangements?

Or photographing a 3D structure a from above?

This photo is the aerial view of the second photo in this post. Oh, don’t scroll back, here it is again:

This structure has a few pipe cleaner attachments.

If there are no pipe cleaners in your life there’s lots of ways to improvise: I leave that to you.

Now all I need is a group of kids to play with….


geometry and paperfolding · Paper Toy

Flexagon 2020

I’m ushering in the new decade with a new family of flexagons.

The first flexagons originated from the fiddley hands of Ph.D. mathematics student Arthur H Stone in 1939. What he discovered was ways to fold paper so that it could flex to reveal hidden faces.

Martin Gardner popularized flexagons in the 1950’s, and Vy Hart made them totally adorable with her videos, which were made during this past decade. There are likely an uncountable number of flexagon configurations just waiting to be discovered. Ann Schwartz , who I met this past summer at MoMath’s paper-folding conference, and whose folded discoveries include a 12-angle flexagon, has told me that she thinks that this one that I’ve made is something new.

My flexagon has a great deal in common with Octaflexagons and Tetraflexagons in that all of these are have square faces embedded in them, and the octaflexes, like mine, are full of isosceles triangles.

Some of the differences between my flexagon and the others is that mine has pockets and fins. It’s also constructed from a different shape than other flexagons, which generally depend on strips on paper. This flexagon starts with a square.

I created these graphically partitioned squares with the idea in mind that I wanted the various surfaces of my flexagon to be recognizable distinct.

Like it’s easy to see that the surfaces above are completely different from the owl-like face below.

Static photos are not the best way to view flexagons. Videos are much better. Here’s the video.

I’m saying that my flexagon is part of a family of flexagons because I’ve realized that if I make slightly different decisions in the constructions of these flexagons that different variations, which have their own distinct characteristics, emerge. There are at least three more variations in this family. I’m looking forward to sharing everything about them in this coming year.

I’ve done a bit of production-making with these. Just made 20 of them. Most of what I’ve made are spoken for but I have 9 that I’m selling on Etsy. Why nine? I finally ran out of my stash 11″ x 17″ Strathmore 25% cotton writing paper that these are printed on. These 9 flexagons that’s I’m selling will be the last of the ones that are made in 2019, and are signed and dated.

These have been great to have all over my desk, but now they need a new home. Etsy.

Art and Math · geometry and paper · Paper Toy



gyrobifastigium (plural gyrobifastigia or gyrobifastigiums)

  1. A polyhedral solid formed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is noted for being one of the few regular polyhedra that packs in three dimensions.