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.
folding · Paper Toy

Six-fold, flat-fold, Paper-fold

Paper Folding the Ferozkah Jaali
Paper Folding the Ferozkah Jaali

I found a fold.

If paperfolding graps your attention, prepare to be overwhelmed.  There’s three things to unpack here: the fold, the pattern on the fold, and how they interact.

I had been wondering if I could fold a tetrahedron out of a rectangle.

Tetrahedrons and other shapes
tetrahedrons and other shapes

Turns out, yes. I can make a tetrahedron with a square base or a triangular base out of the same piece of paper using the same folds in different ways.

Looks like a fish
Looks like a fish

Then I started seeing that I could make other shapes out of the same pieces of paper using the same folds differently.

Some shapes are flat, others are dimensional.

I’ve been playing with these all week, and I am still finding different shapes that these folds create.


I’ve also been drawing this six-fold pattern from Islamic Geometry called the Ferozkoh Jaali. It occurred to me that it would go perfectly with the folds I was making.

detail of Ferozkah Jaali
detail of Ferozkah Jaali


This is just a small portion of the pattern. I’ve been coloring copies of these in all week, trying to get to know the shapes.

Here’s the fold that I’m using:


Mountain and Valley folds
Mountain and Valley folds

It’s four mountain folds (diagonals) and two valley folds (horizontal and vertical) that are created around equilateral triangles. Oh, and there’s a slice in the middle. One horizontal slice.

Now here’s the first wonderful thing about using this image with my folds:

No matter how you use the creases (which are around the equilateral triangles) , the pattern lines up. In the photo above, a corner is peeking through that slice in the paper, and, look, the pattern lines up.

Equilateral triangle(s)
Equilateral triangle(s)

I printed the design on the fronts and backs of my papers, and look, when the paper wraps around itself, the pattern lines up.

Now there is one more thing to mention. Hold on to your seats. This is wonderful. But, first, here’s the foundation of the image I created, first by hand, then on the computer, because I needed the precision of the computer image.


Okay, so as I’ve been folding and refolding and refolding again, and finding different shapes all the time, the last final amazing thing that I noticed (and this makes so much sense) ….

Some heart shapes?
Some heart shapes?

…is that every shape I make with these folds is echoed somewhere in the lines of the  geometric drawing that is printed on the paper.

This makes me so happy, well, I can’t even describe it.

Another heart shape
Another heart shape

Well, there you have it. Hope you love it as much as I do.

covered with NOT geometry
covered with NOT geometry

Oh, and just in case you’re wondering, I think this fold looks good with just about anything on it.