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.
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.
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.
This is officially way cool! The paper sticks are very sturdy. Although the seams came undone on a few, but they still retain their shape. I just used fuzzy sticks (aka pipe cleaners) to attach at the vertices. Easy way to explore polyhedra with basic materials! #playwithmathpic.twitter.com/jcuf55EFxR
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.
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.
Its name enchanted me. I had this moment of deciding whether or not to follow this curiosity. It seemed silly, even to me, to dedicate a good bit of time to a shape just because of its name. While I was considering just ignoring it I felt an arthritis-like pain inflame my left thumb. Arthritis is not a thing for me, but for a moment it was like I was being reminded that there will come a time that if I want to hold my own gyrobifastigium that it will likely be hard to impossible to get someone to help me with it. No putting it off. Can’t trust the future for this. The exploration began.
Gyrobifastigium-Net (note: that web address brings you right back to this blog page)
It took quite a number of tries to settle on a net that suited me. It was a tricky structure to glue together so my net went through a number of unanticipated changes. Then I had to find paper that was just right. Some papers were too thin, others too thick. Couldn’t find one that was just right. Finally decided to use one that was more weighty than regular copy paper, but not a cover weight paper. Used ARJOWIGGINS KEAYKOLOUR VELLUM SEAL 80T 27.5X39.3 184M (120 GSM 700X1000), which was too light to make the feel of what I wanted, but that was a solvable problem.
View of Gyrobifastigia
These shapes have eight faces. I glued extra paper to half of the faces. This gave me the feel that I wanted. Naturally, I had to color in lots and lots of squares and triangles so I could choose what to use.
I colored four times as many units than I actually used. This sounds reasonable to me.
A Gaggle of Gyrobifastigiums
One of the many interesting things about this shape is that it tiles space, meaning they can all fit together to fill a space without any space between them. Think of a cube or a brick: they can completely fill up a space without leaving gaps. As this space-packing thing is big deal, it became evident early on that I needed to make many of them so I could check out how they fit together. I didn’t go overboard with this. Made somewhere in the area of twenty of them.
The crazy thing is that these gyrobifastigiums aren’t particularly cooperative when it comes to building. Not only that, but the shape of the space they fill is completely defined by the fact that it is this shape that is filling it. I know I didn’t say that in a way that is easily digestable, but think about it. Don’t let me be alone in this space.
I could nudge these shapes into some alternate fun kinds of patterns that have nothing to do with tessellating space, but, still, the shapes are surprisingly uncooperative.
I am shipping two or three artworks off to Dana Hall in Massachusetts this week, for a December show centered around women who are involved in both math and art. They don’t know it yet, but, along with other work, I am sending them these 12 gyrobifastigia to have as a hands-on, build-with-them-yourself piece that students can engage with. I am not going to worry about them getting damaged, not when I can give students the opportunity, the only opportunity that they may ever have, to play with gyrobifastigiums themselves.
Playful Bookbinding and Paper Works 