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

Noun

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.

Six-fold-geometry
Six-fold-geometry

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.

 

 

 

 

geometry and paper · geometry and paperfolding · Paper Toy

A Cool Adobe-Illustrator Artboards Thing

To be two cubes
To be two cubes

I wanted to transfer this image to a big piece of paper. Way too big for my printer. It’s just under 24 square inches.

One the way to being two cubes
One the way to being two cubes

I made the pattern with the intention that it would fold into two cubes. BTW, I recently learned that the correct term to use here is net:” A pattern that you can cut and fold to make a model of a solid shape. This is a net of a cube.” (quoteth from the internet)

While I was scheming how to break the net into prints that I could piece back together, it occurred to me to just overlap the artboards in Illustrator. Set them up to be negative one inches apart. Here’s a snip of what the Illustrator workspace looked:

Six overlapping artboards in the Adobe Illustrator workspace
Six overlapping artboards in the Adobe Illustrator workspace

All I did, after setting up the six artboards was to overlay my net onto the artboards. No figuring, no scheming, just laid it right on top. Honestly I didn’t know what would happen. Would the overlapped parts not print? Just didn’t know.

Amazing. Everything printed everywhere. What I mean is that the parts of the image that were on the overlap printed on both papers. This made it really easy to piece together. Of course the best use of this technology is to print Happy Birthday banners. But what I did was piece them together, cover the back of the paper with blue crayon, and, using a ballpoint pen, trace over the lines to transfer to my larger paper.

net of the Cubes, cut out

I didn’t take any more photos of the process, but here’s my fully cut out net.

On the way to cubeness

The blue crayon showed up just enough, but what was really great is that the force of the tracing created score lines, making this easy to fold.

Weighted by a train
Weighted by a train

Here’s the cube. Hard to imagine how that image becomes these two two-inch cubes. So I made a video: