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.

Book Art · geometry and paperfolding · origami · Zhen Xian Bao


Threads, a Zhen Xian Bao, Paula B Krieg
Threads, a Zhen Xian Bao, Paula B Krieg

I just sent this piece out to be in a show in Massachusetts. Included with the piece is an invitation for the book to be handled and for the viewers to take a piece of it with them.  As you might suspect, there’s a bit of a condition.

I’ve been making models of this folder of expandable boxes, known as Zhen Xian Bao, for quite some time. I’ve been so busy deciphering the structure and creating designs for the papers that I make them out of that I haven’t thought too much about what to put into these boxes, which. traditionally, were used to store thread.

Here’s the chronology of thought then. First structure, then embellishment, now content. Finally I’m ready to think about content, now that I am satisfied with some of the solutions to my first and second considerations.

Here’s what I’ve put in the boxes:

There’s about 64 paper tiles stored in the various boxes of this structure. Each tile is threaded with a loop. The back of each tile has words or phrases that I repeat to myself, the threads of thought that help me get through my days.

I had wondered if I would be able to come up with 64 things that I tell myself, so I asked my a couple of friends for some of their thought threads. I included some from Jocelyn, especially liked “Bring a book,” and Susan’s “Mend a thing.”

Funny thing, though, after I got started, it was easy to come up with scores of things I tell myself.  So many thoughts woven into a day.

Now, here’s a box of blank tiles that I’ve sent along with my work. There’s three of these boxes. They are meant to sit alongside my Zhen Xian Bao. There is also a pencil in each box. I’ve sent word that I am inviting viewers to add one of their thoughts to one of my boxes. Then, after they’ve made their contribution, they are invited to take one of my thoughts with them.


I don’t know how this will work out. As there are tiles in each one of these 13 expandable boxes, I am hoping/anticipating that my Threads book will return with wear and tear showing. I will consider evidence of handling as the finishing touches.


Now it’s out of in the world, out of my hands!

design · geometry and paper · geometry and paperfolding

Pattern by Rotation

Twist Boxes
Two Twist Boxes

A bit of eye candy tonight.

Here’s the birds eye view of two collapsed twist boxes . What’s wild about these is that they are made from nearly the exact piece of paper.

Patterned Paper
Patterned Paper

Here’s the paper they were made from. Can you see the difference between them?

What’s going on is that the pattern is shifted by just a small amount.  This small shift makes a big difference in the pattern that is created by the folds that rotate the design on the paper.

Here’s another fun rotation.

I have four of these little squares. I can put them together so that the design is rotated around a specific corner. Like this:

Rotation #1
Rotation #1

A design pops out! When I rotate the design around another point, another design pop-out.

and another…

and since this square has four corners there is another design for these rotations to make, but this is all for now.

I should be prepping for the Center For Book Arts class that I am teaching on Thursday and Friday, on how to make Zhen Xian Bao.,,

Good thing I’ve got tomorrow. Have been spending way to much time rotating things.


geometry and paper · geometry and paperfolding · Glue · Math and Book Arts

Sunday Night in the Studio

Building a dodecahedron
In Progress

I’ve been juggling a number of projects lately, working out different problems, nothing yet fully resolved, so this is going to be a post that shows a bit of this and that. As I’m working little things that I want to share keep coming up, so here’s a glimpse at what’s going on here.

Printing paper
Making Copies

One thing that’s going on in the background is printing papers for a workshop that I will be teaching in a couple of weeks. I will need to make a few hundred copies, and I don’t want to leave this until the last minute, nor do I want to be fully out of the room as the copies come out of the machine. Doing these a few at a time

Cutting a Net

The main event of the evening is making a black 12-sided shape, specifically a dodecahedron. I made this template in Illustrator so that it will have certain details that I want, such as a door that allows access to the inside of the completed shape. The template has to be printed on a light colored paper.

I clip my black paper to the yellow with binder’s clips. First step is to press in the score lines, making sure that there is a surface that gives under the pressure of the stylus. For the stylus I use a glitter pen, because I like using glitter pens. Next step is to cut out the shape.

Black Paper Shape
Platonic Solid in progress

Here’s how it looks with cuts and score lines made.

Next I will be gluing colorful pentagons to the black paper. Pentagons were colored with Sharpies, copies were printed at the copy shop, and then I cut them out.

Blackening the Edges
Blackening edges

My wise friend Jane recommended that I go over the edges of the cut out pentagons, using an India Ink based marker. This is a Coptic Brush Multiliner.

Books, Boses, and Portfolios by Franz Zeier, Pages 86,87
Books, Boxes, and Portfolios by Franz Zeier, Pages 86,87

I read this great book by Franz Zeier a few years ago, and, ever since, I’ve been trying to bring my glueing skills up a notch. He is using straight PVA to glue his models together, which is what I am doing as well. Since my grain directions won’t be lining up perfectly I like using the straight, unthinned PVA, as it has so little moisture in it, and dries so quickly.  I have given up on using a brush with this project, as the glue dries quickly on my brushes, too.

My Modest Glue Station
My Modest Glue Station

I have found that it’s neater to use stiff piece of paper to do my gluing. First I pour a bit of glue in the bowl then dip an edge of my paper scrap into the glue.

Putting Glue on a Pentagon
Putting Glue on a Pentagon

I can put a really thin layer of glue down quickly, which is exactly what I want.

I love the look of the paper once the pentagons are glued down. Sort of hate to keep going.

I do keep going though. Such a friendly looking Platonic solid.

Now that’s it for tonight. Glad I don’t have to clean any brushes.