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.
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.





geometry and paper · holiday project · Ornament · Paper Ornament

Paper Sphericon, a shape that wobbles



Just in time for the holidays, a shape that you probably never heard of. Maybe that’s just as well. If you want something that’s kind of awesome & easy to make, you are in the wrong place (go here instead). This shape is awesome and endearing, but takes more finessing than is comfortable for the average bear. I do not recommend making these with children. Or adults. Proceed at your own risk.

I was introduced to this shape by Vincent Pantaloni, who has a knack for distraction. Near the bottom of this post I will link some of our sphericon-related twitter threads.

Sphericons, nets and shapes
Sphericons, nets and shapes

What’s most endearing about this shape is that it wobbles as it rolls. You have to give it a flick right near one of its edges and it will roll like a drunken sphere. Its net (the flattened out version of the shape) reminds me of a duckbill platypus. I bet they wobble too.

If you have a template you can probably figure out how to make one of these without any help from me. Here are the templates, in black and white, in color, and in two different sizes. I’m pretty sure these will work with A3 and A4. (Let me know if I’m wrong about this!)

Spericon color 8.5 x 11 A4

Spherhicon B&W 8.5 x 11 A4

Large Sphericon B&W 11 x 17 A3

Large Sphericon Color 11 x 17 A3

After you fail a few times you can watch my video to see me struggle through making one. I do have a few good tips to offer.

Score and fold the curves and the edges
Score and fold the curves and the edges

I recommend printing these on heavier papers than standard copy papers. They don’t do their rolly-wobble really well when made from lighter papers. Really, who would want a non-rolly sphericon? I use 67lb cover paper.

Once you cut out the shape, score the curves and the straight lines within the net.

Adhesives matter
Adhesives matter

To get everything to stick together I recommend using some kind of double-sided adhesive.

I’ve tried regular tape and white glue and glue sticks: I do not recommend using regular tape because it messes up the rolling edge. I do not recommend using white glue or glue sticks because I have to hold everything together while the glue dries and this takes too long.

I had some big glue dots around. I like the way they worked, especially as I could stretch them over a larger area that one would expect. When I ran out of glue dots I discovered adhesive LINES. Very cool.

Stretching Adhesive LInes
Stretching Adhesive LInes

See the the adhesive line stretching in the photo above?

I put the adhesive on all the glue surfaces before I actually try to adhere anything to anything else.

The first sticking
The first sticking

I find that it’s best to stick the long tab to the one straight edge before doing anything else. You will likely disagree with me and try to glue one set of the teeth first, then the other set, then do the tab. Then you will realize that this was a mistake. Oh well. Told you so.

So Adorable
So Adorable

I think these look so adorable at this stage that I had to post both of these photos. Now you just have to somehow get those teethy things to stick to the inside edge of that arc (which, according to Vincent, is about about 127.3 degrees of a circle. About)

Getting these to stick together perfectly is just not possible. But good enough is actually good enough. You have to do a 3D print for perfection.

Here’s a video of me struggling through making this. It’s worthwhile to watch but there are some reaaaalllllly boring stretches. The whitish on the bottom of the video never goes away. Sorry.


Here are are some interesting twitter threads to look at. Click on them then scroll up.