folding · origami

Refiguring a Fold

About 7 million people and I have watched Jo Nakashima’s Orgiami Fireworks (Yami Yamauchi)tutorial video of how to use 12 pieces of paper to make an exquisite, kaleidoscopic rotating object. I’ve made this object with teenagers and with teachers at a conference. It’s great doing this with a group because then I don’t have to fold the pieces myself. Even so, it has been a frustrating project to complete as assembling it has not gone as easily for me as it does for Jo Nakashima.

This past Easter weekend, hunkered down in isolation mode, feeling particularly blue, I decided to sit in my chair for three days and just draw and fold paper. My goal with the paper folding was to figure out how to streamline making this Fireworks model, even if it meant using glue, which, happily, I did not have to resort to.

BTW it took me more than three days to get where I wanted to be with this. It was worth it.

It  took making about 15 different models, using all sorts of different papers, different techniques, different workflows before I figured out how to refigure the folds to get something I like, that works well, and that I enjoy making.

What I’m doing is making the origami firework from one long piece of paper that is colored on both sides, instead 12 pieces of paper, so it’s not exactly like the Yami Yamauchi model, but otherwise it’s the same folds.

I’m going to go over what I did right here, but I’m guessing that most people won’t be able to figure it out from this, not unless you’ve got some paper folding skills already. Like, I have no doubt that my friend Catheryn at Byopia Press has the skills to make this, and my friend Mark K has the tenacity to work it out, but this certainly isn’t a casual folding activity.

First thing is to accordion fold a long piece of paper. Specifically, the paper needs to be 6.5 times long as it is wide. Or it needs to be 8 times as long as it is wide, which is easier to make into an accordion, but then a few sections have to be cut away.

To make this kind of accordion you need to have a super power….or you can watch the video below.

Addendum: What’s not mentioned in the video, which I figured out later, is that, after you make the accordion, reverse half of the folds so that they are all valley folds, so that the gold is in the valley.

Next, make this wafflely pattern on the paper… (All of these X folds should be mountain folds.)

…which can squish down like like this:

If you look at in just the right way it looks like a dragon,

but I digress. The whole things should be squished, including that flattish tail above.

Now it’s time to make this dragon consume its tail, creating a circle.

One end slips over the other.

It may not immediately seem obvious how to do this, but there’s not a lot of ways it can go, so I trust you to figure this out.

This next bit is familiar to anyone who makes v-fold pop-ups. Make isosceles triangle folds on the edges then….

…invert the V fold. Do this at each of the 12 points of the star.

[Addendum, May 9,2020 Annie Perkins made a video, clearly showing the step above and below, of the v-fold and the turn-in ]

Nxt step is to fold up the edges.

How much you fold up the edges becomes clear when you do it. There is a natural edge that’s suggested by the inverted v-folds.

Now, the connecting folds are on the outside edge. The inside edge is trickier only because it’s on the inside. This can be changed!~

These outside edges can be rotated inwards.

Nice, huh? Now do all those connecting folds again.

Then you’re done.

So stunning.

Here’s another video showing what the same thing as above, but the folds going the other way, so that the gold and white are inverted.

Now I’m going to go make some more. I just love these.

Addendum: May 4 2020

Fireworks by Mark Kaercher

Addendum May 9 2020

Annie Perkins wrote a very wonderful post about this process at

#MathArtChallenge Day 53: Origami Firework from ONE piece of paper

Geometric Drawings · origami · Zhen Xian Bao

Late for Black Friday: A New Kit

Over the summer I made a 3-hidden boxes packet of these folded paper Zhen Xian Bao with an oceany paper for a workshop in NYC. I put up for sale, then sold out of, the packets that I didn’t use.

In the fall I made more papers for another workshop, but used autumn colors with a design that reminded me of falling leaves. Those are all gone.

Now, here we are, barrelling into winter, so I designed papers that remind me of snowflakes. Just put kits up for these on sale at Etsy, at

I must be delusional. It thought that it would take me only a couple of days to put this project together, thinking they’d be ready for Black Friday. I should know better by now. It has never taken me less than three intense days to design a new paper. Why did I think this would be any faster this time? In fact, it took longer. I couldn’t get the colors to work right, and I struggled with lots about the design.

But I stayed with it, and in the end, FINALLY, today, not two days ago, it finished coming together.

There’s so much about these papers with this structure that I love. It’s full of shapes and colors and patterning that work together in unusual and surprising ways. Really had to work at letting the patterning reveal itself while still keeping it within the boundaries of my own vision.

There’s a bit of a story here.

I had been looking through Clarissa Grande’s twitter post, searching for inspiration. It was the second design in this thread, the one in that is green and gold, that caught my eye.


I didn’t realize it at the time, but it is a design I had studied and created quite some time ago while learning from Samira Mian. Samira pointed out that this was one that she had taught as the Chinese Qur’anic Manuscript Arcing Motif.

Now this shows what I like the most about these geometric constructions: the same underlying structure can create such wildly different results. Going more in the direction of Clarissa’s work, I came up with this pattern:

This led me to the final patterns in the top of the post.

What a journey!

I will have these papers in a kit for sale for just a couple of weeks at




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!

Geometric Drawings · geometry and paper · geometry and paperfolding · origami

Pentagons, Paper Folding, Stars & Origami

I came across a lovely way of folding stars. It was in a youtube video by someone named Tobias.

As lovely as these stars are, what really caught my attention was the way Tobias showed how to use paper folding to make a pentagon from a square. This square-to-pentagon transformation was in a separate video, and since it will take me about two days to forget everything I saw in the video I drew out the directions.

How to fold a Pentagon from a Square
How to fold a Pentagon from a Square. For the Video of this that Tobias made, go to


After the novelty (but not the thrill) wore off of making a pentagon from a square I began to look at the angles that I was making and figured that I could make the star with less steps (and perhaps with more accuracy) if I just started out with the net of the shape, so I made this map of the paper star’s fold lines:

Lines for a Folded Paper Star
Lines for a Folded Paper Star

If you make Tobias’s stars, after you get the hang of which lines fold in which direction, I highly recommend printing out lines above, score the lines with an inkless ink pen, and make that same star using just its essential folds.

The back of the paper sta
The back of the paper star

The photo above shows the backside of these stars. Quite a nice backside!

I’m sure that there are all sorts of things to do with pentagons, but something I want to mention is something that is fast and impressive, sort of the pentagon version of snowflake cuts. If you cut off an angled slice at the bottom of the folded up pentagon (step 12 in my tutorial drawing) there are all sorts of star possibilities.

36-54-90 triangles, with cutting lines on their tips
36-54-90 triangles, with cutting lines on their tips

These little beauties turn into:

Stars in Pentagons
Stars in Pentagons

The stars inscribed into these pentagons were made by cutting through all layers on the tips of the folded shapes.


And look, below there’s something extra for my friends who teach Geometry, and who might like a holiday themed angle activity. Part of the working out the folding pattern for the star was deciphering certain angles.

Find the Angles with degrees of 90, 45, and ~72, 18, 36, 54, and 108
Find the Angles with degrees of 90, 45, and ~72, 18, 23, 36, 54, 63 and 108

I had a good bit of help with the especially tricky parts of understanding the angle relationships. I’m sharing two twitter threads here, just because it was such a pleasure to get help from my friends.


That’s about it for now. Oh, and if you need to directions on how to fold a square from a rectangle, take a look at

addendum March 2018

Here’s someone making this star. She makes it looks so easy!