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

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.


Art and Math · Geometric Drawings · group project

Rotational Symmetry project with 5-9 year olds and Moms

I got to spend some time with a group of kids and moms this past Sunday. They had asked me to plan a math/art project for them. Last time we did this we played with shapes scaled according to the golden ratio. This time I wanted to help them make images that are made by rotating a graphic around a circle. We used a circle that was divided into twelve equal sections, and we got to talk about how rich the number 12 is, in that it comes up often in measurement of time (hours, months), quantities (dozen), distance (Inches) and so much more.

Images were made in two ways. One was to connect the dots around the circle according to a rule, such as connect the first dot to the fifth, connect the fifth dot to the tenth, connect the tenth dot to the dot that is plus+5 further around the circle, then continue until you are back where you started from. A star emerges!

Connecting the points around a circle

We started the afternoon by sitting in a circle of eight people, and doing the skip-counting activities that I described above. This was actually a thrill to me, as it’s something I’ve wanted to try out for a long time. As the star shape grew within the circle of people, who were the “points”, everyone was thrilled. They had no idea a star would emerged. I knew, but I was thrilled too.

I had PDF printout of circles and shapes.People cut out shapes that they wanted to rotate around the center, then colored them in if they wanted to.

I think the young man who did this image is about 8 or 9 years old.
I think the young man who did this image is about 8 or 9 years old.

The moms seemed to like this activity at least as much as the kids.

I never know how these projects will go. A couple of the boys didn’t want to be coloring any more after a while. One boy in particular really liked cutting paper, so I got him started with another kind of rotational symmetry: making snowflakes!

Snowflakes have rotational symmetry
Snowflakes have rotational symmetry

I hadn’t thought about snowflakes beforehand, but liked the way I was able to link to something that was already familiar to this group.

After awhile one of the girls was finished with coloring, I showed her how to make an origami pockets that were sized for the drawings to slip into.

Lot of pockets
Lot of pockets

She really liked making the pockets, and made them for everyone. This also let me segue into showing her how to make a square from a sheet of paper.

In the end, we had made lots of images, pockets, snowflakes and our work area was delightfully messy. Everyone helped with the cleanup, especially with the tiny pieces of paper on the floor.

At the end we put our tiles out on display.

Our tiles
Our tiles

A couple of hours later one of the mom’s texted me saying that, on the way home, her kids were asking to do more of these. YAY!


math · Math and Book Arts

Fraction & more Fractions

Many Parts of a circle
Many Parts of a circle

I’ve been working with 9 different grade levels, nine different projects, this month, which is kind of wild, and even more wild because of all the snow days and other unexpected shifts in schedules. Most of the projects we’re doing are things I’ve written about enough on these pages, but I have managed to slide in a couple of new things with the fourth graders.

I had some extra time with some of the students  because they chose to stay after school for some extra time with me. Am still racing to finish prep for tomorrow, but want to quickly post about these two extra projects.

Dividing up a circle project
Dividing up a circle project

I brought in circles and sheets of regular shapes. Student cut up the shapes, and rotated them around a center point. The circles were marked with 12 evenly spaces dots around the circumference. We talked about other cyclic things that are divided up into 12 parts (clock, months) and talked about how 12 has so many divisors.

Rotating Shaper around a circle
Rotating Shape around a circle

I printed the shapes on heavy paper. I hadn’t done this with kids before so I didn’t know if they’d have trouble with this. It was no problem for them at all. They were excited, worked creatively, asked questions and were totally engaged.

Student rotations
Student rotations

Here’s the PDFs that I created for this project.

Circles with 12 dots

shapes to rotate in circles

I casually mentioned that ANY shape can be rotated. Well, they didn’t have to hear me say that twice before they were making new shapes.

Crazy Shape rotation
Crazy Shape rotation

The trick is to retain points that can still line up with the center and with a point on the edge of the circle.

Another Crazy Shape Rotation

During class time, I worked with students on a fractions/ bookmaking project that I’ve written about previously on my Books Are Fractions  post.

Fractions book
Fractions book

I knew some students would finish up early, so I showed them some images I had printed up some twitter posts. (If you want to see many more images like this, type in the words Fraction Museum in the twitter search bar and you will be well rewarded)

The kids were enthusiastic about creating fraction museum pieces, which I then photographed.

Fraction Museum hearts
Fraction Museum hearts

The idea is to collect items, see them as part of a whole, then write fractions that describe the collection.

Fraction Museum books
Fraction Museum books

There was some deeper thinking going on than I expected.

Mixed Fraction Museum
Mixed Fraction Museum

I’ve assembled all their images on to 2 large sheets of papers, and will present them to the kids tomorrow….but only if I stop this blogging and get back to work,


Addendum March 26 2018

During my fractions conversations with these kids (who, by the way, had a good grasp of fractions before I ever showed up) I talked about the confusion that can happen when trying to understand why, when the denominator is a bigger number, the unit fraction is smaller. I showed them a piece of paper folded into four sections, then said if I had to fold the same paper into eight sections (which we did) that the number of units had to be smaller to accommodate the larger number of sections. Then I asked “Imagine if we had to divide this paper into 100 sections, how small would those sections have to be? 


Well, that was it. They begged to see a page divided into 100 sections. Each time they saw me, they reminded me. Finally, today, I brought in TWO papers, and asked which one of them had fraction units that were each 1/100. Led by an particularly independent thinker, they figured it out. And figured out why, even though the divisions looked different, that they were all 1/100s.  It was a great conversation. Here’ the PDFs of you can ask kids this question yourself: hundreths

So much fun.