Book, peeking out of its box by Paula Beardell Krieg

Book, peeking out of its box

When Miriam Schaer was assembling her teaching collection to send to Telavi University in the Republic of Georgia, I very much wanted to contribute, but nothing I had on hand seemed right. In the nick of time, some thoughts came colliding together. Polygon Fractal book by Paula Beardell Krieg

This structure started out with an exploration of a shape which I wrote about a few weeks ago after watching family math video made by the Lawlers.

Inside Outside Book by Paula Beardell KriegThe book  opens in an accordion-like fashion, but front and back are structurally different.

Polygon Fractal book by Paula Beardell KriegThe colorful pages rotate open to create these double layered corners. The polygon fractals on the pages here are harvested from Dan Anderson’s openprocessing page then toyed with in Photoshop.

To see the fractals in their full radiant radial symmetry one must rotate the book. There are six completely different images to be seen. But it gets more interesting, because there is a whole other side to see.

The folds of those double layered corners completely reverse to form a cube!

You can’t imagine how excited I was when I saw this cube emerge from the folds!

This folded structure totally suggested that, whatever I use on it, that it be about the dual nature of….something….a suitcase (no, too obvious), a politician’s statements (ugh, too boring)…actually wanted to use images that didn’t imply any hierarchy, hiding, agendas, or judgement about contrasting inner and outer manifestations.

It was this thinking, about duality but equality of visuals, which led me to using Dan’s code along with the polygon fractals that it creates. So perfect. Code and images are perfectly linked, simply completely different ways of seeing the same thing. You know, like Blonde Brunette Redhead 

Now, I do have a lingering unresolved issue with this book. I’m not thrilled with the paper that I’ve used. It’s 32lb Finch Fine Color Copy paper. It takes color beautifully, folds well, but I’m thinking that the folds might be more prone to tearing than is comfortable. Not sure what else to use…am open for suggestions. Miriam’s copy has been shipped, but I’m still happy to check out different papers to use.

I can’t help but wonder if people will be able to figure the transformation of these  pages without seeing this post or reading the brief explanation I’ve provided on the back page of the book? Dunno.

Oh, and here’s my favorite variation:

Hanging a tea light from a pencil so I can see the inside and outside at the same time.


Kaleidocycles and Tetrahedrons

February 19, 2017


I looked up the definition of a tetrahedron today, I figured out how to spell kaleidocycle a few hours ago. Just saying.

kaleidocycle-colors-1 Paula Krieg

Pattern for one kind of Kaleidocycle

Sometimes an exploration pursues me. It’s always a gift to be preyed upon by ideas, but if my desk is already full and messy, and I think I can’t bear adding one more layer I pretend to kind of ignore the newcomer. No, this strategy doesn’t work.

I didn’t know that tetrahedrons were following me around. Like I said, just this morning I finally looked up the definition (a solid having four plane triangular faces; a triangular pyramid).

The image above is where this all started. This is not such a startling set of pictures until you know that the image shows the same, unchanged structure viewed from front and back. It’s on the facebook page of someone whose name is written in an alphabet I don’t understand. This is the link to the page on facebook . Take a look if you can. It’s such an amazing bit of transformation, which I have yet to figure out how to do. What’s going on here is that this structure to made up of connected 3D shapes that rotate together to reveal different surfaces. It’s very tricky and fun to see the shapes turn, revealing new surfaces.

The next piece of this story is that a teacher just a bit south of me in Upstate NY posted some directions on how to build a certain  geometric shape, and he asked, via twitter, if anyone would be able to test drive his tutorial. It looked simple enough to me, so I thought I’d try it out the following Saturday morning. I thought it would take about 2o minutes. Ha ha.



Looking back, I think if this teacher, Mr.Kaercher, had done a tutorial on a simple tetrahedron it might have gone more quickly and I might have finished up knowing what a tetrahedron was. But, no, Mr. K provided directions for a tensegrity tetrahedron, and since I didn’t have much of a clue about the definition of either term, I didn’t really have much of an idea of what I was doing.

Tensegrity Tetrahedron

Tensegrity Tetrahedron

Even so, after a megillah of failures, I got it done and was quite pleased with myself.

In the meantime I was still thinking about those images from that facebook page.

I showed the FB clip to book artist Ed Hutchins. He told me that what I was looking at was a type of kaleidocycle.

Oh, and Ed just happened to have a hot-off-the-presses copy of what is probably the world’s most amazing example of a hexagonal kaleidocycle, designed by Simon Arizpe. (This is a fully funded Kickstarter Project, which you can view to see the book in motion.)

This structure tells a story as it rotates. Since these rotating sides can turn forwards or backwards, the sequence of the story is determined by the direction the viewer rotates the kaleidocycle. The way that I choose to turn it, it begins with a bear peeking out at a stream…


…the bear opens his mouth, a salmon jumps out…

The Wild by Simone Arizpe

The Wild by Simone Arizpe

The Wild by Simon Arizpe

The Wild by Simon Arizpe

… and then the salmon jumps into the river. There’s one more frame, but I’m not going to be a spoiler and show it to you.

So what does this have to do with tetrahedrons? I’m getting there.

As it turns out, the last couple of times I’ve gone lurking at the Lawler family math page, they’ve been looking at, yes, tetrahedrons.

This shape that the Lawler’s were considering was beginning to look familiar to me. Part of the reason for this was that, ever since Ed had given me the gift of the term kaleidocycle I had been Googling around then assembling kaleidocycles.

Kaleidocycle, unhinged

Kaleidocycle, unhinged

Here’s one of my first attempts. Notice that I forgot to attach the ends together before I closed things up. This turned out to be a good thing, because, wait! these shapes appear to be repeated echoes of the shape that the Lawler family was exploring.

Just to pile it on, it certainly helped that just yesterday a package came in the mail, all the way from France, from Simon Gregg. In the package was, can you guess?… a tetrahedron.

Tetrahedron from Simon Gregg

Tetrahedron from Simon Gregg

That Saturday a few week ago that I tried, time after time, to create my tensegrity tetrahedron, I had been posting my failures publicly on twitter. I imagine that Simon thought that it might be merciful to send me some bamboo, as the straws that I was using would sometimes collapse. Included with the bamboo rods, Simon also gave me a collapsible tetrahedron, held together by stretchy cord.

With all of these pieces floating around me it,  I finally made the connection that units of kaleidocycles are series of tetrahedrons. 



So cool.

Now to reward you for making it all the way to the end of this post, here is a pattern for a kaleidocycle that you can make yourself.


click the image to get the printable PDF

Just cut it out, use it alone or attach it to the one near the top of this post, but, in either case, do  make sure you attach ends to make it circular. Here’s a pleasant little video to show you how it’s put together.

I still intend to figure out how to make the kaleidocycle that I saw on FB. When I do sit down and try it out, at least, now, I feel like I’m starting with some helpful understandings.

I have no big attachment to figuring it out for myself, so if you are inspired to decipher it, please let me in on its secrets!


That’s it for now. Thanks for staying with me through these meanderings.

Addendum 2/20/2017

truncated tensegrity tertrahedron

truncated tensegrity tertrahedron

Used bamboo sticks with bobby pins in the ends to make another one of the Mark Kaercher project.  The bamboo worked out great! If I was to make this again with straws, I think I’d try to first put stirrers, like what Starbucks provides to stir coffee, inside the straws. But love the bamboo!



Balloon with Moire

Balloon with Moire

Just before Valentines Day the usual group of suspects who capture my attention with their outrageously playful exploration of ideas once again were posting images that stopped me in my tracks.  I have to say that when I saw what they were doing I made a decision not to participate because I was busy-with-other-things.  But then they started using HEARTS in their images and I just couldn’t resist. What they were doing was  creating moire patterns, mostly digitally. Moire patterns are that cool effect you see when two identical screens are laid on top of each other, then shifted.  I started out making some with my Illustrator program, then my attention shifted towards making them in the real world, engineering paper rather than working on the computer.

Moire patterns can start with a repeating tile, a tessellation, which is a shape that can be repeated forever. Here’s a photo of one of the explorations that I saw going on, by Mike Lawler’s family:

Tesselating in the Living Room

Tessellating in the Living Room

Then Dan Anderson started doing a series of outrageously beautiful interactive images on his Open Processing page:

Then Martin aka GHS Maths started making moires with straight lines in the on-line graph program Desmos.

You really must visit some of these links in order to get the full sense of how stunning these images are.

My first response to these image was to make a  few gifs myself.

My Illustrator Heart Moire

My Illustrator Heart Moire (click on image to see the gif)

What I wanted to do, though, is make some movable paper structures. I didn’t really know how to do this, but I know someone who does: book artist extraordinaire  and paper engineer Ed Hutchins. The biggest deterrent for me was that it seemed that it would include lots and lots of paper cutting and I don’t feel like doing that right now. Destiny interceded: I came across some transparency paper for copy machines at the local thrift store (50 cents!) and now I was almost ready to proceed.  Most of the rest of this post is one way of making the copy, cut and paste moire pictured at the top of this post.

Moire Materials

Moire Materials

I wasn’t truly sure that using transparency paper and prints would work, but, as it happened, Dan Anderson invited me to visit his tech lab at the high school that he works at, which is just a short drive from my house (amazing good luck for me, considering the other two conspirators collaborators live thousands of miles away). He printed up some of his images on transparency papers and we were able to immediately see how well this worked.

A Repeating Pattern : Tiling; Tesselation

A Repeating Pattern : Tiling; tessellation

I came home and tried out, oh, about 15 different kinds of images, some in color and some in black and white, and settled on a hexagon kind of tiling. Dan had done some colored moires, which , when on the computer screen, knocked my socks, but the yellows and oranges faded out in real life. There were things I could do to work with color but I chose to work with black and white for now, but then shamelessly decided to use screenshots of Dan’s work as part of the background for my moire.

Moire from D.Anderson's OP

Moire from D.Anderson’s OP

Nice, huh?!!

Circle Rectangle Card Compass

Circle Rectangle Card Compass

First thing I did was cut a four-inch circle from the paper that my tiling was printed on, then I cut a my card from the glorious image above (about 4″ x 8″), and cut a 3 7/8″ x 6″ rectangle from the printed transparency paper.

Shapes to start with

Shapes to start with: 1 inch circle, 1-3/8″ square with a 1″ hole cut out, 2.5″ circle, and an egg cup to hold my straight pin.

I am also unbelievable fortunate to live near Ed Hutchins, who graciously agreed to show my how to do the paper engineering for moires. I thought that it would be a quick kind of thing, that he would just be able to say “snip here, glue there” and we’d be done. Three hours after sitting down with him I sort of had the idea of what to do. Honestly, what I am writing here is mostly so that I can remember how to create what I went to learn. Ed’s skill with cutting tools is far beyond my ability, so I’ve altered what he showed me. There is one major concept that remains intact,the hub; exactly  how to make and insert it is a matter of preference.  So I started with the tools above. The hub is the  little bright green circle in the center. This is the basis of a spinning hub. My hub is a one-inch diameter circle.


The hub fits into a smaller hole, which in this case is 5/8″ wide. There are four snips in the hub, cut just so that it fits snugly into the hole  and can turn without wiggle room or too much friction.

Hub Balloons

Hub Balloons

I made some hubs with a square, some with balloon shapes. There just needs to be enough room around the hub so that the larger piece can be glued down without interfering with the ability of the round piece to turn.

One and five-eighths away from the edge

One and five-eighths away from the edge

After the hub is together I put a straight pin through the center of the circle to help me get everything else centered together. I also cut a hole through my card, large enough to allow the hub to show through, but small enough so that the piece around the hub can be glued to the back of the card.

The Hub showing through

The Hub showing through

Here’s how the inside of the card looks. My egg cup is waiting there for my straight pin, so that it doesn’t land on the floor then in my foot. But for now, the pin stays in the card, waiting to pierce the center of the dark circle. The dark circle will be glued on the hub only, which still turns freely.

Adding the Moire

Now the 4 inch patterned circle will be glued on to the smaller circle


A word about the papers I am using: for the hub I need something that is strong , and that folds and glues well. I started out using regular copy paper but was unhappy with how it behaved, so I switched to using some thin but sturdy wallpaper paper, from a sample book that I had around. The dark circle is also a strong, lightweight paper. This piece may not even be necessary, but I decided I wanted a lighter paper to glue to the hub, because my printed paper, which is heavy Hammermill 80 lb color copy digital cover paper, seemed like it would stress out the structure. I could be wrong, but this was my work flow.

Almost Moire

Almost Moire

Now the printed pattern of hexagons is glued on. I used the straight pin to make sure all the centers were lined up. The pin in now back in the egg cup. You can see I added a cut-out on the front of the card. I like the way a cut-out shape frames the pattern when the card is shut.



Now, with some white glue I glued down the transparency paper. You can’t see the full effect of the moire in a static picture. Here’s a link to the video I uploaded of this. You can’t hear much of what I say, but don’t try: I’m giving instructions to my husband on how to hold the card while I am holding the camera. If you don’t want to watch a video, here’s the front, middle, and back of the cards…


…and here’s another of my gifs:

Click image tp animate

Click image to animate

And, finally,

here’s the video that started this round of visual explorations, a Numberphile video called Freaky Dot Patterns

another fine example of moires in desmos by Martin

and a video that shows what the Lawler family did with their moire-that-wasn’t








An Out of Pocket Experience

October 14, 2015

more dots

Pocketed Groupings

My thoughts do not naturally go towards developing projects for students who are very very young. But, when I wrote about using an unending accordion for number lines John Golden left a comment saying ” Ooh! It’d be fun for young learners to put pictures of things in that pocket that come in a number on that page. Could make a guessing game out of it, or just a way to record number observations.” I liked the suggestion and I thought that I could develop a response in just one day, but nothing happens fast with me. TEN days later I finally got it worked out in a way that I like.

Out of pocket experience

Number Groups with Hidden Numbers

You may notice that I used dots, not pictures. I like dots. Everyone likes dots. Granted, dots can get kind of, well, repetitive after a while, so it’s a good thing that there’s a one side and another side and that dots are malleable.

Unpacking the pocketed number groups

Unpacking the pocketed number groups

Here’s the thought behind this project. First, it’s a pockets project. Students seem to like pockets even more than they like dots. Anything at all that would benefit from a peek-a-boo kind of experience would be fine subject matter for the pages.  I just happen to be on a numbers kick at the moment. I have gleaned from the #MTBoS math people who I follow on Twitter that associating numbers with groupings can be a good foundation skill: creating groupings of, say, three things could help support the understand that the symbol for three is an abstraction representing three somethings.

Made with Standard 8.5

Made with Standard 8.5″ X 11″ paper

 As usual, the challenge I set out for myself was to make this out of standard copy paper. This structure is similar in many ways to the structure in my previous post, the one difference being that I’m not linking the papers together as I don’t see an advantage to this being a continuous number line, though it certainly could be.

I made some PDF templates for this project. (The good news is that I finally figured out how to make PDF’s in a small file size!!! I will eventually be going back and make all my pdf files smaller). I did not however, provide colorful dots and pictures. I see this as a class project, where students can either color in the dots or turn them into balloons and ladybugs. (Let them color it and they will own it.)

Here’s what you can print out if you like:

open dots jpeg

A class color-it-in- yourself template

8.5 X 11 open dot groups

Dot are filled in here.

Dots are filled in here if that ‘s what you need.

8.5 x 11 Filled dot groups

Simple Pocket Template

Simple Pocket Template. Print several.

8.5 x 11 simple pockets

I haven’t included directions on how these pieces fold, but if you fold on the dotted lines and look at the pictures, I think it’s decipherable (?).

Number Groups

Number Group

Maybe the next step after this is, well, dice….which, in a few years could lead you to fedricomath’s Weird Dice.  (which I link to here so that I can keep track of it).

Just for the record, this is what’s been keeping me distracted lately.



It’s a magnificent autumn in Upstate New York.

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