Geez, had I known the math people have such a knack for playing around with materials to make exuberant visuals, I would have found my way back to math years before I did. I just recently got back from the Museum of Mathematics (MoMath) MOVES conference, celebrating recreational mathematics, which was finally scheduled after pandemic postponements. A few years ago I had been at another one of MoMath’s conferences, focused on paper folding, which was simply stellar. This one focused on puzzles and games, and, wow, what a visual feast.
Didn’t take many photos, as I was mostly taking it all in. But did get a few…
The amazing designs of David Plaxco on Rubik’s cubes
I got a close-up look at David Plaxco’s designs on Rubik’s Cubes. Mind boggling and stunning. I’ve been watching David’s work on Instagram, cubes_art but being able to hold some of his cubes and see all their sides was quite a thrill. I had spoken to him years ago after meeting him at previous conferences, and was intrigued with how he was thinking about seeing knots on cubes, but seeing where he’s taken this work was such so uplifting.
Chaim Goodman-Strauss with his hyperbolic creation
I also got to be in the room with Chaim Goodman-Strauss, who I’ve been hanging out with weekly this summer in MoMath sponsored Polyhedra Party sessions, where we’ve been building with paper. Here, though, what Chaim has done is used interlocking mats to create wildly big and beautiful hyperbolic surfaces. I love that his blackboard scrollings got into this photo, too.
Chaim with Suzanne and family
Chaim brought a huge amount of modified rubber mats to New York, and ,was heroically assisted by the museum’s director, Cindy Lawrence who (if am remembering this correctly) carted through the city, via cab, so they’d be ready for us to use to make these enormous waving surfaces.
There were so many fun details to enjoy. For instance, when I asked Lauren Seigel for her card, here’s the spread I got to choose from.
Image by Bob Hearn
The above photo is a print that Bob Hearn gave out after his talk called The Fractal Beauty of Compound Symmetry Groups, where he showed us one stunning image after the other, and how they evolved through orderly overlapping of shapes. His was the last talk of the last day, so I was pretty happy just to sit in the auditorium and watch the pretty pictures.
I had proposed a family workshop for the event, but this time around, probably because of the times we live in, there weren’t many families at this conference. Still, I had a small but mighty group come to do a tricky make-and-take project,
Thanks for the tutorial on making a (mini-)Jacob’s Ladder at MOVES @PaulaKrieg. I do already have glitter all over everything though… pic.twitter.com/j4STZ2UfGV
What we made is very much like what some people know as a magic wallet, though what I showed is able to be extended in the traditional toy, the Jacob’s Ladder, hence I call it Li’l Jacob.
To practice what I’d be doing at the conference, I made a video of the steps. Pretend you are in the room with me, and give it try!
As much I as I liked being at the conference, and enjoyed getting back to the city that was my home for nearly twenty years, the conference days happened on some of the hottest days of the summer. Outside, I fried.
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.
You are likely familiar with the fact that the game of dominoes is played with tiles that are like two squares on top of each other. Much to my delight and surprise, about a year ago, mathematician Justin Lanier enlightened me to the fact that this rectangle, which is twice as long as it is wide, has a specific name and that name is DOMINO.
This may not seems like such a grand reason to celebrate, but, over the past year, by being able to pull this definition out of my back pocket, both my teaching and my structural problem solving have improved and deepened. There is something about calling something by its proper name that has value.
For decades, with thousands of children who have done bookmaking with me, I’ve been trying to figure out how to get students to decorate their books using geometric shapes in a meaningful way. A meaningful way means, to me, getting them to really understand and exploit the characteristics of the shapes. Thousands of children for decades means I can try out one idea after the other, attend to nuance, and maybe even figure out a thing or two.
Cover Made by First graders 2010
I’ve known for a long time that starting with squares can help reign in the chaos of making decorations for books. I actually use to cut out and distribute piles of squares to students so that they could work with squares.
Lots of cut squares
Yeah, I would spend hours cutting little squares out brightly colored card stock. Even now it’s painful looking at these squares, remembering those late nights of cutting cutting cutting… and then the students would want to smear glue all over their books and sprinkle the squares like candy on them and, when the glue dried, half of the squares would fall off because the gluing was sub-prime. Ok, I’m sort of exaggerating: Many turned out beautifully, but I still didn’t feel like the kids were getting the most out of the activity: they weren’t making the connection that I was hoping for with the square.
Dominoes, cut into squares, then half squares, which turn out to be a scaled down domino, then halved again to make small squares
Eventually I realized that I could give students the double-square shape, though at this point I didn’t know that it was called at domino. I show students how to line up two of these double-square perpendicular to each other, from which they cut “on the line” that separates the colors, and snip, snip, they have four squares. If they cut the square again, this time by eye, they have scaled down rectangles, and they can then cut these in half to make baby squares. Yes, baby squares. These are young children and young children like baby squares. I also talk to the about rotating the square, cutting it from point to point to make two triangles, and then cutting the triangles in half to make baby triangles.
Dominoes, cut
Now just this year, during this teaching season, I have a made a point to introduce the double-square shape by name. The students I work with are shape savvy: they know what a rectangle is and they know that the square is a special kind of rectangle, but their eyes light up when I tell them a that a double-square shape is a special shape too, a domino. Does it make a difference for them to call it by name? I answer that with a resounding YES.
Lily’s Emperor Penguin Book
It’s been like night and day. Students seem to honor the shape and special qualities of the domino. I know this because, with the hundreds of students that I’ve worked with this year, I saw, for the first time, the overwhelming majority of students being at ease with the idea of working with just the shapes that I talked with them about. There was so much less arbitrary cutting of paper, so much less just slapping down whatever shape that emerged from a hastily cut paper strip.
Thinking hands
Across the board, with the introduction to the domino, students approached this part of the project with a sophistication that I had never seen before.
pages in progress with cut paper decoration
Instead of having to wade through photographs to show the type of work that I think is most valuable to show, I am finding it hard to choose between all the interesting work that these students are making.
Mostly Green
I look at these and I can sense that the student is connected to the underlying structure. If you don’t know what I mean by that, just look again at these photos.
It seems to me that I even have metric which tells me that this is not just something I’m imagining. One of the other techniques that I show students is how to make paper spring. This is a challenge for most students, but a worthwhile one, as they love how it gets used in their books. Though not a domino shape, it is made by having an awareness that the two strips of paper they start with are of equal width, and I show them how to weave and rotate the strips down into a square.
Working hands, paper spring made with paper strips that are 8 times as wide they are long, which, is I am told by mathematician Paul Salomon, an octomino
After first showing them the techniques of decoration with the domino it was astounding how quickly the students understood how to do the spring.
Paper Spring, completed Interesting fact: my page on How to Make a Paper Spring has been visited more than 16K times
In fact, in each class, these parts of my presentation went so quickly and smoothly that I kept checking my notes, thinking I had missed some part of the lesson because there was so much extra time.
Needless to say, I have much respect for the domino.
For the last few months I have been casually, now more seriously, studying a folded paper structure in Chinese Folk Art, called the Zhen Xian Boa. My plan is to write a number of posts on this structure. It’s not hard to figure out that part of my fascination with the structure is that, from beginning to end, there’s the domino.
Playful Bookbinding and Paper Works 