So I wasn’t actually alone with Van Gogh’s Starry Night. But I went to MoMA this morning and got to tour the museum with other math teachers before the museum opened. Our sherpa? George Hart, mathematical artist. A few months ago, I got an email from two different teacher friends letting me know about this opportunity to take a master class on Geometric Sculpture put together by the Academy for Teachers. What an opportunity indeed!

I show up at 8:30 am and me and a gaggle of math teachers (a gaggle is eighteen, right?) are raring to go. We have fancy namecards and everything. (Note to self: at the book club I’m hosting in a bit over a month, create fancy namecards.)

Beforehand, we were assigned a tiny bit of homework. We were asked to go onto the Bridges website (it’s an international annual math-art conference, organized by our sherpa), look at submitted papers for their conference proceedings, select three papers, and then read and reflect on them.

My Paper Choices and Thoughts

1. Prime Portraits, Zachary Abel

This mathematician was able to construct portraits using the digits of prime numbers. The digit 0 was black and the digit 9 was white, and the other digits were various shades of gray. The digits of a number were put in order in a rectangular array (e.g. 222555777 would be put into 3×3 array, where 222 is the top row, 555 is the middle row and 777 is the bottom row) and an image results. For most numbers, the image will look like noise. But this author was able to use prime numbers put into a rectangular array to create images of Mersenne, Optimus Prime, Sophie Germain (using Sophie Germain primes), Gauss (using Gaussian primes), and others. I was blown away. This intersection of math and art doesn’t quite fall neatly into any of the categories that George provided us, but it is close to “mathematics used in calculating construction details necessary for constructing an artwork.” In this case, the portraits themselves are the “art” and the author was using numbers to reconstruct that art. What makes it interesting is that the math version of these portraits feel unbelievable. Senses of awe and wonder and curiosity filled me when seeing the portraits for the first time because how could it be? It was like a magic trick, because nature couldn’t have embedded those portraits into those numbers. And before reading the paper on how these were constructed, I had a nice few moments thinking to myself how this could have been done.

(If you’re curious, the answer is to start backwards. First take an image, pixelate it, and then turn those pixels into a number. Take that number and check if it’s prime on a computer. If it isn’t prime (which is likely), slightly alter the image by the colors by +1% or -1% (some imperceptible noise), repixelate it, and turn those pixels into a number. And again, check if that number is prime on a computer. If it isn’t, do this again. It turns out that you’re going to need to do this about 2.3n times [where n is the number of pixels]. With a computer, this can go quickly.)

Thoughts/Questions:

(a) Math: I recall faintly from college classes that the distribution of primes is related to the natural logarithm. Which explains why the 2.3n comes from something involving a natural log. But what is this relationship precisely, and how does it yield the 2.3n?

(b) Content: I think prime numbers are very rarely taught in high school math in a meaningful way. Number theory is ignored for the “race to calculus.” However there is so much beauty and investigation in this ignored branch of math. Where could I fit in conversations of prime numbers in an existing high school curriculum? Could ideas from this paper be used to captivate student interest (by letting them choose their own image), while showcasing what various types of prime numbers are?

(c) Extension: Are there other things that we teach that have visualizations that look impossible/unbelievable, but actually are possible? Can we exploit that in our teaching? I’m thinking that often numbers in combinatorics are crazy huge and defy imagination… Perhaps a visualization of the answer to some simple combinatorial problem?

(d) In order to fully appreciate this work, the viewer needs to have an understanding of prime numbers. Without that understanding, this is just a pixelated image with some numbers superimposed. All wonderment of these pieces is lost!

2. Modular Origami Halftoning: Theme and Variations (Zhifu Xiao, Robert Bosch, Craig Kaplan, Robert Lang)

I chose my articles on different days, and I didn’t even notice that this article is very similar to the first article! I chose it because I love the idea of a gigantic public art project in a school (I tried once and failed to make a giant cellular automata that students filled in). But this article basically shows how to fold orgami paper (white on one side, colored on the other side) in five different ways to make squares where all of the square is colored, ¼ of the square is colored, ½ the square is colored, ¾ of the square is colored, and none of the square is colored. A number of each of these origami pieces are constructed.

Then an image is converted to grayscale and scaled down to the number of origami pieces you want to use. Then the image is scaled-down image is pixelated with “origami piece” size pixels, and each pixel is given a number based on brightness [0, ¼, ½, ¾, 1].

Then this origami image can be created by putting these five different origami pieces in the correct order based on the brightness of the pixelated image!

Just like with the previous paper, this intersection of math and art doesn’t quite fall neatly into any of the categories that George Hart provided us, but it is close to “mathematics used in calculating construction details necessary for constructing an artwork.” In this case, the portraits themselves are the “art” and the author was using numbers to reconstruct a variation of that art.

Thoughts/Questions:

(a) Math Classroom: I really love the idea of having kids take an image with a particular area (w by h) and figure out how to “scale down” the image to use a particular number of origami pieces. It is an interesting question that will also involve square roots! It seems like a great Algebra I or Algebra II question.

(b) Extensions: How could this project be extended to the third dimension? 3D “halftone” origami balloons? Unlike a photograph which can be easily pixelated, can we find a way to easily pixelate the “outside”/”visible part” of a 3D object and create a balloon version of this? Similarly

(c) This is not just a low-fidelity copy of an existing piece of art. If we took a random non-professional Instagram photograph, we might call it “pretty but not art.” But if someone made this Instagram photograph out of origami sheets, we would be more likely to call it art. But why? Just one thought, but there is something about the intentionality of the artist (and the craftsmanship that goes into creating the origami piece) that isn’t in the original photograph. It also is likely to evoke something different in a viewer – a viewer will instantly wonder “how was that done” when seeing the origami piece (so the art piece evokes process) while a random photograph might not do the same (they just pressed a button on their phone and got a cool photo).

3. A Pattern Tracing System for Generating Paper Sliceform Artwork, Yongquan Lu and Erik Demaine

I chose this paper because of the beautiful sliceform image on the first and last page. I had only seen them once before, but forgot what they were called! I wanted to learn how to make them. In this paper, the authors share that most existing sliceforms are created in separate pieces (e.g. the image on the first page, a bunch of hexagons created separately) and then pieced together afterwards. The authors wanted to instead thread the paper slices together so they could create the same intricate patterns—but with the paper slices interconnected. So instead of individual hexagons placed together, a giant connected sliceform was created (e.g. the image on the last page). The authors came up with a way to do this for designed created in polygonal tiles, like in many Islamic star patterns, and then created a program to “print” the strips of paper needed – with red lines indicating where folds are, and blue notches indicating where cuts need to be made so the paper slices can be fit into each other.

They accomplished this in two steps. First, they came up with a way to notate the internal structure of a paper slice within one polygon. One notation captured lengths (where slices of paper intersected other slices of paper and where slices of paper needed to be bent/folded), and another notation (not provided) recorded angles that needed to be folded. The second step was more tricky. An algorithm was created that looked at the edge of a polygon (where a paper strip initially ended), and looked to see if it could be extended into another polygon. In that way, one strip could start in one polygon and then enter another, and then another, etc. This is the threading that the authors wanted to get. The authors created a three-step algorithm for deciding if a paper strip could enter another polygon at all, and if there were multiple possible paths for this strip to take, which one it should choose.

After doing all of this, the authors then created a program that could take in an image, calculate out the different strips of paper needed to create the sliceform, and with the notation they created, print out the appropriate slice (see image on page 370 for an example).

Thoughts/Questions:

(a) There were two big things I didn’t totally understand when reading this paper. First, how were angles recorded/notated? Second, where did the 3-step algorithm for extending paper slices come from? How do we know if we follow it that all segments in the figure will be created by the paper slices, and no segment will be repeated?

(b) Besides just being “cool,” is there an application to this in a high school math class? What higher level research does this connect up to? (Just like origami was simply beautiful but then it also was exploited to create new and interesting questions for mathematicians, what does this bring up for us?)

Note: When I went to research these, it turns out that Lu and Demaine created a website to help amateurs out: https://www.sliceformstudio.com/app.html

(c) I was wondering what a 3D version of this might look like, but it turns out that this exists! https://www.sliceformstudio.com/gallery.html

Back to the Master Class

After getting coffee and pastries, and introducing ourselves to each other in small groups, we all were taken on a tour of MoMA, where George led us to certain pieces to spoke to him as he looked at them through mathematical lenses. There was one sculpture in particular that George stopped us at — a sculpture he remembered seeing as a kid visiting MoMA — that I would have walked right by. It was a figure cast in bronze (?), that had a lightness and movement despite it’s medium. To me, it screamed that it was a figure in tension. Rooms later, I was still thinking about how it was a collection of oppositions, form and formlessness, fluidity and stability. For George, describing what drew him to it was ineffable.

Here are more photos of George taking us around.

The whole walkthrough, George kept on saying “I’m not an art historian, but this is what I see in terms of my perspective as a mathematician…” which was just what I needed to hear. I know so little about art history and contemporary art, but hearing that let me feel a bit more “free” in looking at something and thinking about it with my own lens, instead of me passively waiting to hear what the piece is “supposed” to convey or what philosophical/conceptual trend it is a part of. In general, I feel ill-equipped to make statements/ judgments about art in museums that go beyond “I like this” or “I didn’t really like this.” But listening to George talk about what he sees as a mathematician and mathematical artist was liberating. Because I can see mathematical ideas/principles (intentional and unintentional) in some of the art too! This walk and talk reminded me a lot of what I imagine Ron Lancaster’s math walk around MoMA would be like!

And as the title of this blogpost suggested, there was something so special and magical about being able to have the run of the museum before the general public was let in. And a random fun tidbit: I also learned that there is no simple mathematical equation for an egg. I (of course) had to google that when I got home, and came up with this webpage.

We Become Card Sculptors

We get back to the room that was our home base, and some people share out interesting things from the articles they read. I was going to share mine, but I noticed that even though the ratio of men to women was low, more men were taking up airtime than women proportionally. So I kept my hand down.

George gives us a set of 13 cards with notches in them. We only needed 12 but you know how we math teachers really like prime numbers… (Okay, that wasn’t the reason for the 13th card, but I want to pretend it was.) We were asked to crease them like so:

And then… we were asked to put them together somehow, into a freestanding sculpture. No glue, scissors, tape, etc. We were given a hint that you can start with three cards. So I figured we needed to create 4 sets of objects that each take three cards. So with my desk partner we made this:

This was the core object we needed to build the final thing together. It was interesting how it took different pairs different amounts of time to get these three things together. Without instructions, it was a logical guessing game, but it felt so good once we hit upon it.

Then came the tough part. Putting these four building blocks together. That took a long time and some frustration, but the good kind. It was one of those problems that you know is within your grasp, and you know that you can come out on the other side successfully, but you don’t quite know how much time and how much angst the journey will cause you. It’s that sweet spot in problem-solving that I love so much. And lo and behold:

Many people got it! I would post a picture of mine, but all my photos look terrible. You can’t see or appreciate the symmetry and freestanding nature of this beast. But it was a moment of such pride when we got the last card to slide in the last notch! (And of course when my partner and I tried building hers after finishing mine, it went much faster and we had a better sense of things.)

Oh yeah, this card sculpture is isomorphic to a cube. I was blown away by that. It was hard for me to see at first, but realized that to get my kids to see it, I would give ’em purple circular stickers to have them put on the “corners” and blue circular stickers to have them put on the “faces” and green circular stickers to have them put on the “edges.” It would help me not only count the different things (maybe put the numbers 1-8 on the purple stickers, 1-6 on the blue stickers, and 1-12 on the green stickers?), but also “see” how they are in relationship to each other. (And George told the class he liked the suggestion and would think about trying it out!) George asked the class what the “fold angle” is for each card (what angle the card was bent at in the sculpture). I loved the question because it’s so obvious when you look at the sculpture from just the perfect angle! (The answer: 60 degrees.)

We See Art and We Build More Art

Lunch was delivered from Dig Inn, and we ate and briefly chatted. And then George took us on a picture tour of his sculptures and their construction. Some choice quotes:

“Kids need to have an emotional connection to math.”

“Math and art are both about creating new things.”

Finally, we ended our day building our own mathematical sculpture. We had 60 pieces of wood that we set up in trios. And we combined those to create a hanging sculpture.

What’s neat is that this hanging sculpture is going to travel to all the schools of the teachers who were at this session for two week periods. It will come to us disassembled and we’re going to get a group of kids (or teachers!) each to build it up and hang it. And then after two weeks, send it on! I love the idea of this same set of 60 pieces being in the hands of young elementary school kids and my eleventh-grade kids.

Takeaways and Random Thoughts

I have recently been into math art. Last year, I helped organize a math-art exhibit in our school’s gallery. I get excited when kids make math-art for their math explorations that I assign in my precalculus class. (In fact, years ago I had two kids make some sculptures and now I know they came from directions George provided on his website.) For me, it isn’t about “art” per se, but about seeing math as more expansive than kids might initially think, and seeing math as a creative and emotional endeavor. That’s why this resonated with me.

At the start of the year, I had intentions of starting a math-art club. Because my mother was sick and I was not taking on any new responsibilities, I decided to put that idea on hold. But now I’m feeling more excited about trying this out. To do this, I want to create 5 pieces on my own based on things I have found online. Things that will kids to say “oooooooooh.” Heck, things that will get me to say “ooooooooh.” (Like the origami image I saw in the second paper I wrote about above.) And then show them to students and get a core group of 4-5 who want to just build stuff with me on a regular basis. Maybe as a stress reliever.

What can we make? Who knows! Maybe stuff out of office supplies? Maybe some of the zillion awesome project ideas that George and his partner Elizabeth have put together. Maybe something inspired by the awesome tweets with hashtag #mathart that I’ve been following (and sites like John Golden’s). Maybe something on geogebra or desmos? Maybe something else? The idea of a large visible public sculpture appeals to me. One that random people walking by can add to also appeals to me. (I tried last year to get a giant cellular automaton poster going at my school, with two students in the art club, but it didn’t quite work as planned.)

Maybe this happens. Maybe it doesn’t. I hope I can muster the energy to start thinking this summer and making this a reality next year.

Random thought: Based on all the photos that George posted showing him bringing his math art to little kids in public spaces, I wonder if he’s talked to Christopher Danielson who organizes Math-On-A-Stick? Or if he knows Malke Rosenfeld (we had talked about math and dance earlier in the day)? I’m hoping yes to both!

Random note: George said that among his favorite mathematical artists were Helaman Ferguson, Henry Segerman, and John Edmark. Bookmarking those names to check out later.

Random thing: At MoMA in an exhibit about the emergence of computers to help create art was fabric that was created by the artist to hold information in it. What was pointed out to me, which made me go HOLY COW, is that the punch card idea for the first computers came out of the Jacquard loom. So loom –> computer –> loom. What a clever idea. I wish I knew what information was encoded in the fabric I saw! Additionally, this reminded me of one of the artists we had exhibited at the math-art show I helped organize: the deeply hypnotic and mathematical lace of Veronika Irvine. And that of course got me thinking about this kickstarter that I’m so sad I didn’t know about until after it was done: cellular automata scarves!

Random last thing: totally unrelated to this workshop, last night someone posted on twitter that Seattle’s Center on Contemporary Art is about to open a math-art exhibition, and my friend Edmund Harriss is one of the artists in that show! Along with the work of father-son duo Eric and Martin Demaine who both do amazing paperwork (and amazing mathematics). So awesome. Wish I were there so I could go see it.

This is a milestone for me. I have been at my school for ten years, and this is the start of my eleventh. It’s the only school I’ve worked at. That’s a testament to my school, but more specifically, to my colleagues.

Last year, my school’s awesome director of communications contacted the math department to let us know that the one issue of the magazine she publishes four times a year was going to focus on math. And she wasn’t kidding! The cover of the magazine had most of my multivariable calculus kids on it (thinking deeply at the math-art show I helped put on last year)!

One of my favorite things is that the feature article with an alliterative title, Making Math Meaningful, was simply the transcript of a roundtable discussion we had. A bunch of math teachers got in a room around a big table, and we were led by our director of communications who had done her research and come with some questions. There was a digital recorder in the center of the table. And through talking with carefully crafted prompts, we got to think deeply and collectively about our own practice. I can’t even tell you how interesting it was to listen to my colleagues during that facilitated conversation, and how proud I was to be in a school with such like-minded folks that I have the opportunity to learn from. (If you’re a department chair or academic dean, consider doing this!)

I wish I could just post a PDF of the article for you to read, but alas, the whole magazine is online but can’t be downloaded. Here are two quotations to whet your appetite:

…

So if you want to read about a department that is doing strong work moving towards inquiry-based learning, and read the words of real teachers having a real conversation playing off of each other, I highly recommend you:

1. Go to this site
2. Make the magazine full screen
3. Read pages 18 to 29

That is all!

“Fundamentally, this is a question about power in society,” said Daniel Zaharopol, BEAM’s director. “Not just financial power, but who is respected, whose views are listened to, who is assumed to be what kind of person.”

***

My friend Dan said this to a New York Times reporter, in the context of an organization that he started called BEAM. It is a pathway for underserved middle school students to gain exposure, interest, and opportunities to see how amazing the world of mathematics can be. This is important. Why?

Guess how many math and statistics Ph.D.’s were awarded in 2015 to black students?

20.

***

So how do we change that? Dan and I were both at a lecture at Teachers College at Columbia yesterday given by Erica Walker. Her thesis? That you need mathematical socialization, spaces, and sponsoring (mentoring) to build positive, strong math communities.

Dan’s organization is doing that.

I highly recommend checking out the BEAM website to read about the ways it is trying to change the status quo.

But more than that, I recommend reading the article the New York Times reporter wrote about BEAM. She took an intimate, in-depth look at the program through the eyes of its participants — from riding the subway together to their discussion of the Black Lives Matter movement to their families. You may get welled up, as I did.

Yesterday, I told one of my precalculus classes how it was an exciting day. I was setting them up because it was election day, and kids at my school are heavily interested in politics, so I thought they’d say “yes! Election!” And I would say: “Actually, it’s because one of my best friends from college is having a baby.”

Of course that setup didn’t work, because of course a kid asked “why is today exciting?” Thanks, kid. But I told the class about my friend’s baby.

Yesterday evening, as the election results came in, I got more and more anxious. And when it was clear that Trump won, I was destroyed. I am not going to use this blogpost to explain my love for Clinton, or why Trump makes my blood boil. Instead, I want to just share how my day has gone.

I teach at an independent school in Brooklyn, and the population of kids and parents we serve are (for the most part) liberal. The kids are politically active and aware and interested. Today, I came to school and kids were destroyed.

In my first class, I talked to kids a bit, and then asked them what they wanted to do. After hearing them, I came up with the following plan. The kids who woke up to the news and wanted to learn more and get informed could read articles online. (There were about 4 of those kids.) I just asked that before they started reading, they take 3 minutes to type out all the questions that they have — so help them start processing. (Like “How could this happen? What was wrong with the polling? Who was voting for Trump? What does this mean for issue X?”). For the others, we formed a circle with the desks and I let kids talk. At points, kids cried. I didn’t join in — I wanted this to be a space for them. They expressed real sadness, hopelessness, optimism, anger, frustration, embarrassment, terror, empathy. I really heard my kids, and when talking about this election, they were speaking their truth, about their hopes and dreams (and how those hopes and dreams were altering). It destroyed me inside to hear them. To see how much this election has affected them. I guess I hated the fact that my kids are feeling what I’m feeling. I don’t want that for them.

I went to my second class, that precalculus class that I told about my friend’s baby. The first thing a kid said to me was inquiring about my friend’s baby. That small gesture — that this student would remember that — lifted my spirits. In this class, more wanted to read the news, and a handful of us talked. This discussion tended to a bit more political punditry — about the what’s and the how’s and less about their emotional state. I suspect they got many of their feelings out in their previous classes.

In my third class, we watched Hillary’s concession speech.I teared up twice during the speech. One kid left to gather themselves for a few minutes after the speech. I didn’t know what to do after. Kids said they didn’t feel like discussing things anymore — they were discussed out — but they also didn’t see how they could focus on work. I made the executive decision to spend the last 20 minutes of class having my kids watch the pilot of the West Wing. I hoped that some optimism in politics might help.

I have one more class to go. It’s a 90-minute block. I’m drained, right now. I don’t have much more in me. I suspect kids are also drained, but I don’t know. I’ll suss out how things are, and try to get through it.

I’m exhausted. Yesterday I woke up at 5:30am to vote. Yesterday I didn’t get to bed until very late (maybe 1pm), and then woke up at 3am to watch Trump’s victory speech. I then read articles until I forced myself to sleep from 4-6am.

Teaching is hard work. Yes, there are lesson plans and grading and meetings and a zillion other things. But days like today, days like today keep me in check. And reminds me how hard the hard work can really be. Because the hard work is being an emotional support. To let kids cry. To let kids know you cry. And to get through the hard times together.

Update: My last class came in with bags under their eyes. I was also tired. I asked them what they wanted to do. A few wanted to continue talking, a couple wanted to do some math and do some talking about the election (a mix), and one just wanted to do math. I decided we would go over the nightly work first, and then talk about the election.

When going over the nightly work, kids were actually focusing better than expected. They asked questions. They were able to answer questions. It was going well. I then ended up going on a fascinating tangent about fractals (related to one of the questions we talked about). And when I realized kids had never heard of fractals before, I showed them a youtube fractal video. Then they wanted to know how it was made. So I gave a short 10-minute lecture on the complex plane, and how the Mandlebrot set is formed. Kids were entranced by the video. I gave a 5-minute break before we sat down to talk about the election. (During the break, kids were in the hall watching more of the fractal video on one of their phones!) When we returned, everyone was silent. No one spoke. I just let it hang there. Eventually one voice. Then another. It wasn’t a rowdy discussion. Not everyone was in it. But most kids had something to say. And then when the day was close to ending, and there was a natural lull, I used a comment about “voting systems” to show a video about alternative voting systems. And then I let kids go home.

I just made the first four slides for class tomorrow. They’re not fancy. I’m tired. But I think they encapsulate what I’ve taken away from today.

***

Now I must end. I now have to change all my lesson plans for the upcoming days, prepare for parent visiting day tomorrow, and write narrative comments. This feels impossible. But I needed to process today.

***

UPDATE: A student gave me a paper flower she made today, to thank me for facilitating a conversation about the election in our class on Wednesday. And that flower is going to stay on my desk all year to remind me of the other things we do as teachers that can be meaningful.

Last week, Technically Beautiful opened. It is a math-art show that I helped put together (with another math teacher and a museum educator) at our school. We have a small teaching gallery at my school, and we wanted to do something special for this space which capitalized on our expertise. That is how Technically Beautiful came into being.

The poster for the show. The name, by the way, came from a #MTBoS tweep!

In this post, I wanted to share with you the gallery virtually. (In a future post, whenever I get a chance to breathe, I’d love to talk about the programming we made around the show and possibly a bit about how the show actually came into being.)

Here’s a walkthrough video of the gallery:

Here’s our vision statement for the show:

The website for the show is here. The five artists featured in the show are: George Hart, Edmund Harriss, Veronika Irvine, the Oakes Twins, and Paul Salomon.

And lastly, here are photos of many of the pieces: