The Last 10 #MathArtChallenge

Only 10 left until we reach the goal of 100 Math Art Challenges.

A thing I’ve noticed is that since I took a pause after George Floyd was murdered (there were just more important places for my and everyone else’s attention) is there’s a notable slow down of engagement. Which is totally fine, of course. Part of my capturing and recording all of them here on this blog is so they’ll be around and available whenever you want to play or when schools start up again in the fall. But I would like to encourage folk to engage again and to capture a bit of the magic that I felt when folk were spinning them into something new nearly every day. So the last 10 will come out over the next 10 days. The finale will be August 5th.

Another part of the slow rate they’ve been coming out recently has been that I’m trying to be a bit more thoughtful about them, which has led to a big of paralyzation on my part. I spend a lot of time thinking, “Is this even worth people’s time? Is it meaningful enough? Is it connected to enough things?” And while those musings are important to me, there comes a time when I’ve mused enough and need to just create and publish. So for the last 10, I wanted to expound a bit on what I’m prioritizing and what I’ve been musing about as I’ve planned these last 10.

I can’t stop thinking about how much of math is trapped by the expectation that math needs to be calculation centered – involving symbols and paper scratching. Of course there is freedom that comes with exploring the meaning behind those symbols and scratching, but there are so many other ways to experience and expand math. I think a lot about the de-colonization and re-humanizing of mathematics and how worship of the written word is wrapped up in preventing us from expanding our perception of what mathematics is.

Mathematics is beautiful. That’s important to me.

Mathematics is powerful. It can persuade for good and evil purposes.

I want to help us explore mathematics in ways that celebrate the historical importance of mathematics in a variety of cultures, and in ways that expand your idea of what math is and can be.

Finally, mathematics can be used to exclude when it’s too mystified. Part of me thinks that by keeping the planned #MathArtChallenge-s to myself, I’ve been a bit exclusionary. So I’m posting at least the titles of all remaining 10 below. Feel free to engage early if that’s exciting to you. I have put a lot of work into this, but it’s not my endeavor alone. It wouldn’t be anything if y’all hadn’t engaged.

So the last 10 will be:

91. Monday, July 27th: 3D Cube letter shadows suggested by Sam Shah

92. Tuesday, July 28th: W.E.B DuBois’ Data portraits

93. Wednesday, July 29th: Polyhedra creation based on vertex descriptions

94. Thursday, July 30th Friday, July 31st: A final Islamic Art design

95. Friday, July 31st Saturday, August 1st: Magic Squares & Circles

96. Saturday, August 1st Sunday, August 2nd: Rational Tangles & Candice Price

97. Sunday, August 2nd Monday, August 3rd: Creative origami sculpture

98. Monday, August 3rdTuesday, August 4th: The most beautiful proof I’ve ever seen

99. Tuesday, August 4thWednesday, August 5th: Hidden in Plain View quilting patterns, oral history, and the Underground Railroad

100. Wednesday, August 5thThursday, August 6th: Celebration (So this one’s still a bit of a secret – indulge me, but if you want to play along you probably want to get a hold of some twisting balloons, which is maybe enough for you to guess what I’m planning.)

#MathArtChallenge 90: Sand Piles

The Challenge: Explore a toppling sand pile.

Here is my explanation (note, I’m pretty sure I have a mistake in toppling 20)

Here is a video that Numberphile put together describing them as well:

Reflection questions:

  • Is it possible for a pile to topple differently? Does it matter if different stages appear different or is it necessary for the toppling to be consistent?
  • What would an isometric (triangular) grid toppling looks like? How about a hexagonal one?
  • How might you meld this with computer programming?
  • Is there a less or more illuminative coloring for the sand piles? What kinds of coloring may make the sand pile structures more obvious or less?

#MathArtChallenge 89: Voronoi diagrams

The Challenge: Throw some random points (or carefully selected ones!) on a plane. Identify the parts of the plane that are closest to each of those points.

Voronoi Diagram Sketched in Notebook

I made mine (above and below) using perpendicular bisectors between the points shown, but are there other ways? How might you reason your way into making one of these? (Hint, start with just 2 points, then 3, then…)

Part of my reason for selecting these is their versatility for the classroom. There are so many ways to use them for deep mathematical discussion that can leverage your student’s knowledge. In the map below, I marked the large Minneapolis Public High Schools with a star and then created a Voronoi diagram. THINK of the conversations you could have about this map.

IMG_1847

FOR EXAMPLE:

  • Which high school, just based on this map, do you think is the largest?
  • Are the high schools reasonably distributed throughout the city?
  • Which high school is easiest for the most people to get to? Which is the hardest? Just THINK about what your kids could bring to this discussion. Their knowledge of roads, bus maps (MPS students use the public bus system for high school), safety, residency.
  • If we were to add a new high school, what area makes most sense to place it in?
  • Where do you think students are most densely concentrated?
  • How else might we make this map? Would it make sense to instead base the lines on a street grid map? (So you’re closest according to driving distance, not how birds fly?) Or based on population?
  • What are the benefits of true community schools (defined here as “closest school to you”)? What are the drawbacks?
  • If we laid a map of the elementary and middle schools over this, what would you expect to see? What would you want to see?

And naturally, there are easy ways to make these digitally. Here’s a quick version made in Geogebra:

#MathArtChallenge 88: the Recamán Sequence

The Challenge: Create a visual (or audio?) of the Recamán sequence, created by a Colombian mathematician, Bernardo Recamán Santos (who seems to have very little biographical information out there??). I was first introduced through Alex Bellos and Edmund Harris’s book.

The formal rule for the sequence is:

recaman

In other words, start at zero, and then add 1, then add 2, then add 3 and continue using the next integer to get the next number BUT if you can go back, go back. So after you’ve added 3, you’ll be at 6, and will then be able to go back to 2. This video explains it all much better than myself. I also love that the video explores the sequence in audio form. I’ve been challenged recently to think of how we might think of ways that math is exclusionary, and I think that insistence that it’s written down is a constraint we could lift. I love that this explores it’s audio manifestation.

I also love that from relatively simple rules, we get such complexity. If you watch my video up above, you can maybe pick out the spot where I had to stop counting and needed to work the sequence out on paper, which nearly led me to asking whether or not the Online Encyclopedia of Integer Sequences was incorrect as 42 appears twice, but sure enough, it is in the sequence twice, once reach from above and once from below. Which makes me wonder what the sequences of numbers is that’s doubled in this sequence? Maybe tripled? Is that even possible? I wonder. Infinity offers a lot of room for exploration.

Reflection questions:

  • How else might we represent this sequence?
  • What is always true about the sequence values? What could never be true?
  • How does the audio version help us learn more about the sequence?
  • What sequences result from tweaking some of the rules for this sequence?

#MathArtChallenge 87: Burr Puzzle Origami

The Challenge: Fold yourself a 6 piece puzzle that comes together as a 3D star shape. You’ll need 6 pieces of paper.

The Background: This is a bit of a hodge-podge of things. First, it’s origami. I followed these PDF instructions, and then made this video for those who may need the visual aid.

Where the puzzle originated is a bit of a mystery. The name “burr” is likely to come from the finished shape of these puzzles – resembling a seed burr. There are examples of it in Chinese print, and an English engraving, and similar puzzles making a resurgence in Kerala called Edakoodam. I’ve had this one on my list for months now, and was surprised to see the variety of burr puzzles out there. I know that I have some (in my quarantined and inaccessible classroom right now) that I have yet to solve. I love the idea of the puzzle being in how to make wooden pieces (a more traditional medium than origami for these) manipulable while appearing, when put together, to be immovable.

Reflection Questions:

  • How does the folding transform a 2D shape into a 3D one? What supports are there?
  • Are there other ways for the 6 pieces to go together?
  • How can this teach us about the benefits (or drawbacks) of precision when we’re folding the pieces?
  • What do you notice about the final structure? How could you describe the structure to someone who could not see it?
  • These puzzles are frequently (traditionally?) wooden, what is gained or lost by making them from origami?

Sources:

http://www.robspuzzlepage.com/interlocking.htm

https://www.artofplay.com/blogs/articles/history-of-the-burr-puzzle

#MathArtChallenge 86: Tla’amin Basket Weaving: Coding and Indigenous teaching

The Challenge: Play with the widget here to explore your own Tla’amin basket weaving patterns.

The Background: A student of mine, Willa Flink, wrote a brilliant paper using this website on Tla’amin basket weaving this year. The Tla’amin nation is an indigenous culture in British Colombia. Part of Canada’s Truth and Reconciliation Commission calls to action, calls for culturally appropriate education. This collaboration between the Tla’amin nation and Simon Fraser University’s Math Catcher Outreach Program is one attempt by the mathematics community in Canada to do just that.

The creation of Tla’amin baskets, requires “precise measurement, the creation of appropriate shapes and adhering to certain well-established patterns.” The collaboration which resulted in the widget and Jupyter notebooks within the program online learning platform Callysto.

Per usual lately (thanks Siddhi!) this challenge has very little to do with my creation. Really, just go to the website you don’t need me anymore. But if you’re still here, I want to make note of several things this project does well.

  1. Person and Land acknowledgment. They start off by acknowledging those who taught them this skill and that this work was done on unceded indigenous land. acknowledgements

2. They make it “math teacher friendly”.math teacher friendly

Using language like I’ve highlighted, they help math teachers see the academic merit in investigating these patterns. There’s a larger conversation to be had for why this is maybe not the way we need to go forever and always (I’m up for it, comment away below or hit me up on twitter), but right after the acknowledgment, they start talking in terms that make math teachers comfortable and more likely to use the content. I yearn for a day when we can all see meaningful mathematics in places outside of textbooks and tests, but until we get there, I want to notice their effort to explicitly name the mathematics.

3. For those who can code, you can play with the code. I am a baby in this regard, but I love that this page is easily manipulable by those who have those skills. I am SUPER excited by the possibility that a few years down the road, I could have my students play with indigenous mathematics and coding simultaneously. (I’m sure there are many other ways to get it done, but it’s so great that this already exists as a starting point.)

coding

Finally, what strikes me as fascinating about this project, is that I have spent soooo much time in my life as a math teacher teaching things like reflections and translations, and I have never (certainly not in recollection) connected it to culture. That’s ridiculous! There are so many examples of artwork involving tilings like what’s present in these basket weavings. And here, we have a beautiful, already prepared example of how to do this and implement it. AND if you’re distance learning, this is an excellent project!

(FYI, the widget didn’t work for me right away, but that could definitely have been just me. I followed the page’s suggestion to: “click Kernel then click Restart & Run All.” and it worked beautifully for me.)

Reflection questions:

  • What kinds of patterns are not possible? (If you know much about coding my guess is you can play with the code on the page to get it to do more)
  • Is it possible to achieve the same result from different operations?
  • Is there a place you see similar patterns in your own culture? How are they similar? How are they unique?
  • What skills do you think are necessary to translate these digital patterns to physical baskets?
  • What does it mean to have mathematics reflected or embedded in culture?

#MathArtChallenge 84: Tangrams

The Challenge: Get yourself a square of paper (or something else?) and cut yourself a set of tangrams. Then create away!

The Background: Tangrams have a bit of a muddled history – they’re definitely Chinese in origin, but I’ve found accounts of them existing as early as 1796, some argue 1815, and of course, there was a European guy who made up The Eighth Book of Tan, which has been proven a complete hoax.

There are OOOOODLES of tangram “puzzles” out there, but I’d argue you may have more fun with students just creating their own creations, and THEN you introduce the “tangram paradoxes”. For example, the three shapes below appear to have the same outlines, and yet definitely not the same. Subtle changes in how they’re arranged definitely encourage students to start wondering and noticing and calculating areas.

There are, of course ways to just play with shape. You can make cats, or the middle shape here is fun because it looks to me right now like a bird, yet if you rotate it, I start seeing flowers or snakes. The last shape invites me to start considering area and proportion among the triangles shown.

Reflection questions:

  • How do the areas of the shapes compare?
  • How might you describe the “size” of each shape?
  • How many equal edges are available? How about edge combinations?
  • Which of the shapes are most alike or most different?
  • Can you create other “paradoxes”?

 

#MathArtChallenge 83: African Fractals

The Challenge: Create a self-repeating pattern – a fractal. You may choose your own design, or perhaps you recreate some of the ones from Ron Eglash’s survey studying the fractal formation of African villages. I did both of these looking at the applet at his website.

The History: Ron Eglash noticed aerial footage, in 1988, showing that the layout of African villages built in fractal formations. He got a grant to travel the continent for a year studying these fractal formations, and wrote a book (a PDF available here). This article does a nice job of outlining what he discovered, and his TED talk is below:

I really love the depth with which he’s studying how the creators of these villages consider them and their structure. I must admit, I have not yet read his entire book – just passages here and there. It is certainly on my “currently reading shelf”.

His conclusions are fascinating, and raise all sorts of new questions for me. For example, he hypothesized that all (or most) indigenous cultures would replicate these fractal patterns, but found that to not be the case. These fractals are largely unique to Africa.

“But if fractal architecture is simply the automatic result of a non-state social organization, then we should see fractal settle­ment patterns in the indigenous societies of many parts of the world…we will examine the settlement patterns found in the indigenous societies of the Americas and the South Pacific, but our search will turn up very few fractals. Rather than dividing the world between a Euclidean West and fractal non-West, we will find that each society makes use of its particular design themes in organizing its built environment.” (Eglash, p. 39)

Chicago’s WBEZ has a great piece (~50 minutes) about this from last year that led me to Eglash’s fantastic computing website https://csdt.org/ There, you can play with fractals, read some about the history of various ethnomathematical concepts and explore how and where mathematics appears in culture. I made this using an applet on the website:

csdt applet

The website and WBEZ piece go into bit about African hairstyles and pulls in Nnedi Okorafor‘s brilliant book Binti, which is an Afrofuturistic novel about an African girl who is a mathematical genius. The mathematics behind cornrows also appears in Dr. Shelly Jones’s book, Women Who Count.

In sum, I’m grateful for the depth of Eglash’s work highlighting these structures, although I do take issue with at least one of his premises:

“Unconscious structures do not count as mathematical knowledge, even though we can use mathematics to describe them.” (p. 12)

I think of these “unconscious structures” as the foundation of human mathematical knowledge. I want to celebrate my student’s inborn mathematical tendencies as the first layer of mathematical understanding. To honor the idea that teachers should never be the sole “knowers of things” in a classroom we need to celebrate the inborn mathematics our students bring to our classrooms. We want to invite and encourage our students to build upon their inherent mathematical tendencies. That said, I do appreciate the distinction Eglash makes between those structures built mathematically without conscious effort and those that are built conscious of mathematical structures. They are worth studying differently.

So, go, explore the website, read the book, and play with fractals. And ABSOLUTELY, when you bring fractals up in class, make sure that if you’re sharing that Mandelbrot “discovered” them in the 1970’s, share that he definitely wasn’t the first. Let our Black students revel in and explore their mathematical history.

82 #MathArtChallenge meets #MathPhoto20 in “What is Math Art?”

The challenge: Define math art for yourself, capture an example of it in a picture (or a series of pictures), and share with #MathPhoto20 and #mathartchallenge.

This week #MathPhoto20 and its organizers, Carl Oliver and Erick Lee, are also going to be engaging with the #mathartchallenge by creating some math art challenges and posting those with the same hashtags. Our hope is to spark a discussion about math art, photography, and where and how we see math in the world.

The context: Math art is a poorly defined term. Definitions can be challenging, but clear, precise definitions help us communicate better. I’ve occasionally found heated disagreements quickly resolve once its noticed folk involved are using different definitions. For example: I usually hold that lines do not need to be straight, but for the sake of proving some geometric properties in Euclidean geometry, I’m happy to situationally accept the requirement of “straight”.

80+ challenges in, I would hope we have some *feel* for what math art is (if you need more inspiration, check out the Bridges galleries), but as yet, we are not working from a common definition. Much as art is likely to have a differing definition among many people, I’m perfectly comfortable with us each arriving at our own definitions for math art. That doesn’t make this exercise fruitless – I think in grappling with what we believe math art to be, we can strengthen our connection to the art and the mathematics.

Personally, I will say that it bothers me when we’ve done a math art activity in class and students gleefully exclaim, “We did art instead of math today!” I know some teachers have used #mathartchallenge-s “instead” of math. While I appreciate the enthusiasm for the tasks, this framing grates on me. If you’re doing math art, you ARE doing math. It is as though an Advanced Algebra class, said “we did geometry today instead of math”! Completely nonsensical to me. That said, I can acknowledge that if a significant chunk of my “students” have the same conception, there’s something in my teaching that’s not getting through. 

So, I’ve been digging into the question: “What is math art?” I know it when I see it, but can I wrap a definition around it? There are times when I’ve found a piece of math art, but upon reflection, I discard the label for that work. The reverse has also happened. Something I had dismissed as not math art becomes vastly more interesting when someone draws me in by highlighting the mathematical ideas embedded in the piece.

If we can define it, perhaps we can help our students, colleagues, and friends notice it as well. No worries if they take the definition and bend it to their liking – the discussion is what we’re after in this challenge. I accept that some look at the math art I create and think it fanciful to imagine I’m engaged in mathematical ideas. I have yet to meet and converse with someone that was unwilling to revise that opinion upon discussing the pieces with me, however. Unsurprisingly, I have little tolerance for those who think that being “art” makes it somehow less rigorous, less intellectually challenging, or less mathematically worthwhile, although I concede that some discussion may be necessary to help people see their error.

So go forth, my friends, and grapple with the definition. What makes something math art? What elements are necessary? Compare your definitions and images. How does your definition differ from your friend’s? I’ll post along with you. I have a working definition right now, but it’s open to revision upon meeting a reasonable argument.

UPDATE: Here is my contribution for the week:

#MathArtChallenge 79: Knot Surfaces & “Why” Diversity

Inspired by this tweet (below), the goal here is to make a knot surface*.

To my great shock, although it seems sensical in hindsight, these knot surfaces (at least this one) function like a möbius one sided strip**. There’s only one edge! Seems like a rather egregious failing on my part that, knowing these strips exist, I hadn’t even considered other surfaces that may have a single edge. Here is my crocheted version.

I followed these instructions to crochet this. Will definitely be trying for some general knot surfaces soon. Replications of some of Hanne Kekkonen‘s work is likely soon to follow.

This particular challenge came about because I’m spending some of my summer working with a friend on knot theory. We spent a fair bit of our last conversation talking about how our different “natural” ways of approaching knots complement and can build on one another. To make sense of knots myself, I immediately needed to hold and manipulate them – I crochet them so I can stretch and move them to see their various projections and start “knowing” them in 3-space. Donnie, on the other hand, imagined an electron traveling the length of the knot, and how it might be acted upon by forces from the various crossings. It was a wonderful discussion and has expanded our ability to consider knot crossings (which is what we’re investigating).

That, right there, is why we need an actual diverse field of mathematicians. Two different angles for the same problem that cracked it more open for both of us. We decided that there has to be a way to discuss the relationship between the over/under crossings in a knot, which led us to the space between them…hence knot surfaces.

And that’s just two people. Who already share a lot in common. Imagine where we could get with 3, 4, 10…50 people, especially if they bring vastly different lived experiences and perspectives to the table.

I am routinely in class with 30+ other humans, but I don’t often have these kinds of conversations in class. Do kids come up with “new” ways to solve problems? Absolutely. But the “different” methods in class are often just different algebraic paths – rarely as disparate as crochet and electrons. I want to dig into why that is. Why there aren’t broader solutions/methods coming out of class. I can think of 3 main contributors to that:

  1. Expectation of where knowledge already exists: One thing that encourages open conversation between Donnie and myself is that neither of us know knot theory that well, so no one has an upper hand – I’m not waiting for him, nor is he waiting for me to “reveal” the path. That’s a hurdle for us to overcome in math class. Students are often (I know because they tell me this explicitly) waiting for me, the teacher, to “reveal the path”.
  2. Limitations rooted in what is “known” & what the goals are: Donnie and I have no set goal beyond “learn more about knot theory”. That means we can move in whatever direction we want to go in. In math class, I may get kids exploring symmetry, and it’s often because I want to get them to describe the difference between rotational and reflective symmetry. End of story. There’s a fixed goal that limits our ability to stretch beyond it.
  3. Time: With as much material as we’re supposed to cover, we rarely let students actually play with math unless we see that play headed in a productive direction. That stinks.

So the reflective questions I’m currently considering with this knot surface are:

  • Why do we name mathematical ideas after people? Rather stupid to imagine they’re the first or only person who independently explored that idea.
    • How might our ideas about mathematicians and who “does” math change if we were to abandon these names?
  • How do I better set myself and my students up to challenge the ideas of what we “know”?
  • How can I introduce more play into class and actually leverage the diversity of my student’s experience and knowledge?

*They’re often referred to as “Seifert surfaces”, and, fine, credit where it’s due, as best I can tell Seifert was the first to describe them, but I’m going to call them “knot surfaces” because that’s why I’m interested in them here, and I’m frankly it’s exhausting to always credit the European “discoverer” of something that naturally exists in the universe. Especially when so many of those “discoveries” eventually are found to have also been discovered by someone else before or independently of the man who gets credit. 

**Blech. And of course möbius strips are named after a guy who “discovered” them in 1858, except that of course he didn’t discover them