#MathArtChallenge 100: Balloon Polyhedra

In the first, allow me to thank each and every one of you who has participated in the #MathArtChallenge in the last few months. This is my “last” post. Meaning, I don’t promise to make more Math Art Challenges, but there’s always the chance that something will come up…

All of the #MathArtChallenge-s will continue to be up on this blog, and I really hope that you’ll make use of them in your classes or in your fun time or however brings you joy.

Today, you get Balloon Polyhedra. There’s actually several papers written about this, so go check them out.

Materials Needed: twisting balloons, pump, patience
Connected Math Concepts: graph theory, geometric structure

I intended to make all the Platonic solids and all the Archimedean solids, but frankly, after what I did do today, my hands are sore from tying and the balloons didn’t fit in my house any more.

I also intended to take some videos of these, but there are few unbreakable rules in the universe and one of them is: when the neighbor kid sees your balloons, you have to give them to her. (She wore them all in outfits and was just gleeful. It was pretty great to see.)

Depending on how you use this activity, you may engage with different standards. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments!

1.) Make sense of problems and persevere in solving them. How many balloons are needed for each structure? How will you connect them such that you don’t have the balloons double-backing on themselves?

6.) Attend to precision. In creating the models, you’ll have to pay close attention to how each balloon is segmented and which segments connect in which ways to other balloons.

7.) Look for and make use of structure. These often require some serious planning, but their creation leans on your understanding of and the understanding you build of the structure of the edges and vertices of the various solids.

#MathArtChallenge 98: The Most Beautiful Proof

A blue watercolor with gold lines and highlighting.

The Challenge: Pick YOUR favorite proof, or mathematical fact and illustrate it. What’s beautiful about it? Why do you love it? I really, truly want to know.

And it doesn’t have to be fancy. It can be simple. The runner up for me is probably the triangle area formula. The following, however has to be the winner as even ~10 years after I learned it, if I’m reminded of it, it still stops me in my tracks.

MY favorite proof, and still, to this day, the most beautiful thing I think I’ve ever experienced in mathematics, is a proof for how many spaces are created by n-crossing lines. If you just want the prettier part, here you are. This is the proof: IMG_2092

Here’s even a time lapse:

But (insert math teacher persona here) the REAL beauty is, of course, the idea. Which I did my best to explain here:

#MathArtChallenge 97: Tomoko Fusé’s Bird Tetrahedron Origami

The Challenge: Make an origami sculpture using Tomoko Fusé’s text Multidimensional Transoformations Unit Orgiami. (I used instructions on pgs 134, 138-139.)

Materials Needed: Paper (origami paper is handy, but any paper will work) and scissors/paper cutter.
Math Concepts: 3 dimensional building, angles, space filling, rotations, proportions

I love this particular type of origami because you get to determine the shape and scope of the final piece. You’ve basically made yourself some origami legos. If you have a lot of time, you’ll be able to fill space (well, there’ll be holes) with this unit & connectors, and could therefore, theoretically build a really dramatic sculpture, much like one could also do with legos. Just this is with paper!
For mine, I opted for the relatively simple tripe helix with a double rainbow. It took about 8 hours of folding and construction. We’re all a ways a way from this, but someday, post-pandemic, this would be a REALLY lovely project to do as a whole class or in groups.

To build this sculpture, you need square paper, scissors and 3 different folds:

  • Bird tetrahedron (technically a di-triangular pyramid)
    • 3 pieces of nxn square paper per di-triangular pyramid.
  • Horizontal connectors.
    • If your bird units are made with nxn paper, these are n/4 x n/2 paper. (1/8th the area of the nxn paper) 2 connectors needed to connect each 2 bird units. 6 needed to connect 3 units.
  • Vertical connectors.
    • If your bird units are made with nxn paper, these are n/2 squares. (or 1/4th of the original nxn paper) 3 connectors will connect 2 bird units.

Folding the bird units:

Putting the bird units together:

Folding the horizontal connectors:

Putting bird units together horizontally:

Folding the vertical connectors:

Putting bird units together vertically:

Questions to consider:

  • What forms are possible? What are excluded?
  • How much space is actually taken up?
  • Is it possible to get a square grid going here? A cubic one?
  • How many transformations are actually happening?
  • How many triangles appear in each of the tetrahedral units?
    • Each di-pyramid is actually 3 square faces folded along their diagonal to make 2 45-45-90 triangles. What do you notice about that construction? What might it tell you about the angles between the faces?

Depending on how you use this activity, you may engage with different standards. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments!

6.) Attend to precision. (This gets pretty annoying if you’re not at least working toward being precise when folding.)

7.) Look for and make use of structure. How can these units be put together? What is possible and what is not?

8.) Look for and express regularity in repeated reasoning What do you notice about how the pieces do and do not fit together? What generalities can we determine?

#MathArtChallenge 92: W.E.B. Du Bois Data Portraits

The Challenge: Following the style of W.E.B. Du Bois’ Data Portraits, update or create a graphic demonstrating current data. For example, below on the left is Du Bois’s portrait comparing Black and white occupations in 1890 and on the right is my recreation using the closest set of matching data I could find in 2018.

I love his image because it plays with the traditional circle graph and units (hello, #UnitChat!) while highlighting the disparities in job sectors. My data is not a perfect match, but it is relatively close. I invite you to examine both and offer whatever critiques, notices or wonders that occur to you.

I first came across the Data Portraits created by W.E.B Du Bois and a team at Atlanta University from this conversation between Marian Dingle and Chris Nho, and I immediately ordered my own copy of the book.

The entire project (many more images of his work at that link and this) is breathtaking for a variety of reasons. Du Bois did, as Marian explains, create this set of data to convince contemporaries of the injustices still very much realities for Black America. From the essays, it seems he was largely unheard. We are fortunate, however to have the plates from his exhibition because they can serve as such a rich resource for learning, discussion and mathematics.

The mathematician in me is struck by the beauty and creativity with which Du Bois drove home his messages. I know that I have been guilty in critiquing whether students use independent or dependent variables as x or y axes, and a teacher may have criticized the image below, but why? Insistence that there is one correct way to organize a graph is stifling and strikes me as a place mathematicians hinder themselves creatively. IMG_1942

In the image below, the angles chosen to connect proportioned lines are arbitrary, but do draw the eye. “The text paired with each segment reads more like a narrative than a typical key.” as the book plate notes. His design draws the eye to the large red spiral where we can ruminate on the large portion of the population of Black Americans living in rural areas in 1890. The immediate question that occurs to me is: how exactly had their lives changed since emancipation? What does it mean that the graphic would be so different today? Would it, in fact? How different? What happened in the intervening years?

IMG_1943

Du Bois and his team played with size and scale in fascinating ways that also helps to illustrate much of the situations of Black America in 1890. Notice his mastery of a true unit. Though Black America was “one-eighth” of the population ins 1890, notice how many more people that represents compared with the “one-fifth” from 1800.

There’s so much more in the book, and I really encourage you to get your hands on a copy.

The essays that accompany the plates are thoughtful and really dig into the history of the Paris exhibition, which “presented a global stage for nations to strut their sense of national pride,” (Morris, 23). Du Bois’s exhibition speaks to “the gains that had been made by African Americans in spite of the machinery of white supremacist culture, policy and law that surround them. In this way, the data portraits actually challenged the dominant framework of liberal freedom and progress that characterized both the American Negro Exhibit and the Paris Exposition,” (Battle-Baptiste & Rusert, 22). If that’s not a quote to make you question America’s current situation with respect to Black America, I invite you to entertain it.

I, personally, am struck at the timing of the exhibit. Too often, white America thinks of the end of enslavement as a point of pause in American history, but

“As the twentieth century approached, these ex-slaves found themselves exiled in their own land, where their unpaid slave labor had constructed one of the world’s greatest empires. Rather than benefitting from this bounty, freemen and -women found themselves homeless, penniless, stripped of the vote, unable to seek education, and patrolled by whites. Indeed a new racial order was forged.”

Mathematics is, and always has been, a tool to question the world we live in. A tool to highlight what you see that others may not. A tool to obfuscate what you want hidden. I invite you to learn more about Du Bois’s exhibition – I linked these articles above, and more is written here and here. Then perhaps attempt to recreate or create your own visualization.

UPDATE: Naturally, I’m not the first to do this, check out this from Mona Chalabi (via Mark):

#MathArtChallenge 91: 3D Cube Cut-Outs

The Challenge: Find some materials that will allow you to create one of those beautiful 3D 3-letter (or shape? Follow your heart) cut-outs. I used a potato.

This one was suggested by Sam Shah (although for the life of me I can’t find the tweet suggesting it), and while there are no obvious historical or mathematician connections, I really love how challenging it was to find a way to do this. I thought about legos (don’t have enough at home), did actually try it by cutting foam, but it was suuuuper messy and difficult to see the shadows, and considered cheese to precious (the potato made for good hash browns, I assure you I wasn’t wasteful).

Reflection questions & Math Practices Used:

  • The tool I used was a pumpkin carver – what else may work?
  • How might your students interpret this activity?
    • What might they do besides just their initials? I used mine: AKP (Anne Kristine Perkins), but what shapes might work?
  • Is there anything that wouldn’t work here? Something that’s impossible to carve on the 3 sides/perspectives/orthogonal axes (?) of a cube?

How else can we invite students to engage mathematically? Here I believe I engaged 4 of the 8 math practices:

  • Make sense of problems and persevere in solving them – To get the shadows correctly, I needed to play with 3-D space and engage in how the letter shapes would and would not impact each other.
  • Use appropriate tools strategically – I’ve been sitting on this math challenge for a while as I perused all the resources I had – it wasn’t until I remembered the pumpkin carving tools (bought probably 6 years ago) sitting in my basement that I was satisfied with what I was able to create. Scissors in foam just didn’t work effectively enough and foam itself was too difficult to carve appropriately.
  • Attend to precision – To make the letters visible, I needed to design them appropriately so they’d be recognizable after being carved in potato, and so their interactions with each other wouldn’t warp the shapes too much.
  • Look for and make use of structure – Both in selecting potato and in recognizing the sides of the cube I needed to carve and then in how to twist the carved potato to get the requisite shadows in the correct order.

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 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?