#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.

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.)

#MathArtChallenge 99: Quilts & the Underground Railroad

“My mother was told… that slaves would work out a quilt piece by piece, field by field, until they had an actual map, an escape route. And they used that map to find out how to get off the plantation.” -Mrs. Elizabeth Talford Scott, octogenarian African-American quilt maker

Several years ago, after I’d picked up quilting as a hobby, my mother gave me Hidden in Plain View: A Secret Story of Quilts and the Underground Railroad, which explores the possibility that quilts were used as signaling, planning, or storytelling tools to help with the escape of enslaved people on the Underground Railroad.

The account in the book is based on the the story of Mrs. Ozella McDaniel Williams, an African American quilt maker based in Charleston, South Carolina, who was descended from enslaved people:

“Did you know that quilts were used by slaves to communicate on the Underground Railroad?” – Mrs. Williams

That quote started a years long search by Jacqueline L. Tobin, and her co-author Raymond Dobard, Ph.D., to attempt to track down evidence of just such a code. The authors are clear throughout that what they have is not proof, but rather strong conjecture based on oral history and research. They refer to quilting as a “geometric language”, and their story, while perhaps technically unproven, is incredibly convincing and compelling.

There’s a lot about this book that appeals to me, but I’d like to highlight 3 things in particular:

  • It makes a strong case for the need to rely on histories beyond the written word. I wonder how we can apply this to our student’s lives and experiences in our classrooms. How can we value, oral, artistic, and other types of history? How can we let that guide us into other ways of “knowing”?
  • This story makes for an excellent push back on the very false version of history wherein enslaved people did not resist. In fact, much of the book is dedicated to refuting this, while interweaving the involvement of quilts. There’s a long section dedicated to the various types of rebellions of enslaved people, and another specific to spiritual songs and the need for secrecy among the African American community so as to protect their various rebellions.
  • Because the quilting designs are so geometric, it offers us, math teachers, an easy way to honor a far too often overlooked part of the history of the U.S. All of our students (and teachers) should be better versed in the resistance of enslaved peoples and the ways they were active in attaining their own freedom.

I love the exploration of quilts are a form a language. I’ll let several quotes do the heavy lifting for me here:

“Quilts represent one of the most highly evolved systems of writing in the New World.” (Hidden, p9).

“Patchwork quilts were readable objects in nineteenth-century America.” (28)

“Communicating secrets using ordinary objects is very much a part of African, culture, in which familiarity provides the perfect cover.” (35)

“Did the American quiltmaking tradition, in which geometric patterns were given names and meaning, invite appropriation by black slaves and their descendants seeking ways and means to continue their own encoding traditions?” (31)

A huge part of the book, naturally, is dedicated to the specific code that Mrs. Williams shares with Tobin. It is relatively brief, and alludes to 10 common quilt blocks, all illustrated here.


In case you can’t read that, the code says,

“The monkey wrench turns the wagon wheel toward Canada on a bear’s paw trail to the crossroads. Once they got to the crossroads, they dug a log cabin on the ground. Shoofly told them to dress up in cotton and satin bow ties and go to the cathedral church, get married and exchange double wedding rings. Flying geese stay on the drunkard’s path and follow the stars.”

Each of the bolded phrases represents a quilt block shown in the pictures. (The exception being the double wedding rings, which wasn’t popularized until after the end of the Underground railroad – the authors believe this may have represented the shackles of slavery.) For my contribution to this challenge, I made 6 of the 10 blocks (if you’re keen eyed, you’ll notice they’re the simplest of the 10, but forgive me, my quilting skills are  beginner and a bit rusty.)


Part of the frustration in the book is the inability to prove this code directly. The authors do go through and attempt to explain the significance of each block (I will summarize as best I can, there’s a lot of complexity to each of these as detailed in, you know, a whole book):

  • Monkey Wrench likely refers to a skilled, knowledgable person, perhaps guiding the preparation for escape
  • Wagon Wheel references traveling
  • Bear’s Paw may reference actual bear trails needed to be taken
  • Crossroads likely references Cleveland, Ohio, a major hub of the Underground Railroad
  • Shoofly is thought to reference a person, perhaps a contact, who may assist along the way
  • The bow ties cautioned people to dress formally so as to better blend in
  • The log cabin, specifically mentioned to be “dug” into the ground may refer to drawing symbols in the dirt so as to identify persons to be trusted
  • Flying geese likely meant traveling north…
  • …along a Drunkard’s Path, which winds and turns so as to avoid slave catchers
  • And finally the Stars likely meant ways to navigate north or perhaps simply signaling freedom.

Again, that is really oversimplified from the detail the book goes into. I highly encourage you to get a hold of it. Part of the reason so much of the above is couched in “likely” and “may have” is because we do not have direct records of the code, only Mrs. Williams’ accounting and Tobin and Dobard’s research. Deprived, intentionally, of book literacy, enslaved people relied on oral histories, and the code Mrs. Williams shared with Tobin is one, “passed down orally in her family for generations; Ozella received it from her mother, who received it from Ozella’s grandmother.” The authors also lamented the inability to find quilts made by enslaved people, likely because, in addition to possibly helping with the Underground Railroad, quilts were useful objects, often made from poor cloth, and washed with harsh lye soap which caused them to disintegrate. Part of the publishing of this book (1999) was in hopes that more materials or stories may turn up.

For us, as mathematicians, I think the strength of this story is its beautiful connection between mathematics – the geometric language described by the authors – and such an important aspect of American history. I know I have personally heard complaint from my students that all they learn of African American history is of passive enslavement – and while obviously this is deeply tied to slavery, it highlights the people’s rebellion. It’s unclear exactly how the quilts may have been used – perhaps all 10 were set out sequentially to prepare people for escape – perhaps they were literal maps. There’s a fair bit of evidence for the latter. Not addressed in the book, but it occurred to me that perhaps this quilt code was specific to Mrs. Ozella Williams’s family, which might additionally explain why there is not a wealth of record. That’s just speculation, of course.

“Many times in the discussion of the code, Ozella stated that it was the ‘mathematicians’ who devised the code in the first place and that these mathematicians were similar to what we know of today as fraternities.” (p.103). The authors speculate this references Masonic orders, most likely the Prince Hall Masons, an African-American fraternity. How powerful to help connect our students to such an important use of mathematics and geometric language in the heritage of African Americans. And I don’t just mean for Black students – for all students, this can be a powerful lesson in who does mathematics and how mathematics can be used.

On a practical note, I realize that not everyone wants to take up quilting. I chose to do so because

  1. I have some quilting ability
  2. There is soooooo MUCH mathematics in piecing quilts together.
    1. Precision measurement and cutting
    2. Calculations of angles and fabric allowances
    3. Geometric organization: which pieces sewed first, which next, how do the seams come together?

But I imagine you could easily engage students with any of the following tasks, (and perhaps you’ll suggest others?)

  • Designing their own geometric block with its own meaning to them.
  • Literally constructing, with ruler and compass, the blocks shown above.
  • Coloring or piecing with construction paper or something similar, the blocks above.
  • Exploration of areas of positive and negative spacing in the quilts.
  • Possible arrangements of the blocks: I did a 2×3, if I’d made all 10, how could they be best arranged? Is there a possibility outside of 1×10 or 2×5?

On this, the 99th challenge, my REAL challenge to all of us is to make certain we’re no longer abandoning culture and history in our classes. We may very well screw it up, but I would argue that we’ve already done that by pretending math is neutral for so long. Would love to engage in comments or discussion in the comments or on twitter.

UPDATE: Was reminded by @picaresquity of mathematician Chawne Kimber’s excellent modern quilting. Check out her work at completelycauchy.com. I first heard of her work on this episode of My Favorite Theorem.



#MathArtChallenge 98: The Most Beautiful Proof

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

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?

#MathArtChallenge 96: Not so much rational tangles as… a 2 link challenge.

Soooooooo, I planned on having this be rational tangles, but that didn’t work out. If you want to skip to the challenge, scroll down to THE CHALLENGE (it’s obvious, I promise).

Still, I wanted to do rational tangles because I desperately wanted to carve out some time for myself to play more with knot theory and because I think that rational tangles are spectacular (even if I don’t fully understand them yet) But, as they say, I bit off more than I could chew tonight. (Although seriously, if you want to play with me and rational tangles, let me know, I’d love to give poor Professor Noodle Arms a break from my incessant questioning.)

I also wanted to have this be on rational tangles because I’d get the chance to introduce you to Candice Price and Mariel Vázquez, who are both working on biological topology with DNA and knot theory (my heart!)

I was introduced to Dr. Price because she features in an episode of My Favorite Theorem wherein she explains rational tangles and the Conway’s rational tangle theorem. She also happens to be a co-founder of MathematicallyGiftedAndBlack.com which is a wonderful resource for helping to diversify your knowledge of mathematicians. Dr. Price also mentions Dr. Vázquez in the My Favorite Theorem episode, which led me to this gem, and honestly, be still my heart, if I’m ever able to meet either of these women, I’ll act like any normal person meeting Beyoncé.

Unfortunately, I don’t think I can make rational tangles work as a math art challenge tonight (maybe a surprise bonus challenge once I get my feet under me) BUT I can get you closer. At least knot theory is involved.


So here’s the challenge: Using two links – these are CLOSED loops (Use an extension cord for each – you can plug them into themselves, and they’ll work beautifully) – see how many totally unique arrangements you can make. I’ll start you off. Below are a bunch of 2 link knots…(someone’s going to yell at me for that language, just do so nicely in the chat and I’ll adjust)… each unique from the others. Can you come up with others? 

2 link knots

Things to ponder:

  • Can you prove there are an infinite number of 2-link knots?
  • Aren’t Dr Price and Dr Vázquez amazing?
  • If you removed one of the links, (cut and pulled away) how many of the arrangements would leave an unknotted link behind? (This, admittedly, is directly related to my Bridges submission this year. Totally worth your time to peruse the whole gallery.)

#MathArtChallenge 95: Magic Squares

The Challenge: Create a magic square. Bonus points if you make it a physical thing.

Magic Squares go WAY back. I’d heard of them, but hadn’t really dug in until I started reading George Gheverghese Joseph’s The Crest of the Peacock. Here’s the gist:

  • Using a square grid, place the counting numbers (equal to the number of spaces in the grid) in a manner such that every row, column and diagonal sum to the same value.
    • So for a 3×3 grid, you want to place the numbers 1 through 9. The sum of each row, column and diagonal will be 15.
    • For a 4×4, place the numbers 1 through 16, the sums will be… ? (It’s more fun if you work it out.)

Part of my attraction to this is that Joseph shares Yang Hui’s description of how to form a 3×3 magic square, and it felt like a story that begged to be made with legos. So I did that:

Yang Hui’s work on magic squares was published in 1275, and even he said he was “passing on the works of earlier scholars and would make no claim to originality,” (Peacock, 209).

Spoiler alert…. there’s only one 3×3 square. I would love to lego up a 4×4, but I don’t have enough legos currently, but you’re all far more creative than I and I’m sure you can sort out a way to explore a bit with the 4×4 and the 5×5.

Magic squares have been around for a….while. There are examples of them in China back to 190 BCE and in India as early as 587 CE. The Chinese examples were connected to an emperor Yu (Joseph calls him “semimythical”) and are just beautiful.

Yu magic squares

Hui went so far as to create magic circles. Joseph notes that some of his explanations for how he derived the higher-number magic squares and circles were “either absent or cryptic to the point of obscurity”. I would love to see some of you illustrate his magic circle… but I’ll also be thrilled just to see some magic squares.


Things to ponder:

  • How many 4×4 squares are possible? What about 5×5? Can you create an algorithm for deriving them?
  • What impact does it have on the creation if there is a center square (like in a 3×3 or a 5×5 grid) and when there is not (like a 4×4 grid)?
  • How could you use my video (and, please, definitely legos or blocks that the students can play with themselves) to help students think about means/averages?

#MathArtChallenge 94: Twelvefold Islamic Geometric Rosette from Samira Mian

The Challenge: Make yourself a 12-fold Islamic Geometric Rosette.

All of the instructions I followed here can be found at Samira Mian’s website. If you aren’t already familiar with how much I respect and admire this woman, you probably haven’t been following me particularly closely. I couldn’t do the last 10 without nodding to Samira. Her work is an absolute gift to the mathematics community. You should probably definitely go take her first and second online classes.

12fold rosette full

12fold rosette detail

Here’s the time lapse of me making it:

Things to ponder:

  • Check out how different the rosette looks when it’s in a 4 fold or a 6 fold tiling (you can see both at Samira’s page). What shapes change? How are the tilings similar or different?
  • What other ways could you highlight shapes? Is there a way to have a laced pattern with alternating shapes colored or empty? (Positive and negative space)

(p.s. I know this is a day late. I was at the EduColor summit yesterday and it was just necessary for me to spend my time there. I’ll still try to post another later today.) 

#MathArtChallenge 93: Archimedean Solids from Vertex descriptions

The Challenge: Using a vertex description, build yourself one, two… up to all 13 of the Archimedean solids.

Materials needed: Card stock and tape (painter’s tape is great, or masking. Other stuff will work, but I’ve had more success with the paper-y tapes.) OR Magnatiles, but those can get pretty pricey.


Platonic solids are 3D shapes with congruent regular faces. There are 5.

Archimedean solids are 3D shapes with regular (all congruent side lengths and angle measures) faces and identical vertices. There are 13.

So the idea is to get yourself a bunch of equilateral triangles, quadrilaterals, pentagons hexagons (and octagons or decagons if you’re feeling ambitious), and start building!

Below is a timelapse of me building the {3,6,6}, {3,4,4,4} and the {3,5,3,5}.

I think it’s fascinating how challenging it is to predict the finished sizes and number of faces. There are definitely ways to do it, though, so for students who want a challenge, see if they can figure out how many of each face they’ll need without looking it up or building it first.

Here are the vertex descriptions:

Platonic Solids

  • {3,3,3}
  • {4,4,4}
  • {5,5,5}
  • {3,3,3,3}
  • {3,3,3,3,3}

Archimedean Solids

  • {3,6,6}
  • {3,4,3,4}
  • {3,8,8}
  • {4,6,6}
  • {3,4,4,4}
  • {4,6,8}
  • {3,3,3,3,4}
  • {3,5,3,5}
  • {3,10,10}
  • {5,6,6}
  • {3,4,5,4}
  • {4,6,10}
  • {3,3,3,3,5}

To make it a bit easier on you, I have a sheet of 2 inch side length shape PDFs for you:

2in equilateral triangles

2in squares

2in pentagons

2in hexagons

One caveat to this vertex notation: the pseudo-rhombicuboctahedron:

This is actually an activity I have done with students before. Megan Schmidt and I got to run a summer camp for a week last summer and it was just glorious. It was so much fun watching students try to puzzle their way through making these shapes and then drawing connections between them. Some questions to consider:

  • How many faces or edges will each shape have?
  • How are the shapes related to each other? What connections do you see?
  • What other materials might you use?
  • Can you identify the shapes that are chiral? (They have a right or a left turning?)
  • Why are there only 13 Archimedean shapes? Why only 5 Platonic shapes? Can you find more? Why or why not?

#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?


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.