#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.

#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.

 

#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

#MathArtChallenge 78: Ferozkoh Jaali & perspectives

This one is a bit philosophical – but does get to the heart of my passion for math art.

I use math art as a way to open doors normally closed in math classrooms. In my experience, often those students who carry “high status” in a math classroom (think about who bestows that, how or if it is challenged, what the implications are for each student surrounding that “status”) are those students most intimidated by math art. Traditional math classrooms are often centered around a correct or best answer, and not the path leading to it. Art is a way for us to slow down, think deeply, and create avenues for our student’s brilliance to shine that are not as available otherwise.

I’m currently reading Francis Su‘s extraordinary Mathematics for Human Flourishing and geez is that title accurate. One of his mathematical problem vignettes comes from this image from Catriona Shearer.

I immediately recognized the pattern as similar to the Ferzokah Jaali (it’s in Samira Mian’s first online course you should definitely sign up if you have any interest in playing with Islamic geometry).

It’s a well known design among Islamic geometrists, likely because of the relative simplicity of the underlying grid, although I really struggled to find an origin picture. This image (bottom) the best that I’ve yet found, which came out of this online PDF. I cropped it to include the page number so you can find it for yourself.

Ferzokah jaali

Based on internet sleuthing, “Ferzokah” refers to the capital of the 12th-century Ghorid Dynasty in Afghanistan, but I couldn’t find any images of this construction dating to then. Unclear to me if this pattern originated then or if a style/location is simply being referenced. The image in the book above references the pattern’s appearance in Agra Fort (India), Iran, and Cairo. I wonder if this pattern originated in the 12th century and spread to these other locations, or if it was discovered independently in any of them. Makes me want to dig in a lot further to the origins of these designs. I refer to them as “Islamic design” or “Islamic geometry”, but the history of them seems far more complex.

After discussing some of those origins with students, I would invite them to create the underlying grid for this pattern and then let them loose. With a little exploration, you can work it out from the images below.

One of the reasons I like this pattern is that it’s easy to mess up with some interesting results:

And, as the video at the top shows, there are so many ways to interpret the underlying grid. There’s a lot of room for students to play here, and mathematical play is one of the best ways to invite students to dig deeper into deep, rigorous mathematics.

Reflection questions you may ask:

  • How many different ways can you design this pattern and still have it be recognizable?
  • What are the core components of this construction? At what point is a design no longer “true” to the construction and becomes something else?
  • Does your brain naturally intepret this a a 3, 6 or 12 fold rotational pattern? Or something else?
  • What is an ideal ratio for negative to positive space in this? Why do you think so?
  • In the images that turn this into linked rectangles, is there a best way to order the intersections where 3 links overlap? Is it possible to make the links alternating?
  • What other ways can this grid be interpreted?

#MathArtChallenge Day 77: Chaos Game and Fern Hunt

Today’s math art challenge was inspired while I was perusing the excellent Power in Numbers by Talithia Williams. Her chapter on Fern Hunt indicated one of her research interests as Chaos Theory – something that always grabs kids attention, and leads to one of the more fascinating probability related math-art creations.

Dr Hunt’s main focus was in “dynamical systems and applied probability”. Nevertheless, the connection to chaos is enough to open the door for a probability lesson where you can teach about Dr. Hunt and her work before leading into the following activity.

  1. Draw a triangle.
  2. Number each corner such that rolling a die indicates them each equally. (I numbered mine with {1,2} {3,4} and {5,6} on each corner so each vertex had a 1/3 probability of being selected with a fair 6-sided die roll.)
  3. Place a point in the triangle randomly.
  4. Randomly select a vertex (I rolled a die for this) and using a ruler, mark a point exactly halfway from your original point to that vertex.
  5. Repeat…. MANY times.

Ideally, this is an excellent activity to do on transparency paper in a class, and then overlay everyone’s images. The end result is rather surprising. Here is what I was able to do on my own (with about 2 hours of podcasts).

Simply because it takes a long time for the image to resolve itself if you’re doing this by hand, this may be a good one to try to code (not a current skill set of mine). This numberphile video does a nice job explaining and unpacking the chaos game, but if you start it where I have tagged it below (3:10) you can see the form taking shape:

 

Questions you might ask yourself or students participating in this:

  • Can you predict the shape that will form?
  • Were you surprised by what appeared?
    • What is surprising and what may not be surprising about the final image?
  • How is “chaos” apparent here or not?
  • What about the “rules” for creating this imposed structure here?
  • How might you alter the rules to get different results? (The wikipedia page actually explores some neat answers to that.)
  • What do you imagine are actual applications for “applied probability”?
  • What do you notice or wonder about the accuracy of by hand (mine above or yours created) and computer generated images? Is there a boon to one over the other? When may “by hand” be more useful? What about “computer generated”?

 

#MathArtChallenge 76: Decagon & Pride Flag

Here is my first attempt at a more thoughtful #mathartchallenge. Pushback, thoughts, additions you’d like to see all welcome. You can comment below, or here, or talk to/DM me on twitter.

June is Pride month. I try to always acknowledge and honor that in my classroom. Last year, I put up this display in my classroom window:

This year, it made sense to create a rainbow #mathartchallenge to post for my kids to see. Having seen pride flags with Black and Brown stripes, noting it as a nod to a more inclusive pride, I picked a decagon design that I wasn’t totally sure how to create from Arts and Crafts of the Islamic Lands , and voila.

Except, of course, that I messed up. I didn’t do my research. ANY research. The original rainbow flag is 6 colors: red, orange, yellow, green, blue and violet. There are new designs that include Black and Brown stripes, and as of 2018, designed by Daniel Quasar, there is this flag:

Daniel Quasar flag

This flag includes not just the Black and Brown stripes, but also nods to the transgender pride flag. You can read about it here. 

I also want to speak the names of Tony McDade, a Black trans man killed by police, and Nina Pop, a Black trans woman killed in her apartment, both within the last few weeks. 

When I posted my ill-informed rainbow, I hadn’t necessarily planned it to be a #mathartchallenge. However, seeing the collaboration of Tina Cardone and Xi Yu, I think it makes for a good exercise. If you follow the tweet, you can see their solution – the tutorial Xi refers to is also here.

You may also notice they actually tagged their creation with “Queer #BlackLivesMatter”, which I failed to do.

Here is a better version.

MAC 75 Decagon pride flag

Better, I think because it honors (as best I could match) the actual colors of the pride flag, including those represented by the transgender flag. Not great yet, because as any decagon has 10 parts, I just ran out of space for all the colors (an 11 sided shape is rather challenging to construct). I would like to note that in the trans flag, the white color represents gender neutral or non-defined gender. Given the white paper, it seemed the easiest shade to not include. By no means do I mean to exclude gender neutral or non-defined gender people in this representation.

If you choose to participate in this #mathartchallenge, here are some questions I’d like you to reflect on:

  • The larger shape here is a decagon (10 sided shape). Where else do you see the theme of 10 appear here?
  • How many? What did you count and how did you count it?
  • What angles are represented here?
  • What shapes do you see that are not decagons?
  • How do the shapes interact with each other? What can you say about the relationships between them?
  • Can you find an 11 sided design that would better represent Daniel Quasar’s 11 color-flag?
  • How do symbols, like flags, interact with identity at large? With your personal identity? What thoughts about representation are present or missing here?

Current & Future Plans re: Math Art Challenge

The Math Art Challenge has been on hiatus for about a week now. Mostly because it’s jarring to see folk happily engaging in math art while protestors are getting arrested. I couldn’t conscionably post things about the Hilbert curve, knowing it would divert time and energy that we need focused elsewhere.

I am keenly aware that a lot of white educators are doing more harm than good right now. Often because we’re moving too fast in an attempt to assuage guilty feelings that are hard to sit with. I am trying to let myself sit with and consider those feelings while also making sure that I am taking thoughtful, productive action and planning to be in this for the long haul. Because we need to be here beyond this week. Especially white folk. Especially white educators.

I’m just now starting “summer”, and I want my time this summer to be spent thoughtfully planning for how I can do better, and how I can help support other educators to do better this fall. As I’m considering what I can contribute to the dismantling of white supremacy, specifically that within schools, right now, I’m trying to hold these things in my mind: 

  • I am not an expert in community organizing, nor in the dismantling of police brutality. I can listen to, support, and lift up the voices of those who are. 
  • I am growing my expertise in math education, and how math art can be used in math class to refocus education on patience, deep thinking, and connection making. 
  • Whiteness pervades math curricula and the structure of math classes in America, and it must be dismantled. 
  • Math classrooms are deeply segregated. Both the students that comprise them, and the way curriculum is presented. If we are to change that, I believe the fundamental structure of math classes has to change. 
  • Many of the followers of the #mathartchallenge are white educators. It is not lost on me that when I post pretty math art, it gets many likes, while posts about #BlackTransLivesMatter gets crickets. I can marry the two better so that when you come to me for math art, you also are challenged to consider how you are upholding or dismantling white supremacy. 
  • Given COVID-19, there has never been an easier time for us to change what math class looks like. We already know that the fall will be different. Let’s seize on that. 

The whole impetus for the #MathArtChallenge was a desperate need to stay in touch with students after we left the school building. I knew worksheets and traditional lessons would not engage them, but that math art could, and it could do so in a meaningful way. Like pretty much everything else that happened this spring, the MathArtChallenge was put together haphazardly, without a lot of deep thought going into the specific outcomes nor goals beyond getting students to engage somehow in mathematical thinking from afar. 

I know many teachers are doing great work in their classes, and some people may really have broken out of the factory model of education, but too many math classes (including mine) are still primarily focused on getting through a list of standards rather than championing our student’s brilliance and giving them space to direct more of their own learning.

Considering it now, had I thoughtfully planned from the beginning, here are the goals I’d like to see accomplished by the #mathartchallenge, all of which are aimed at disrupting the factory model of education.

  1. Create a bank of resources, that anyone, but teachers specifically, can draw on, to inspire creative mathematical thought. Aware that teachers commonly only use resources directly connected to their curriculum, I will do my best to go back and tag each of the Math Art Challenges with content standards. I understand that ease of use will make the resource more likely to be used. I also know that simply by virtue of being art, math art opens possibilities for students to creatively explore mathematically. That is simply less true (if at all) of traditional standards-based learning. 
  2. Research and share the historical context for as many #mathartchallenge-s that I can. If we’re to decolonize the curriculum, we have to stop pretending that all math comes from Europe and that the only mathematical thinkers worthy of celebrating are old white dead men. That takes active, intentional work and planning. To that end, I’m committing to also spending much of my summer updating, revising and adding to the Mathematicians Project. With respect to the #mathartchallenge, that will also mean doing things like connecting the Islamic geometry challenges to actual mosques where the original designs were found. Highlighting mathematicians, specifically non-white mathematicians, who do work related to each challenge. We need to connect math to history. And specifically to the history that has so long been denied Black and Brown students. The history that centers the brilliance of their ancestry. 
  3. Encourage patient, slow thinking in classrooms. Art takes a while. That’s a huge benefit. The factory model of education has not worked for way too many students, and it doesn’t take much thinking to know why. When I seek to learn a new math concept, I mentally prepare to take enormous amounts of time to deeply engage. I give myself room to explore paths that may very well take lots of time and lead to dead ends. I don’t place a schedule on when I must have reached mastery. Though I get why people create and use pacing guides, if we’re to center math class on actual mathematical thinking and on our student’s humanity, we need to throw those in the trash heap. Yes, definitely, some standards will have to go. Tough. We should be focused on teaching students how to learn, not making sure they know every math concept we can cram in. If they know how to learn, they can teach themselves whatever concept they didn’t get in school when it becomes necessary for them. I truly believe this shift makes math class more welcoming and reflective of what math actually is. 
  4. Plan for necessary reflection and connection making. I have, thus far, utterly failed at this, trusting that in the process of creation, we end up naturally reflecting and making connections. But if we’re to effectively use math art not just as a healing source (though that is a wonderful and necessary thing that art can provide), we need to be intentional in the ways we’ll ask students to process their creations. To this end, I plan to spend time this summer going back to formulate reflection questions that can be used broadly, for any #mathartchallenge-s, and specifically, to be used for individual ones. 

I want to be clear that I understand that planning to disrupt a factory model of education is not going to happen overnight, nor am I the first to encourage or work toward it, nor does this work absolve me from doing work elsewhere: supporting protests, engaging politically, being intentional with how and where I spend money.

I am teaching two new-to-me classes next year, and I already hear voices in my head:

“You still have to prep them for the test”

“How will your co-workers react?”

“What will the parents say?”

“You need to hit all the chapters.”

I want to be clear. Those voices are the ingrained white supremacy that pervades education. I firmly believe that slow, deep mathematical thinking, as opposed to a factory model, will benefit students. So I need to research and prepare to teach this way, and to address these voices, focusing primarily on my student’s brilliance and humanity. 

This summer, I do plan to finish out the 100 Math Art Challenges. They are not likely to come out one per day, because I hope to actually take the time that summer affords me to do more intentional research. I hope to have each of the coming challenges rooted in some sort of larger context, whether that is through connection to a non-white mathematician working in a related field or through historical connections. 

As with anything, collaborative work is stronger than individual work. If you are interested in helping me with this, I would love to collaborate. I am also keenly aware that my plans may need to change as I progress through this. Push back on any and all of this is welcome, especially if it centers student learning and making math class more equitable for students. 

 

 

#MathArtChallenge Day 75: Black Lives Matter

People keep asking how they can help. So here are some ways you can help.

  1. First, please say out loud with me right now, “Black Lives Matter.” Don’t just tweet it, say it out loud where ever you are. Believe it.
  2. Please do not ask any person of color to do the work for you. That includes putting the onus on them to figure out how you can help.

https://twitter.com/sheathescholar/status/1267076252508344321?s=20

3. If you are white, talk with your people: your family, your co-workers, your neighbors. It may get uncomfortable, but it is not as uncomfortable as being Black in America. And again, in those spaces, do not put the weight of the work on the people of color. If you hear something along the lines of, “but the protests are so destructive”, point them to the many examples of peaceful protests. Then remind them that peaceful protests have been tried. Colin Kaepernick’s kneel was peaceful. Then, point them to this thread and article which documents many, many, many instances of the police escalating and inciting violence.

4. You can donate to help out at these organizations:

5. And here is an EXCELLENT list of resources to help support the Minneapolis Protests.

6. You can call Mike Freeman, the DA responsible for bringing charges to the police officers. (612) 348-5550

This is a partial and small list. But if you do any #mathartchallenge do this one. I’d love for you to add any additional suggestions in the comments.

Black Lives Matter.

 

#MathArtChallenge Day 74

IMG_1463

I really cannot promise I’ll continue with the #mathartchallenge. May be a break, may be a stop. It’s pretty jarring to see things not related to #JusticeForGeorgeFloyd on my timeline right now.

I want to acknowledge: I am perfectly safe. I am participating in the peaceful protests – mostly outside Mike Freeman’s house. I was there for about an hour while protestors gathered today. Cars came in and parked in the street. People flooded in, asking for the arrest of all 4 officers. Each time it was totally peaceful.
I am trying to support my students from afar- the pandemic hasn’t ended. Today, there is a #mathartchallenge because a student invited me to a walk by grad party, and that’s a chance for me to set eyes on someone I care about to show that I care for them. Not having cards, I made this to give to them.

Art does help give me space to process. If that’s the case for you, I’m glad to help you find things for that.

Below is a link to instructions. You’ll have to do some figuring out.

https://design.tutsplus.com/series/geometric-design-for-beginners–cms-790

#MathArtChallenge Day 73: More Islamic Design

I tend to treat math art a bit like meditation & it helps to calm me down when I’m stressed or struggling. Last night, I could hear helicopters and smell smoke – please know that I am fine – several miles away from what’s happening in Minneapolis. But I couldn’t sleep, so I got up to make some tea and watched this video from Samira Mian with plans to create it myself in the morning. Didn’t fix everything that’s happening, of course, but helped me to calm down enough to sleep.

You only need one circle and the rest is straight edge work. Samira uses tracing paper, but you can easily just color the creation you make without transferring it to watercolor paper.

Mine: