Artist Sol Lewitt famously created rules that created his art. Making images using math is just that: a rule is created by deciding on an equation or a method, then an image is created by having a graphic program or a patient hand be guided by the rule.
I’ve done plenty work by hand, but that’s not what I’m showing in this post.
I’ve been having great fun making images in various programs. The image above was made in a free on-line program called Geogebra. I wanted to begin to learn Geogebra for a long time, but hadn’t been able to make heads or tails of it. A couple of months ago twitter friend Becky Warren offered classes, 90 minutes a week for 10 weeks. We’re on week 8. I’ve been all thumbs with the learning, but have stuck with it. This week we’ve been doing spirographic images. Check out the #geogebraArt hashtag on twitter to see some amazing work that’s been done by participants in this class.
I’ve also been doing this kind of work in Adobe Illustrator. Using Illustrator is actually closer to making something by hand that the other programs I’ve been using. This could be because I’m so much more familiar with it than I am with Desmos or Geogebra.
More often than not the “rules” have nothing to do with numbers, but, rather, they are about building relationships between shapes. The image above is a sample of following a method of working, following a Byzantine design which I found until the title Khirbat al Mafjar Oculus. I recreated it, thinking I would use it for a folding project that I was working on, but it turned out not to be the right choice. Still, it was great fun to make. The final outline is below.
Once I have this image in my computer, I can color it in all sorts of ways, which is great fun.
A perk of using the graphic math software is that I can set things up so the images transform easily by moving sliders that mess with the relationships of the curves in the drawings. The image below is a variation of the first image of this post, transformed in seconds by moving around a few sliders.
These graphics have been such pleasure to explore.
What I’ve learned about math, what keeps me wanting to keep diving in, is that it allows me to do so much more than I could do otherwise. Not sure if it’s play or work that I’m doing, but it feels both whimsical a valuable.
I’m still making work by hand, enjoying that too. Which is what my next post will be about.
I put these photos, and a video, together for a math teacher friend, Lana, a couple of years ago, and thought I had made a post about it. Lana reported back that she had made it will kids, and that they had enjoyed it.
I’ve been posting projects, weekdays, on twitter, from my blog. Wanted to feature this one today, but turns out I never did write a post. Made a video, took some photos, but never wrote about it here.
It’s a fun structure, not too hard to make. I’m thinking of it now as a fun things to make and send in the mail.
Something about how it is cyclical feels appropriate for for the times right now. Can be made from lots of different kinds of papers. Old pages from calendars, maps, and grocery grocery bag, or just regular copy paper can all be used.
Folding pattern shown above. Video tutorial below.
A sturdy paper can be set up to make this funny little shape below.
During this time that we’ve suddenly become a nation of homeschoolers, I hope everyone will continuing learning. Home learning can be tough. As some of you know, I’ve become a great fan of taking a deep dive into learning about symmetry. I’ve become a fan because it’s become clear to me how, harnessing a formal understanding of symmetry, can be an incredibly powerful tool which can facilitate an understanding of wide ranging concepts. .
In my next post will be talking about extending symmetry to support other learning.
My last ten posts have been about exploring frieze symmetries. If you are looking for a compelling, thorough, mathematically rigorous and artistically beautiful inquiry, please start at my January 24, 2020 post and keep going.
If you are looking for something totally worthwhile and doable with young children, this post of is for you. No special materials needed. OH, but you do need at least two of things, like spoons, macaronis, pencils, shapes cut from paper, envelopes, pennies, paperclips. crayons.
These images I’m showing are done by 4 and 5 year olds. The lesson is about mirror reflections. I put out a curated “mess” of stuff, and the children organize them into these mirrored arrangements. It’s fun to do this with a partner. You can see the yellow yarn in the center of the design. This is the line that the objects have been reflected across. Children take turns laying down an object on their side of the line, then their partner places an identical object on their side of the line so that it reflects in the same way that a mirror would reflect the object.
Using scissors is an interesting challenge. Some students immediately see that there needs to be some thoughtfulness in the orientation of the scissors. How do you explain to a child when they get the orientation wrong? I heard one young boy instruct his partner to flip the scissors so they match.
I have found that it takes very little direct instruction to get even 4 and 5 years olds to create these symmetries, which leads me to believe that they have an intutive connection to symmetry. Formalizing an understanding of this way of thinking about arrangements is something that can help develop a way of seeing deeply into things, whether it be concepts or constructions.
Over the next few weeks I intend to post often, mostly to recommend projects that have both a fun visual appeal as well as rigorous math appeal. My thought is that the connecting thread through all of them will be symmetries of many types.
Third and last post in this group of postings that are meant to help me clarify and remember where I am at in my thinking about the work that I do with students in schools.
After years doing bookmaking projects to make with children, I realized that many of the art and design skills I use every day align with some of the skills that mathematicians aim to develop. Part of the reason this alignment caught my attention is that I have a great affection for the mathematical thinking that I want to encourage.
In the relatively short time I am in schools with students, I hope to have a positive influence. My experiences and interests have led me to an unusual place where I can use colorful, artful materials to help kids create projects that enrich mathematical thinking. My place isn’t to teach art or teach math, but rather to plant seeds of engagement and excitement.
It seems to me that children intuitively understand concepts that are recognizably both artful and mathful. More and more, my thinking is centered around how to engage and encourage that which is already inside of students.
For instance, kids absolutely understand the idea of scale. They realize that their hands are the same, but smaller versions, of dad’s hand. Same with their shoes, their shirts, everything in their world.
There is no room in the school day for formal study of scale until the intuitive connection to it seems to have long disappeared. Turns out that scale doesn’t only have to do with making large models smaller, but it also is intrinsically connected to relationship thinking, predictive thinking and to the recognition of trends.
Discovering that children are naturally inclined to embrace symmetry has been another exciting area for me to explore with kids. When making books or other structures with students, there is nearly always symmetrical folding going on. I have choices when I teach folding: I can introduce what I do as step-by-step directions, or I can nudge the students to see the symmetry of what’s going on so that they can predict for themselves what the next fold will be. The latter way gets them to see the project in a more global way, draws them in because they have understanding which includes them, rather than being like a little robot that is being programmed to do this then do that.
Symmetry is deeply embedded in math thinking, so I have been talking to children about connecting symmetry to what they are learning right now in math. Specifically, I talk to them about how when they are looking at an equality, such as 5+3 = 8, that this expression is balanced on both sides. It can also be understood as 5+3= 4+4. If I add 6 to one side of the equation, then I have to add 6 to the other side so that the symmetry of the equation remains true. Talking to students about equations as balanced forms just might help them, later on, when they will have to maintain balance in an equation to solve for x.
As far as I can tell, the only time symmetry is formally taught in elementary school it’s part of the examination of lines of symmetry in regular geometric objects. I like to be able to at least offer hints that symmetry has richer applications.
Children seem to have an innate sense of parts that make up the whole, which seems antithetical to the reality that teaching fractions is unfathomably difficult. Is it possible, though, to focus on having students work fractionally from a very early age, way before we introduce the numbers that describe the fractions?
Playing with blocks was one of my favorite activities as a kid. I certainly noticed halves, fourths, and wholes, but I didn’t make this connection between the blocks and fractions until I was much older. This makes me value not only exposing kids to artful mathematical thinking, but also, sooner rather than later, to help students connect their hands-on activities to the numbers.
There’s more I have to say about all this. but I reminding myself that I have to get to work getting ready for classes.
Am going to end with my list of ideas that I want to keep in mind, not all of which are explained here. Maybe I will get to writing about all these here and there through my teaching season. If not, at least I will have them here to keep me on track.