Art and Math

Color My Math

Here’s a true story. I was attracted to patterns of curved lines. Then I started to learned how to make them, using math.

That story took about sixty years to be true, from beginning to end.

The woman who lived next door had a spirograph. I would go knock on her back door. She would set up the spirograph for me. I would sit and make curves while she ironed clothes.

My mother bought wrought iron headboards for the twin beds in my room. I would sit and stare at the spirals, run my fingers around the curves.

It wasn’t until my own children were in high school that I learned that spirograph curves could be described by mathematical equations. I wanted to know how to do this. I mean, I really wanted to know how to make curves using math. I learned how. I’m still learning.

Every piece of jewelry a person wears has a story attached to it. I enjoy asking about these items. In the same way, the curves I make have stories. I’ve written some of these down.

A number of years ago an extraordinary woman from down the road started a free summer program for the children in this area. Kids were served lunch, they played, and listened to stories. When it was clear that more and more children needed a place to eat, learn, and play, current and retired educators stepped up to create an exceptional program. Parents drop off their children, the children are fed, offered swimming lessons at a nearby lake, and have engaging educational experiences at the community center. Local teenagers are hired as counselors. Local teachers and artists are engaged to design and teach compelling learning blocks. Kids are treated to visits from people in the community who come by and share their own interests and skills with groups. With help from many sources, this generous program, which charges exactly nothing, is continually growing.

This year I’ve decided to sell something to support our Lunch, Learn & Play program.

I’ve made cards of curves to color. On the back of the folded card is my story of each curve. I’m also including a web link to a graph to so the curious mind can see how math makes these curves.

Are you curious? Hop over to the link below. There is a blue dot on the bottom row left. Move that dot an see what happens to the picture. You will be dazzled.

There are eight different black & white cards to color, and eight envelopes too.

The paper that these cards are printed on Springhill Digital Vellum Bristol White Cover, 67 lb, the perfect paper for crayons and Crayola colored pencils since these marking items work the best on paper that has a bit of roughness to it. Markers work really well on this paper, too, which is thick enough that there is very little soak-through.

These cards for coloring are going on my Etsy site, where, through the month of June, all proceeds from these cards will be donated to the local children’s program.

If you know a someone who would appreciate finding out something about the beautiful curves of math, and you would like to support a truly wonderful children’s program, follow this link:

Art and Math · Art with Math Supplies

Making Art by Making Rules

Rosette made in Desmos

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.

Spirograph image made in Geogebra

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.

Under drawing for Oculus

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.

Art and Math · Art with Math Supplies · geometry and paper

A Spiraling Book


Spiral Book
Spiral Book

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.

Spiraling pages made from copy paper, an old calendar, outdated map, and a pretty orange scrap
Spiraling pages made from copy paper, an old calendar, outdated map, and a pretty orange scrap

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.

The Folds before the Spiraling
The Folds before the Spiraling

Folding pattern shown above. Video tutorial below.

A sturdy paper can be set up to make this funny little shape below.

Side view of a Spiral Book
Side view of a Spiral Book

Try it out. See where it takes you.

Spiraling Snake or Snaking Spiral?
Spiraling Snake or Snaking Spiral?

Art and Math

Symmetry is for Artists, Mathematicians, Students & Five-year-olds

My Symmetry Tiles, designed by Paula Krieg
My Symmetry Tiles

Start with symmetry.

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.