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 with Math Supplies · Fractions · Math with Art Supplies · Uncategorized

About Halfway There

Equivalent Fractions
Equivalent Fractions

I’ve been interested in creating fractions projects for kids exactly as long as I’ve been working with children in schools (decades). This year, after enjoying,messing around with a hexagon/golden ratio project I wondered if I could modify the idea of using scaled hexagons to help fourth graders make better sense of fractions. My first attempt at this didn’t work out so well.

Making 1, or 100%
Making 1, or 100% out of two halves

I gave students hexagons that were scaled to 1, one-half, a third, a fourth, a fifth, a sixth, an eighth, a tenth, and twelfths. The task was to pair and arrange them so they would span the length of a whole, aka 100% across.

1/3 + 1/6 + 1/6 + 1/3 = 1
1/3 + 1/6 + 1/6 + 1/3 = 1

The project went okay, but it just didn’t snap for me.

I’ve been thinking about how to improve this project. Today I had a chance to work with a small group of kids. I tried a new approach that worked so much better. What was especially great was that it included making a simple book. Yay!

Fractions for a Book
Fractions for a Book

I started kids off with a hexagon that was labeled 1/2. I explained about how the lengths we would be looking at would be the horizontal or vertical length of the hexagon (I didn’t use these words, rather gestured what I meant). Then we layered the hexagon with equivalencies. Here you can see two 1/12ths equals 1/6, three 1/6ths equals 1/2, and 1/6th and two 1/12ths equals 1/3.

Equivalent Fractions
Equivalent Fractions

Nice, right? Snap!

The books we made were just two sheets of paper folded in half, bound with yarn using a modified pamphlet stitch. 

Equivalent Fractions Book
Equivalent Fractions Book

What’s great about using hexagons for this project is that you can still see the labels of the lower layers as the equivalencies are built up. The adults in the room had a bit of trouble with accepting that the hexagons were scaled (similar) versions of each other, but the kids had no problem with it. This reinforces my notion that children have a better intuitive understanding of scale than do adults.

This is the way I explain the scaling to adults: We all know what half a candy bar looks like. That’s one way of thinking of one-half. But when we say a child is half the size of the parent, we don’t envision the child to be half a parent, like they were half a candy bar. Instead, we envision them smaller than the parent in their height as well as width.  This explanation seems to work.

The bullseye view of fractions
The bullseye view of fractions

After doing a bunch of equivalencies, this child decided to nest her fractions.

Okay then. Here, what’s obvious is the hierarchy of the hexagons that are scaled by fractions. Nice!

This project can use a bit more refinement, but this is as far as I’m going with it right now.

I’m including PDF of the hexagons. The labeling includes the colors of the paper I use for printing.  Yeah, it’s lots of files. Welcome to my life.


hexagon halves blue

hexagon sixths halve grape

hexagon 3rds 12ths chartreusegreen

hexagon 4ths orange

hexagon 5ths 8ths pink

hexagon 6ths halves grape

zhexagon 12ths full page chartreuse

2 pieces to make 12 inch hexagon

12 inch hexagon

Notes about hexagons




Art with Math Supplies

A Round-Up of Valentines

imon Gregg Valentine 2019
“Every triangle is a love triangle when you love triangles.” Pythagoras

Mathematicians celebrate Valentine’s Day with more unbridled abandon than any group I’ve ever seen. Kindergartners get the silver, for liking the day of hearts, but, hands down, mathematicians take the gold.

I’m gathering just a fraction of the posts I saw coming out of the math community this past February 14. Images started popping up early in the week in anticipation of the big day. Luke Walsh and and Iva Salley were the early birds. Iva with her heart shaped number puzzle…

and Luke with a number of images, some static, some dynamic. Here’s a link, which if you follow it, and scroll down, there’s more:


The end of Valentines Day was bracketed by late day contributions by Suzanne von Oy and Martin Holtham.

Martin’s young daughter even joined in:

Dan Anderson made this lovely series of translucent hearts that create patterns with each other

Then there’s this one, that I started but that Dan helped me out with:

Phil DeOrsey made this heart that looks like it doesn’t know if it’s coming or going, in response to a challenge by Daniel Shiffman

Mark Kaercher added some origami to the mix

Ed Southhall seized the opportunity to educate us:

Ben Orlin schooled us on how to obliquely  sweet-talk our sweeties:

Grant Sanderson wrote a poem and made a video

The National Museum of Mathematics sent out a craft to its mailing list, which people shared.

The New York Times publish a math-heart article, which was shared widely

And if all this isn’t enough, Vincent Pantaloni is doing his own collection, spanning over at least a couple of years, of images he’s harvested from twitter


Colleen Young linked to a collection of heart-themed activities, which includes a couple of Numberphile video that shows heart shape made with a marvelous, unexpected twist.

There’s many more Valentine references in this math community that I could possibly list and still get to sleep tonight. Feel free to add more in the comments.

Now here’s my contribution to the day, a stop-motion crayon on newspaper drawing/construction:


Hope you all feel the love!

Happy Valentine’s Week.