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.

geometry and paper



I had no idea what was a volvelle. I was focussed on how patterns happen when there are rotations around a center point. A few years ago, with the help from a friend, I had figured out how to seamlessly paper engineer one pattern rotating above another, which. as it turns out, is the action that defines a volvelle.

Recently it I tried rotating a Persian motif. Here’s what I came up with. The image below shows a rotated transparency above a copy of itself, which is seen in image above.

Here’s a video of the movement:

After this first attempt at rotating a complex circular geometric design worked out so well I thought that I could easily come up with another one.

Turns out I got lucky with my first attempt. I tried out one design after another, and, one try after another led me nowhere.


I found I could make some wonderful revolving, kaleidoscopic patterns with this 8-fold design. I had a good time playing with colors here. Notice, in the image about that, radianting from the center there are four blue darts. Below, a colored rotated transparency creates green and pink darts, and the blue seems to have gone away.

Now I have no doubt that you want to see what the back of this volvelle looks like.

The Paper Hub on the back of the volvelle
The Paper Hub on the back of the volvelle

It was only after I made these that I realized that these constructions have a name, as well as a long history.

Just a few days ago, watching The Moveable Book Society’s Ocober 3, 2020 meeting pop-up book collector Guan Zhongping showed a volvelle from the earliest known Chinese moveable book piece.

Early Chinese Volvelle, from the collection of Guan Zhongping
Early Chinese Volvelle, from the collection of Guan Zhongping

Then I remembered seeing a volvelle that NASA publishes to track the moon phases, but, even though I had made it, I didn’t know the name of the structure.

This is a fabulous activity. Download of the tutorial can be found at

This example of a volvelle being used to calculate or align information is the traditional use of this structure, which is what I found out after being handed a book written by Jessica Helfand, which contains a wonderous collection of volvelles. The book, Reinventing the Wheel, was lent to me by the Book Architecture Resource Center here Salem, New York.

So far I haven’t found any other examples of this rotating disk structure being used for the kaleidoscopic or moiré purposes that I’ve been making. Oh, yes, that’s right, I made a moiré volvelle too.

This one is so cool. I think I should add a video of this moiré volvelle in action. Maybe I will add it tomorrow.

I’ve loaded up my etsy shop with these volvelles, with the holidays in mind. If you want to go there and browse, please, be my guest.

Math with Art Supplies

Circular Grid made with Straight Lines

Purple and Green Mandala
Purple and Green Mandala

What if you want to make a circular design but don’t have the tools or the templates to make concentric circles? This is something I’ve been thinking about as I think about how to work with students during this time of distanciong.

Outlines waiting for color
Outlines waiting for color

First, a bit of backstory, giving credit where credit is due. This past May I started seeing the most glorious mandalas on twitter. It took me awhile to realize that a hashtag, #maydala, accompanied these images, and it was nearly the end of the month before I fully grasped what was going on. Clarissa Grande wrote a comprehensive, stunningly beautiful blog post about the #maydala project, showing images as well as the links to tutorials that got people started. Visit her blog, as it’s not my intention to repeat what Clarissa has already expertly written.

Drawing in color
Drawing in color

I like rotating things around a center. The rotations I do are always created with tools of one sort of another, including but not limited to spirographs, computer programs, compass and ruler, and templates. Being in the midst of my own fog, it took me awhile to realize how many of the rotating patterns I was seeing were drawn by hand. It had never occured to me to go free-hand with rotations. In just the last couple of days of the maydala hastag I made just a couple of mostly hand drawn mandalas. It was liberating! I did follow the tutorials suggested by Clarissa to make the underlying grid, with my own perfect imperfect designs to create a circular design, but lately I’ve been thinking about working with kids who don’t have access to compasses or other tools, so I want to make these polar grids using just pencil and paper.

Foundation for a Mandala
Foundation for a Mandala (aka Polar Grid)

Here’s what I came up, if I make lines on one piece of paper, and use a second piece to make angles and other markings, I end up with a what I need to make a mandala. It takes me about 10 minutes to make a circular grid using just a pencil and two pieces of paper.

You may be able to get this idea of what I’m doing with a few photos, so that’s what’s coming next. I do think, however, that it’s easiest to see what I’m getting at through video, so that’s posted here too, at the end of all these photo

Have two pieces of regular copy paper on hand, and a pencil. The first piece of paper is the drawing paper. the second piece of paper is a tool. I’m using black marker because it shows up better in photos, but use a pencil!!

The beginning of a Polar Grid
The beginning

Make a mark approximately in the center of the drawing paper.


The Second piece of Paper is a tool
The Second piece of Paper is a tool


Fold in half a corner of the second piece of paper


Align the vertical straight edge of the paper exactly to vertical edge of the drawing paper so that it intersects the dot in the middle of the paper. Draw the horizontal line across the middle of the paper.


Align the horizontal edge of the second paper to the horizontal edge of the drawing paper, making sure to go through the center line. Draw the vertical line in the center of the paper.


Keeping the horizontal edge of the second paper aligned with the paper beneath it, slide it over so that diagonal intersects the center point of the drawing paper. Draw that diagonal line. Figure out how to make the diagonal that goes the other way.

Now that all the basic lines are radiating from the center, make some new lines on the second paper to mark how far apart you want your concentric circles to be. Be sure to make the center.

You will be rotating this paper, around the center mark. 180 degrees so that the new lines can be transferred to the lower part of the line.


This photo is not upside down: it's been rotated 180 degrees
This photo is not upside down: it’s been rotated 180 degrees

Transfer the marks to the lower part of the graph AND mark the top paper to match the first set of line. Now it will be easier to mark all the rest of the lines with these tick marks.

There, all lines are marked!

Want more circle guides? Use the guide to make them. You really don’t need the radiating lines.


Just a reminder, please use a pencil that you can erase. Save the inking in for what you are sure about your lines.

One more tip: some drawings don’t look so good up close. Like this one……but from a distance, usually look really good.

The point is, don’t give up on a drawing before seeing it from a distance@

Here’s the video of how this all goes:

I will end with this: doing these in color is great fun, but black and white is awesome too.


Drawings · Knots

Knot Obsessed

Celtic knot

I didn’t see this coming. A full month without posting. Here’s what happened.

Sometime after the pandemic isolations began, I slowly began noticing postings that were actively and enthusiastically bringing people together to fold, draw and build things together. For instance, Anne Perkins demonstrated a hundred days of projects on her blog and Clarissa Grande introduced monthly twitter hastags like #Maydala #GeometricJuly which were introduced with video tutorial support, and other little groups seem to pop-up here and there to work on something together, What’s happened with me is that I was swept away with learning new stuff from so many other people. Rather than creating content, I’ve been happily devouring it.

I think I will be writing at least a few posts, sharing some of the cool stuff I’ve been looking at, starting with knots.

Celtic Knot, Paula Krieg
Celtic Knot, Paula Krieg

No question about it, I like knots. Have been drawing them, on and off, for decades.

sketchbook page
Knots from 2012

I had seen Celtic Knots, but hadn’t paid much attention. Finally because of Annie Perkins post and which includes this video I decided to try one out. It wasn’t just her post that inspired me, it was also all the people who created and posted knots after they made them that caught my attention. I was still a bit slow : Annie posted in April but I didn’t get to this until August. Thought I spend one day with it.

I started getting carried away. What I realized was this: Even though starting with a standard grid makes total sense to me, I realized I could make knot that had, well, wonky grids. I loved discovering a way into creating my own grids. This means that if I have just a pencil and paper, I could amuse myself indefinitely making these knots. I can’t tell you how much this idea appeals to me.

Wonky Celtic, P Krieg

These wonky knots were really hard to do. I’d spend a great bit of time trying to figure out how to get them to work out just right. Sometimes I’d fail to get them to work out. The knot above has five separate strands.

Here it is, close up.

The next one is based on a larger number of crossings but is just one continuous strand.

I tried lots of ways to color this one in. I was happy with it once I was done, but wasn’t happy with any of the photos, except for this one:

These have been great fun for me. I feel like I am never going to be finished doing them, though I have slowed down a good bit.

It’s dawning on me that it might be a good thing to do a little video tutorial on how I go about making these odd knots. Besides enjoying making these myself, I think that making these would be great activity for kids as they could be as complex or as easy as is appropriate for each child. They are so much fun.

Ok. So fully intend to post a video. Here. Within a day or two.

Addendum: Here’s the video. Good luck with this!