Art and Math

A Day in the Life of a Lissajous Curve

Lissajous Curve with Transformations

Herein lies the story of numerous people in several countries and many time zones finding moments in their day to fiddle around with lines and curves on a number of different programs and platforms.

Jen Sikverman and John Golden's Harmonograph drawings
Jen Silverman and John Golden’s Harmonograph drawings

The story starts with Jen Silverman who posted images that she and John Golden had made using a harmonograph, a device which can be simply described as a spirograph that is controlled by three pendulums.

I’m attracted to images that are made by the accumulation of repeated markings, so I started asking questions. John sent me a GeoGebra link which explained the shape that the  images were based on, something called a Lissajous Curve,. This function can describe something about sound frequencies. Its graphic representation with transformations and iterations also are known for making compelling designs.

 Lissajous Transformation using Illustrator by Paula Beardell Krieg
My Lissajous Transformation using Illustrator

I opened up my Adobe Illustrator and started to see what I could do with a basic Lissajous Curve.

Lissajous transformations by Paula Beardell Krieg
More of my lissajous transformations using Illustrator. Here’s more

After awhile Josh Giesbrecht saw my images on Twitter and suggested that I try to do something using transformations of the graphic representation of the sine function.

Paula Krieg playing with Sine

With this, a dialogue began. Josh opened a graph at and his made his own transformations of the Lissajous Curve.

Josh Geisbrecht's Lissajous Curve on Desmos
Josh Giesbrecht’s Lissajous Curve on Desmos

Ooo! Something new to play with! I started tinkering around with the equations and intervals and sent the following link and image to Josh.

Improvising with Josh on desmos, Paula Krieg
Me, improvising with Josh on Desmos

Desmos took notice of what Josh and I were doing, and posted a note on twitter, encouraging us to to make an interactive graph using sliders, and to make a GIF of what we were doing. I had no idea how to do this, but Josh figured it out, Turns out it’s easy to make a GIF of an animated desmos-created graph, thanks to  Josh added the sliders to his graph and I made a  GIF of his graph.

 John Golden adding more sliders to Josh Greisbrecht's graph
John Golden adding more sliders to Josh Giesbrecht’s graph

In the meantime John Golden jumped back in and added more sliders. John is, self-described, slider-crazy.  Yes.

Now, about the same time Dan Anderson decided to use his lunch hour to write some code for what we were playing around with.

Dan Anderson's Open Processing Lissajous
Dan Anderson’s Open Processing Lissajous

If you click on the link in the caption you’ll see the image being created. Or look at the GIf that Dan gifted.

Now, let it be clear that starting from here, the most I could do is stand back and admire. Coding is, at this point, out of my skill set.

Josh Geisbrecht inspired by Harmonograph in Open Processing
Josh Giesbrecht inspired by Harmonograph in Open Processing

After returning to the internet after sharing dinner with my husband I started seeing wonderful images, like the one above and the ones below, in my twitter feed.

Dan Anderson with Josh G's Code
Posted by Dan Anderson

And this from Simon Gregg:

Simon Gregg with Josh Griesbrecht's Open Processing Simon Gregg’s response to Josh

Josh had expressed that he wanted to add more color to his curves, but I didn’t know how people were coming up with the images I was seeing. I clicked Josh’s links and I would get something different from what I was seeing. Then Josh mentioned that his code was interactive.  The user can right and left click on the image to change the values of magnitude and period (not sure if those of are right words. Would someone correct me here if I’ve gotten this wrong?)

So, this is what I hope that you do: go to Josh’s Open Processing page and interact with an equation. If you can left click and right click you can do it.

So, there you have it, curves being explored using a harmonograph, GeoGebra,  Adobe Illustrator, and

I would like to see this as a flip book, but that project is going to have to wait in line. However, if you would like to see an example of  animating an equation via a flip book, take a look my The Animated Equation Book post.

GHS Maths graphs of tan
GHS Maths graphs of tan

Towards the end of this adventure GHS Maths Department sent out a note reminding us that graphs of tan feel left out. True, I never think of tickling tan, and as far as I can tell it’s a legitimate observation. But it was late in the day and I think we’ll have to tan-play another day.

Now, believe it or not, there’s a couple more links that I want to add into this post, but I couldn’t figure out where else to put them. Here’s a look at how a grown-up can make a spirograph: Craig Newswanger’s Guilloche Drawing Machine.

Also, if you now want to learn more about coding (I do) take a look at Dan Shiffman’s You Tube Channel. I haven’t looked past the intro yet, but it takes about 2 seconds of watching to know that this guy is engaging and probably an excellent choice for on-line code learning.


Chalkdust Magazine
Chalkdust Magazine

just when I thought I had thought enough about the look of functions, John Golden posted a link to this article about favorite functions, freshly published on-line in chalkdust magazine, a new publication out of University College London. So, if you want to look at more visual representations of functions, go have a look.

As for me, I am going to be drawing some leaves tomorrow. Then I’ll be turning my attention back towards flip book functions.

More Harmonograph imagines from Josh Griesbrecht's Open Processing Page

To all who participated in Lissajous Day, thank you.

8 thoughts on “A Day in the Life of a Lissajous Curve

  1. Josh’s page works even without a left and right click — maybe differently, of course, but I can click and drag with my Apple mouse that only has an ambidextrous click. — But for most of this post, all I can do is quote you: “Now, let it be clear that .. the most I could do is stand back and admire.”

    Liked by 1 person

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