Although I intend the title of this post to refer to what I’ve been messing around with this weekend, I’m not really sure it means anything. What’s been happening in my studio is that I’ve wanted to mix up some interesting lines with some interesting folds.

After a person with the handle of GHS Maths posted a group of images made by rotating the graph of a trig equation I got it into my head to see what one of them would look like on a hexagon-flexagon.

If you don’t know what a hexagon-flexagon is, you haven’t watched enough Vy Hart videos. In 2012 Vy offered her own utterly delightful interpretations of what she thought people should know about this piece of paper wizardry in Hexaflexagon, (6 MILLION views!), Hexaflexagon 2, and the sequel Hexaflexagon Safety Guide .

 A Hexagon-Flexagon has three distinct sides, which results in six distinct designs: I’ve written about these here, here, and showcased student work here. I haven’t thought about these in a while, but it seemed to me that the image at the top of this post, and others that I had been working with lately, might be interesting to put on a hexaflexagon. 


I had  ideas for all sorts of images but I became so enchanted by what the variations of the image above that in the end this is what I went with.


My computer did not like this idea at all. I spent half my weekend redoing what I lost when my program crashed, half my weekend watching that blue swirly thing going around, and half my weekend coddling my computer so it wouldn’t crash. I know that I’ve listed three halves, so if that bothers you, here’s what I did with  fourth half: I was able to actually make an image that became a hexaflexagon.

A hexagon-flexagon on 8.5" x 11" copy paper

A hexagon-flexagon on 8.5″ x 11″ copy paper

This is what a hexagon-flexagon looks like on two sides of copy paper. I always want things to fit onto standard copy paper, so I had made this template: hexagonflexagontemplatecolor


It’s a bit tricky to follow, but it actually works really well. I love being able to print these up on my little printer.

This is a dynamic structure, that is not easily appreciated in still photos. I am going to either get my son to make a video of me working the structure or will post the more appropriate stills that I can come up with. Tomorrow. Edited into this post. See you then.

Update: I made a quick video! My first one! Based on the image in this PDF

Dolls by Angela

Dolls by Angela, age 6-ish

This blog became public six years ago today. I had been writing in secret for a while, trying to get the hang of WordPress, before first hitting the Publish Public setting. To celebrate, I’m posting photos of work done by each member of my little family, and pointing out my most visited pages.

Birthday Bird, by John

Birthday Bird, by my son John

The images on my blog that have gotten the most clicks are:

How to Make an Origami Pamphlet : 3750 clicks.  My number one favorite book to makes has the most clicks. This elegant little structure is known also know as a  zine, hot dog booklet, eight page pamphlet as well as by many other names.

Origami Pocket Tutorial:  3647 clicks.  This is slightly different from an origami cup because it is sized to hold a little book.

Modified Pamphlet Stitch Booklet:  3006 clicks.  A good structure to make when needle and thread seems too cumbersome.

How to Make a Six-Sided Snowflake from a Paper Napkin: 2859 clicks. I made this image quickly one evening after trying to explain how to do this on the phone to my good friend Cynthia in Minnesota. There are so many tutorials on how to make a snowflake that I never thought mine would rise to the towards the top of google search results, but it has!

Decorated Box by my husband

Box decorated by my husband

My most visited posts are:

‘Tis the Season to Make Paper Snowflakes    34,689 visitors. These views are mostly seasonal.

Elementary Nature Printing   17,198 visitors. I think it’s Pinterest that drives views to this page.

How to Make a Paper Spring  14,164 visitors. There’s not much competition on search engines for this structure. Even so, I think that the tutorial page on this post is one of my best and I doubt anyone will make one that’s better. Blush.

How to Make an Origami Pamphlet  13,674 visitors. This post, by the way, was published 10 days shy of six years ago. Nearly everyday this post is visited. Yesterday it got 16 views.

The remains on the inside of a Conch Shell, egg tempra, by me, today.

The remains on the inside of a Conch Shell, egg tempera, by me, today.

So, that’s it, my six-year wrap up. Next year maybe I will write about my own favorite posts. Thanks for visiting. It means the world to me to know you’ve stopped by.

Weekend Bookend #5

November 15, 2015

Rotateds sine

Based on Dan Anderson’s Rotated Squiggles aka Trig

Rotating parametric curves:  I barely know what this means, but am happy to have discovered it.


These mostly visual Weekend-Bookend posts are like a snap shot of what’s been going on in my studio during this past week.


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.

And this from Simon Gregg:

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.

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.


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.

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