Sorting out the Golden Ratio

Golden Ratio
Golden Ratio

I’ve been reading this book, The Golden Section. 

The Golden Section by Garth E Runion
The Golden Section by Garth E Runion

Maybe this is a lie, as, after three days I’m still only on page two.

My  notebook is full of cross-outs. Each time I think I finally understand what’s going on I write bigger.  The pages of my notebook looks like a map to insanity. Maybe  a bit of insanity is what it takes, so I can finally disconnect from what’s already in my head.

Page 2
Page 2

I have to let go of preconceptions before I can finally see, sitting so smugly, so adorable but nearly invisible, right in front of me, the detail I’ve overlooked. (Got to be good lookin’ ‘cuz he’s so hard to see….)

This is a scaling issue. As  someone who is interested in paper-folding and book making I am always scaling things. Thought I completely understood scaling. Scaling helps me do things I want to do nearly every day. But there is the wall (or ceiling?) I keep bumping up against. Am trying to work it out, from many fronts. Last few days, trying to understand the golden ratio, is one of the many ways that I am trying to deepen my understanding of scale. Also, there’s something about pentagons that I just don’t get. I don’t know what it is about pentagons that is eluding me, but I know golden ratio and regular pentagons are two peas in a pod (sorry, not a good metaphor. I think I’m hungry).

Yeah, I get it but it doesn’t stick https://en.wikipedia.org/wiki/Golden_spiral

We’ve all seen the golden ratio spiral, embedded in a rectangle which contains successively smaller squares created in the leftover part of the rectangle once the biggest square is made. My mind doesn’t think in spirals, though, so I can’t extrapolate that image into something that I can get cozy with.

This does not work for me either. It makes me think too much.  https://commons.wikimedia.org/wiki/File:Golden_ratio_line.svg

I love number lines, but I’ve had a heck of a time seeing the golden ratio on a number line. Over and over I have read description of a line divided to show the golden ratio, and, although I understand what it’s saying, the words never make the image snap into focus. What I want is an aha! moment, an image I can conjure up effortlessly and know what I am seeing.

Finally opened my favorite graphic program. Started making circles. Took a few minutes but, oh, yeah, AHA!. Maybe I am the only person that this makes sense to but that’s okay. At least it makes sense to me.


Golden Ratio Circles
Golden Ratio Circles

This is a two-step visual. It doesn’t prove anything. It’s just a way for me to visualize what’s going on.

The circles above are enlarged by the same proportion over and over again. They aren’t increasing by twice their size, which would be 200%. They are increasing by 161.8%. And using this specific percentage, makes something happen that seems unlikely….

Golden Ration Circles
Golden Ration Circles

What happens is that the two smaller circles fit exactly into the next larger circle.

Now this is an image I can conjure up and understand.

And play with. Hmm.

Needs color.

Golden Ration Circles colored
Golden Ration Circles colored

Needs to show all the circles fitting into other circles

More golden ratio circle
More golden ratio circle

Needs to see what other arrangements work.

Even more circles
Even more circles

Needs color.

But, even more pressing,  I need to get to sleep. So no more color tonight.

Tomorrow I will be ready for page 3.



design · Geometric Drawings · geometry and paper

What to do about Color Combinations


Steal them.

Since I started decorating my own papers with geometric designs (as well as decorating geometric designs) I’ve been flummoxed about color combinations. Some of the decisions I’ve made have been truly horrible. Sometimes they’ve not been so awful, but, even then, it takes way too long to come up with color combinations that look good to me.

I suppose I could take an on-line class about color theory, but somehow I’m just not drawn to do that just now. Abode has a palette-sharing site, but it’s not supported in the version I use. Recently, having spent way too much time having way too little success, it finally occurred to me to try out dipping directly into the palettes of painters who use color in a way that sing to me.

Geometry and Georgia O'Keefe
Geometry and Georgia O’Keefe

I probably wouldn’t write about this if  I could do this only in Adobe Illustrator, since this info would be totally useless to most people. I noticed, though, that more and more people in my circle are using Inkscape, which is a free graphics program, and it turns out that dipping into the palettes my favorite painters is even easier to do in Inkscape than Illustrator.

Fra Angelico
Fra Angelico

Here’s what to do in Inkscape. Find a painting you’d like to dip into. Save it to your computer. Drag and drop it into Inkscape. Select the shape or area that  you want to color. Press F7 or choose the eyedropper tool (second to the last tool from the bottom on the left side) and click on the color on the painting that you want to use. That’s it.

Vectorizing the Image in Adobe Illustrator
Vectorizing the Image in Adobe Illustrator

To do this in Adobe Illustrator, it’s bit more complicated. Place the image into the Illustrator file, then vectorize it  in Image Trace. I generally use the high fidelity photo setting in image trace. This separates the painting into regions, which if you zoom in really closely looks abstract and totally cool (in Inkscape, getting this close just looks blurry).

Up-close O'Keefe
Up-close O’Keefe

Just like in Inkscape, to harvest the color use the eyedropper tool, which has the key shortcut “I”. I’ve been using the live paintbucket tool (k) to fill in the areas that I want to color, but, like Inkscape, choosing the shape then the eyedropper works too.

Even more O'Keefe
Even more O’Keefe

Now, I just want to mention that even though this is the best method I’ve used to choose colors digitally, there’s still a bunch of trial and error. But instead of me doing trail and error with millions of colors, I’m using this more limited palette. Works for me. Am having lots of fun with this.


Harry O’Malley just pointed me towards http://www.colourlovers.com/, which looks like the internet’s free version of Adobe’s Kuler. Yay! Another color resource! (I can use all the help I can get.)





folding · How-to

The Paper Spring in the Classroom

Teaching kids how to make a paper spring is always thrilling. Children ooh and ahh, and practically jump out of their seats when I show them what we’ll be making.

The only problem has been is that it takes up a big chunk of my teaching time, as only about 55% of the students (who are usually 6-8 years old) in the classes I teach are able to make paper springs without extra help.

I’ve been teaching kids how to make paper springs for probably 20 years. Have shown it to thousands of students. We usually glue something to the top of it it, like a cut-out of their hand, to give the books we are making another dimensional element.

About a year ago, driving to another of my itinerant teaching-artist jobs, I was stressing over the fact that, due to time constraints I needed to cut something from my agenda . Realized the paper spring was going to have to be eliminated…unless…unless I could figure out how to get all of the kids to do make it without any extra help.

A caterpillar of paper springs
A caterpillar of paper springs

The way I’ve been teaching it is to glue two paper strips together to form a right angle, then alternate folding the strips on top on each until the papers fold down into a square. It’s easy to teach this method to adults, but kids keep folding in front then wrapping behind, which sabotages their springs.



Notice the corner is like a square, Draw a happy face on the square.

What if I ask students to fold the other way, to fold it below the glued corner, rather than above it? And to keep them from folding forward, draw a happy face which they are told should not be covered up?

Really, no one wants to cover up a happy face.

So I tried it out. Asked the students to alternate colors folding behind the happy face, said what we wanted to end up with is a little square.

Almost done

Couldn’t believe how well this went when I first tried it out. There is still a bit a confusion that happens when they see these flaps at the end.  I probably should say to cut off these pieces, but…

Last step

…these flaps can be  folded back too, then secured with a bit of glue.

This method of teaching has worked out for me unbelievably well. Unbelievable, even to me. Students have been nearly 100% successful in class after class.  So exciting to have discovered this way of teaching the paper spring.

Here’s a video:


Art and Math

Designing for Considering Boundaries around Infinity

Thinking about squares and infinity

This was bound to happen, that I would put up a post on my book and paper arts blog that appears to just be about math.

Initial impressions sometimes need refinement.

Anyone who follows me has probably noticed my attention to math ideas emerging as a theme. I’ve been paying attention to math and it is shaping how my creative work is evolving.

What I am here to say right now is that I think that math needs more designers and paper engineers.

In calculus there is this thing called Integration-by-Parts. It requires either a fluency with many rules or access to tables that contain these rules. The two problems with this is that it is not typical for the student to be fluent with the rules and the tables are not at all friendly looking and are embedded in humongous textbooks

When I was doing integration by parts problems it occurred to me to make a foldable that organized the information I needed to do these problems into a handy reference page.

I got lots of input and help from folks in the math community. I now have a greater appreciation for people who write whole textbooks, as just this one foldable was a big deal to do.

four page booklet
four page booklet

Here are the PDFs for

I’m going to be making a series of videos integration-by-parts.  As they are done, I will be editing them into this post so that I don’t flood my book arts followers with math videos. Still, I hope some of my non-math friends will take a look at this up to the 6:05 minute mark and tell me if it makes sense to them at all.  I am really enjoying being an artist who thinks about math instruction as a design issue.

Much of my thinking about math, as in my thinking about book arts instruction, centers around the weak links, meaning I search for places where misunderstanding sabotages learning.

This next video tries to address the disorientation of no longer solving for x, after solving for x for so many years.


Here’s the third video. At this point there’s not much here of general interest as it’s getting more specific to this specific method.


Finally, some worked examples. So far I’m showing two problems, but hope to add two more in the near future. Then these will be done!


The one below has a bit of bonus material about Desmos.com near the beginning.