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.




“…so the Muse may enter and not soil her gown.”

Getting to work

Am in the midst of a string of some challenging projects. Can’t seem to get one all the way done before I need to start on the next, so there’s been on overlapping that is uncomfortable. Just keeping my head above water here. Today I absolutely needed to get started on prepping for a residency that starts in a few days. I decided to start by spending most of the day clearing my desk. Although I wrote recently wrote about this need to clear space,  I was happy to read this bit last night from Steven Pressfield’s book The War of Art

It’s great when I read how someone else expresses what I’ve been thinking about.

I am an naturally organized person, but not a naturally neat person. It’s always a struggle to keep my space clear. But that’s a good thing, because it means I am constantly doing what I love, which is work.

Here’s a snippet from a different book that I recently finished, A Month in the Country by J.L. Carr, expressing the thoughts of the a person who is uncovering a painting that had been whitewashed over.

This has nothing to do with anything else: it’s such a lovely snippet I wanted to share it.

Now to bed so I can get started again early tomorrow.


The Best Fold to Know How to Do

Kids laugh when I sheepishly announce that I am going to teach them how to fold a piece of paper in half.

The older the person, the happier they are with this activity. .

It’s one of those things that everything that can go wrong does go wrong.

It seems nearly impossible to get to line up the edges of the paper. Then once they line up, they shift. You adjust.

You press down on the paper; it shifts.

You thought it looked good then it doesn’t.


This video is one minute and twenty-eight seconds about folding paper in half well.  I say everything twice because it’s all worth repeating.

Happy folding!


A question that I got on twitter was about folding a square on the diagonal. Here’s a wordless video to answer the question. It’s really the same steps as above, just slightly modified.

The things to keep in mind are to use two fingers as a backstop near the corners that you are trying to match. Slide the paper until it’s in place, then HOLD TIGHT with one hand, while sliding across the paper towards the fold, with the other hand.


…and why am I doing this?

shapes on a mirror
Space filling shapes: the Weaire-Phelan structure 

I avoid the question “why am I doing this?” The question never serves me well. Seems like the reason I am doing it will reveal itself when it’s good and ready.

My current favorite example of this is about a little black book . It started out because I have so many pieces of this gorgeous black paper that I wanted to do something with.

This little book has three signature, with three flaps, sewn with continuous thread, using a single piece of paper for the cover. This is not typical. Sewing two signatures on to a flap is something more common. It was something that I wanted to work out. It took many days of tying this and that, until I was satisfied. I eventually gifted one copy away, used one for notes, and the others sit on my shelf along with so many other little books. As I was laboring over this little gem, it did cross my mind that I was spending a huge amount of time on something that I couldn’t justify as worthwhile, but I banished that thought as quickly as I could.


This past summer I grabbed one of these little black books and tossed it in with the very small number of items I brought with me to Penland School of Crafts (where I would be assisting Susan Joy Share in teaching a book and box making class). In a conversation with Henrik Drescher, who was teaching in the drawing studio, he said he had been trying to work out a simple sewing for a 3 signature book sewn on three flaps with a continuous piece of thread, but hadn’t come up with anything that he liked, and wondered if I had any ideas about it.

It took me about a half-hour to realize that what he was asking for described, exactly, the only sewn book I had brought with me from home.

I taught Henrik and another bookbinder how to go about doing this sewing, which turned out to be such fun time that I’m laughing as I think about it. Certainly, this reinforced my belief to just following my curiosity and instincts.

Which brings to me how I spent my weekend with shapes.

Irregular dodecahedron
Irregular dodecahedron

Vincent Pantaloni tagged me on a thread about space-tiling that referenced the Wearie-Phelan structure, along with a link to a page which offered a download of the templates (aka the nets ) for the units that make up the structure http://www.cutoutfoldup.com/214-weaire-phelan-structure.php


I had been feeling like I wanted more practice making solids of out paper. After reading the awesome Books, Boxes and Portfolios by Franz Zeier where he, in meticulous detail, writes about creating solids from nets, I’ve begun to appreciate how practicing this could be helpful to me in a global way, in the way that practicing scales helps with piano playing. For more about this book, take a look at Cathryn Miller’s post https://byopiapress.wordpress.com/2017/02/19/books-boxes-and-portfolios/

gluing down only 1/8 inch of a tab
Worth Reading, Books, Boxes and Portfolios by Franz Zeier


Each of the Wearie-Phelan units took at least 25 minutes to make. I made eleven of them. After the first three, I decided they should be made out our bubble paper, so I decorated papers with bubble.


Bubble papers
Bubble paper

It’s generally a good sign if the process pleases me as much as the result.

tetrakaidecahedron (truncated hexagonal trapezohedron)
almost closed

I really liked the funny shape of the box before I glued down the final flap. (Note my GREAT needle point glue injector? My current favorite tool.)


So I got to practice making shapes, I got be involved in a day of play with my on-line community (above is Mike Lawler’s preview of these shapes he is creating with his 3D printer) and I got to be part of a lively conversation that Jen Silverman somehow elevated to the level of hot dog packing…


…and today, quite unexpectedly, I am thinking of some wonderful ways to continue doing some cool things with this shape.

The point of this post to keep reminding myself not to listen to those voices that would have me justify my explorations.