A couple of days ago I folded a really amazing “action” origami model. The model was so intriguing that it led me down a rather long path of investigation to understand why it worked, with the thought of designing a model of my own based on the same principles. So, this post has two goals:

- Expose everyone who hasn’t already seen it to a cool piece of geometric art
- Expose those who are interested in
**designing and creating**geometric art to some general principles and approaches that might be helpful

## Where It All Started

As the title reveals, the model in question is called “Spring Into Action” and was designed by Jeff Beynon, a British origamist. I folded the model from the diagrams in Origami in Action : Paper Toys That Fly, Flap, Gobble, and Inflate, a very fun book by Robert J. Lang. I will not be going into the details of how to fold this model here – there are already numerous YouTube videos on how to fold the model. However, everyone MUST see this model in action to truly appreciate it, so here it is (NOTE: This is NOT my model, this video is from someone else):

HOW COOL IS THAT? So cool that in my case, I wanted to find out how and why it worked, and see if I might generate my own variation. Right off the bat, this leads me to the first couple of principles for budding geometric art designers:

**Principle 1: Be curious**

**Principle 2: Learn from others**

## Starting Down the Path

So, I started madly sketching, cutting, and calculating. One stand out feature of this model is that unlike most origami it does not start from a square, but from a rectangle with aspect ratio 8:15. Clearly those numbers aren’t random; by the time you’re done, you have a a bunch of skinny rectangles that “magically” fold together and flatten into a (seemingly perfect) dodecagon. So that was the first thing I wanted to understand.

Here is a diagram of the whole 8:15 rectangle crease pattern before folding (the numbers will be explained shortly):

Each vertical strip of 12 rectangles/24 triangles folds up into a dodecagon, and the six vertical strips then create the segments of the finished spring. So, I focused in on one vertical strip, enlarged it, and folded it on its own. This verified that those little triangles do indeed fold perfectly (within the tolerances of a human folding paper!) into a dodecagon.

**Principle 3: Break things down into understandable pieces**

Here’s how the single spiral looks going from flat to rectangle to flattened dodecagon:

## Time For Some Math

OK, clearly it works – but why? To understand this, we need to pull out some math, specifically some (basic) trigonometry. It’s a reality that in order to *appreciate* geometric art, you don’t *have* to know math, but to *design *it, you do. So:

**Principle 4: Learn the math you need, and use it**

I started from the dodecagon end. Luckily, the angles involved are fairly nice numbers. Here is a diagram of the relevant measurements (I’ll leave the derivation of the angles as an exercise for the reader!):

So, for this fold to be geometrically perfect, the highlighted triangle would need to have a skinny angle of 15 degrees, which means (here comes the trig!) that the **ratio of the short bottom side to the long left side needs to be the tangent of 15 degrees, or ~ 0.2679**.

Circling back to the original diagram, we have 12 of these rectangles (now turned with long side horizontal) on top of each other, forming the strip that’s 1/6 of the full rectangle. So, in terms of the original 8:15 rectangle (we can imagine it’s actually 8 inches by 15 inches — the exact lengths don’t matter, only their ratios to each other), the long side of the rectangle is 15/6, and the short side is 8/12 = 2/3. Finally, we can calculate **the ratio of short side to long side** to be (2/3) / (15/6) = (2/3) * (6/15) = **4/15 which is ~ 0.2666**. Not exactly the tangent of 15 degrees, but awfully darn close, with an error of only ~ 0.0013! In reality, that small of a difference is *way* smaller than the errors introduced by the thickness of the paper and human folding accuracy. This is another hugely important principle when it comes to the matter of translating a theoretical idea into a tangible piece of geometric art:

**Principle 5: Approximations are often critical for efficient, practical design**

Being a stickler for detail, I often want things to be *perfect*, and so this principle has taken me a **LONG** time to make my peace with. Something inside of me knows that it “isn’t quite exact” and that bugs the bejeezuz out of me. However, over many years I’ve learned that it’s never going to be perfect anyway, so “close enough” is OK if it saves hours of time actually building the thing!

## Time To Get Creative

By this point I had solved the initial mystery of “why 8:15”? Answer: because this shape rectangle yields the desired grid of 12×6 flat rectangles each with the critical aspect ratio of 4:15, so that the triangles formed by the diagonals will collapse down to a near-perfect dodecagon. Understanding the basic construction, I was now curious about how to design my own similar model. I fell back on another principle:

**Principle 6: Generalize existing designs in as many ways you can think of**

There were two immediate ways I could think of to make a slightly different model:

- Add more segments to the existing spring (i.e. go from 6 segments to 8, 10, 12, etc.)
- Change the number of sides in the collapsed polygon

The first change is easy to visualize – just add more segments to the right edge of the model, as many as you want. The proportion of the starting rectangle will change from 8:15 (6 segments), to 8:20 (8 segments), 8:25 (10 segments), and so on. The second change is much more interesting and challenging, so being a sucker for punishment :-), I took that route.

In this case, changing the number of sides of the collapsed polygon changes the angles, thus the tangents, thus the desired ratios — basically, everything! Luckily, the calculations all flow directly from the number of sides chosen. The only other constraint is that for the whole thing to work you need a right triangle in the middle, which means in short that the polygon needs an even number of sides. So, out came my spreadsheet software and I quickly generated the following chart:

The interesting part of this was coming up with decent approximations, which is more of an art than a science. Basically, we’re looking for a balance between an acceptably close approximation and reasonable small numbers for the numerator and denominator. As you can see in a short period I was able to come up with fairly tidy approximations in small integers for all but a couple of the options. I haven’t had time yet to actually create one of these derivative models, but I have my eye on the 16- and 24-sided ones. I’ll post one when I get around to making it!

Once I had the approximations worked out, I added some columns to figure out the actual paper sizes I would need to start with. And that’s when I had…

## The Final Revelation

When perusing the results of my chart above, I started doing something I have come to habitually do, which is to look for interesting patterns:

**Principle 7: Always be on the lookout for interesting, unexpected patterns**

When I looked at the chart, a few things jumped out at me:

- The paper ratio was almost completely flat (1.875, our famous 15:8 ratio) once the number of sides was 12 or larger
- The strip numbers for the 22-sided figure were 22 and 7, and 22/7 is a well-known approximation for π

So, I added a column to calculate the strip aspect ratio, then, realizing this was based on the approximations, I added two more columns for strip ratio and paper ratio that are based on the exact tangent values:

Sure enough, the strip ratio is definitely approaching the value of π. Why is that? Well, if you look back at the photo sequence and the shaded triangle in the dodecagon figure, you can see that the strip width = the long side of the shaded triangle, and the strip height = the perimeter of the dodecagon. If you imagine the triangle getting skinnier and skinnier (as the number of sides increases), the perimeter of the polygon approaches the perimeter of a perfect circle, and the long side of the triangle approaches the length of the hypotenuse of the shaded triangle (which is also the diameter of the same circle). And, the very definition of π is “the ratio of a circle’s circumference to its diameter.” Eureka!

**The bottom line is: you could start with an 8:15 piece of paper, and divide the left side into ANY even number of segments >=12, and still have a spiral that works!** As the number of sides increases, the model will get “smoother” or more “circular.” My mechanical intuition isn’t good enough to predict whether the “springiness” will get better or worse with more sides – I’ll just have to build one and let you know!

## What Principles Do You Use?

For those of you out there who make your own geometric art, I’m curious – do these principles resonate with you? Do you have others that you steer by when creating your own work? If so, please comment below and share them with us!

I love it when pi pops out of a problem unexpectedly.