In my previous post, I said:

Over a few years I’ve been building up a list of subjects I could attempt to design. This most recent lockdown is when I finally decided to bite the bullet and make a proper go of it.

Well, this is the first attempt at doing that. After deciding I wanted to primarily design Australian animals, the question was which one to start with. The title of this post not withstanding, the answer was… a box jellyfish.

I was intrigued by the combination of a small, cube-like body and incredibly long tendrils. (The tendrils of an Irukanji box jellyfish can be over 100 times longer than its bell.) I had an idea in my head of how to do it. But once I started trying it out, the limitations of paper thickness started to interfere with the shape I wanted to achieve.

In the end I got… *some* kind of jelly-like creature, but it wasn’t entirely what I was aiming for.

Realistically I knew that that approach was a bit of a dead-end, so I moved on to animal target number 2.

## Actually a platypus this time

I love platypuses (or platypodes, but *never* platypi) as a distinctly Aussie creature. But I hadn’t seen any great representations of them in origami. This became my main focus. To avoid burying the lede any further, I ended up with something I’m really proud of.

This certainly didn’t just appear fully-formed in my mind. Some parts I had to experiment with and refine many times. Other parts turned up as serendipitous accidents.

Quoting again from my last post:

I’ve got to remind myself that there’s a fair bit of selection bias going on; we don’t see all the failed attempts, only the ones that succeeded. I also have to remember that the incredibly prolific creators have spent, in a lot of cases,

decadespractising and refining.

I don’t want this to just be another “tada!” moment, so I’m going to break down the process of how I designed it. Whether I do the same for every future model depends on how much effort it takes…

## Design process

Note: I’m well aware that this blog that has traditionally been very focused on web tech, so it’s what the majority of readers will be familiar with. Some of this post is going to be heavy on origami terminology. Hopefully it’s explained enough that you can just go with the flow.

One of the beauties of an artistic field like origami is that it can range from almost-impossibly precise to eye-squintingly abstract. Most origami artists have their own particular style; what should mine be? (Or at least, what should I aim for?)

[…] a common theme is picking just one or two features of their chosen subject and emphasising them almost to the point of cartoonishness. I think this is where I want to head with designing models — not super-complex, but with

— Quoting my previous blog post again.personality.

First I should define what I was actually aiming for. I wanted a platypus model that had:

- Roughly the right proportions, primarily viewed from above.
- As few folds as possible down the back (from the head to the tail), emphasising the sleek body built for swimming.
- A colour-change effect on the bill and, if possible, the legs. Thankfully platypuses have black bills and feet, so using black/brown paper would work really well.

### Skeletons and trees

A good way to think about the core structure of a model is via a simplified tree drawing. A simple graph of points and lines representing the basic relationship of parts. It’s very much like looking at a creature’s skeleton, but only focusing on the joints of major features.

If you look at a platypus from above, you’ll see that it’s a ridiculously basic structure:

Seven lines, all of roughly the same length. Now to find an optimal way to arrange those points on a square piece of paper.

### Basic base

Coincidentally, this exact example appears in Robert J. Lang’s *Origami Design Secrets*. I only realised this after working out the platypus tree, and was looking through the book for tips on designing bases.

Lang’s maximally-efficient base produced flaps of all exactly the same length. But it also involved little “gusset” creases that made it more complicated to fold. Additionally, I wanted my model to be folded in “plan view”, meaning that it would open out flat as if you’re viewing the animal from above. Lang’s base wouldn’t quite work for that.

I’m completely willing to sacrifice mathematical accuracy for cleaner folding, so I tweaked the reference points a little. The flaps were no longer exactly the same, but still pretty close, and now it could open in plan mode.

As an unexpected bonus, when I first tried to fold that base flat, one bit of paper flipped over and gave a colour change right where the head would be. A happy accident that gave me the technique for the bill without even trying.

I tried to play around a bit with extending that colour-changed section in such a way that I could make some eyes as well. The results were… a little creepy, actually, and way too large (platypuses have tiny eyes in reality). Much later I realised I could just do some subtle shaping to give the *hint* of eyes, without being explicit.

### Grafting

The base was ok at this stage, but a bit too thin and pointy at the head and tail. Aiming for a smooth back at the beginning allowed me some flexibility around width. I used a technique known as *strip grafting* to widen the middle of the model without overly affecting anything else.

It’s effectively like taking the crease pattern, cutting it in half, shoving some extra space in the middle, then stitching everything back together again.

This worked really well for the design, but came with two extra problems:

- How wide should the extra strip be to get the right body width proportions?
- Adding extra space in the middle, while still keeping the starting paper square, means squashing the patterns on the side. This in turn changes the relative proportions of the flaps, making the body too long between the front and back legs. The location of reference points for the creases will need to change to account for that… but how?

### Massively over-thinking it

*In which the protagonist reverts to type and uses computers to solve a problem.*

By this point, I realised I had 3 independently-moveable reference points that all affected the final proportions of the model somehow. The width of the body, the “head mark” (which equated to length of the colour change on the bill), and the “back leg mark” (which became the toe of the back leg). All of these reference points could be changed without having to change the other two, but every change affected the relative lengths of all the flaps.

My process for drawing the base crease pattern is usually just a pencil, some grid paper, and a bunch of hand sketching.

But now I had a problem that was beyond simple sketching. What would be the optimal configuration of these reference points, to create a model with *roughly* the right proportions, while not being overly fiddly to try to measure on a real piece of paper.

This basically gave me the excuse to play with something I’ve been meaning to try for a while: Observable notebooks. It was also a good reason to learn more about the underlying mathematics and geometry of origami crease patterns.

So I sketched out a more accurate crease pattern with a bunch of named reference points (as seen in the image above). Then I worked out how to derive the exact location of every point using only the 3 variable points from before. I was able to hook up some sliding range HTML inputs to the calculations, then write a whole bunch of messy trigonometry. At first I got the notebook to just draw the basic skeleton to check the proportions. But, well, I couldn’t stop tinkering with it, and it got more and more complex. Eventually it was also drawing the crease pattern and a rough representation of what the folded base would look like.

The code is messy, but in the name of openness I’ll share it anyway. If you want to play around with it yourself, it’s at https://observablehq.com/@gilmoreorless/origami-platypus-base-ratios.

If I end up doing something similar for other models in the future, I’ll build upon the excellent Rabbit Ear library instead. But for this exercise I wanted to make sure I understood the underlying mathematics before leaning on an abstraction library. Hell, I even ended up drawing myself a trigonometry cheat sheet:

### Refining (more grafting)

The design now had legs of *roughly* the right length according to the mathematics. But when folded in the non-abstract real world, there just didn’t seem to be enough usable paper to make good feet and claws. My answer to this was to use the same technique as the body, but on the edge of the paper. This is unsurprisingly known as *edge grafting*. I also tried adding an extra edge graft for the bill, which turned out to give me more flexibility in the final shaping.

Of course, this had added yet another variable into the calculations: how wide should the edge graft be? This time I didn’t bother with the maths and just tried something that was easy to fold. I first tried a measure of 1/16th of the whole paper width. By remarkable coincidence that turned out to be best value for what I needed. I tried other widths but they caused more problems than they solved.

At one point I created another Observable notebook to help me find measuring reference points: https://observablehq.com/@gilmoreorless/origami-grafting-calculator. This was particularly helpful for finding a point on the *inner* bit of paper (the original pattern before adding the edge grafts), while only measuring the *outer* edge of the paper.

So after all that combining of elements and experimenting with different values, I finally settled on the following measurements:

- Edge strips for bill/claws: 1/16
- Body width: 1/8
- Head mark: 3/16
- Back leg mark: 1/4

## Conclusion

Despite all the measuring and re-measuring and re-re-measuring and so on, this whole process has been a lot of fun. I’ve now got the confidence that *I can actually do this designing malarkey*. This model surprisingly ended up being an exercise in finalising the 3D shaping by feel, rather than by maths. I’m pretty damn pleased by the results — I even managed to get an approximation of the venomous spur on the hind legs.

I also now properly understand why so many origami instructions don’t use specific reference points on the final shaping steps; it’s all about best-guess judgement based on what kind of pose or expression you’re after. No two versions of the platypus I’ve made so far have been the same. Though that does have the unfortunate side effect of making it much harder to create step-by-step instructions, if I ever get that far.

As a sign-off, here’s a picture of all the variations I played around with, as a measure of progression.

At one point, there was even a wonderful moment of accidental design. I was folding one of the variations, but had only half-folded the back half. When I put it down on a table, the front half popped up and – with fat back legs not quite finished — it looked a lot like a sitting frog. That’s a variation I’ll have to try exploring properly at some point.

Credits:

- Robert J. Lang’s original base comes from his book
*Origami Design Secrets*. The diagram was re-created by me using Lang’s TreeMaker software and in Adobe Illustrator. - Platypus crease patterns were created using OrigamiDraw then touched up in Illustrator.