A beginner's guide to Origami Tessellations

Some origami is representational, attempting to capture the essence of an animal or flower in a folded piece of paper. Some origami is useful, creating practical wallets, boxes, and even cups. Some origami is an exercise in engineering, creating vast modular structures. Tessellations are merely beautiful. The idea of Origami tessellations is take a piece of paper, and to collapse it into a series of repeated patterns. Tessellations can be created on any shape of paper, most commonly squares and hexagons. And they are seriously addictive!

I don’t regard myself as any kind of expert, as I folded my first tessellation just a few years ago, and I’m still learning and making mistakes. My starting point was a simple twist fold pattern that I discovered in Nick Robinson’s “Origami Encyclopaedia”. From then on, whenever I read a new origami book, I’d look for more diagrams of tessellations. The big breakthrough in my tessellation folding was the discovery of the work of Eric Gjerde, who has designed a number of my favourite tessellations, and who has written one of the few books on the subject.(Origami Tessellations - Awe-inspiring Geometric Designs).

The British Origami society in 2010 as also had a massive impact on my folding, as my first convention in York introduced me to two of my all-time favourite tessellations, the Fujimoto Hydrangea and Roman Diaz’s amazing fractal flower.

Although there are lots of enthusiastic tessellation folders, it’s one of those subjects which are only thinly covered on the web, and there are currently only four books on the field, two of which are in Japanese.

Aside from Eric Gjerde’s primer and book, the only other resources on the web at the moment, are Ilan Garibi and Sara Adams “A brief guide to warping paper” which is available from the Origami USA site, and which is highly recommended. You don’t need to be a member in order to read it. Although the Origami USA introduction is fantastically useful, and has taught me a lot, it includes a fair amount of theory, and I’m personally more interested in the practical aspects, such as how to fold an existing model rather than creating new ones for myself. Since I firmly believe that tessellations are within the reach of most folders, I thought it might be helpful to write an alternative piece which covered the practicalities of tessellation folding. I also wanted to document the lessons I’ve learned from my personal folding experiments, not least because I want to remember them myself.

Basic principles

All tessellations are created in two phases:

1. Pre-creasing

   This involves making a series of reference folds that will be needed to create the final model.

2. Collapsing

   This takes the previously created folds and manipulates them to produce the tessellation.

Depending on the model, it may be necessary to perform several pre-crease/collapse stages to produce the final result.

Pre-creasing

A common starting point is to create a grid divided into horizontal and vertical creases. An 8x8, for example, would be divided into eight sections horizontal sections and eight sections vertically, dividing the paper into 64 small squares. It’s also possible to divide the paper diagonally or into triangles. Other tessellations require you to impose specific creases into the paper, usually by following a printed or laser-cut crease pattern.

Folding a grid

The first step is to divide the paper in half using valley folds in both directions, which results in a 2x2 grid. You then create new valley folds from the edges of the paper to the centre, which will divide the paper into sixteenths (4x4). By repeatedly dividing each section in half, you can produce 8x8, 16x16, 32x32, and even 64x64 grids.

The key requirements for pre-creasing the paper correctly are:

Precision

There’s no point in rushing or taking short-cuts, as the quality of the final model will be totally dependent on the quality of the original creases. Don’t crease more than one layer at a time, unless explicitly requested by the instructions, as no matter how hard you try to avoid it, the layers will creep apart and the lines will become unevenly spaced. You should be especially careful about the first few folds, as these will provide the reference for all the subsequent creases. When I divide the paper in half with the first valley fold, I always run my finger across the top edge to ensure that the two edges of the paper line up perfectly. It’s only once I’m happy that the crease is correct that I reinforce the valley fold. Incidentally, don’t be afraid to make the odd mistake, as that’s inevitable.

You can correct minor errors during the collapse phase, although if you make too many of them, you probably won’t be happy with the result. At some points, you may have to reluctantly throw away the current model and start again, as it’s pointless to waste time on a model that’s never going to fold correctly and you can save yourself a lot of frustration by simply putting it down to experience.

Patience

There’s no getting around the fact, pre-creasing takes time. A 16x16 grid will require 30 individual creases, and the larger grids such as 32x32 or 64x64 can significant time to fold. Personally, I find the pre-creasing stage relaxing, and I generally do it whilst I’m listening to music or casually watching TV. However, if you don’t have the time or the will to spend several hours on an origami model, you can limit yourself to smaller versions. Many tessellations look fine with even a few levels, and if you’re selective with your choice of paper, you can produce excellent results with only a limited amount of pre-creasing.

Practice

The more folding you do, the easier it gets, and after a while you can do it without really thinking.

Collapsing the paper

After you’ve created your reference creases, you need to fold them into the final model. Depending on the design, you might start the collapsing process at the centre or on one of the corners.

The key things to remember when collapsing a tessellation are:

Take it slowly

It’s best to take a slow, methodical approach, as rushing will inevitably lead to mistakes. Any errors will get worse as you continue with the collapse, as each part of the design is connected, and if one of the elements is the wrong shape, it will be impossible to link this to its neighbours without seriously compromising the design. This can be upsetting if you’ve just spend ages on the pre-creasing.

Fold with feeling

You generally need to ease or cajole the paper into the required shape, rather than harshly force it into position. Many tessellations require you to collapse each part of the design gradually, only finalising the creases when the entire model is complete, and it’s important not to be too rough.

Fold from both sides

If you’re having problems with a particular part of the collapse, turn the paper over and see if it’s easier from the other side. It’s amazing how often a simple tweak from the other direction can simplify the folding process dramatically.

There are plenty of origami tessellations to try if you’re prepared to look for them.

Here are some good starting points from the web:

http://www.happyfolding.com/instructions-garibi-adams-bricks

http://www.garibiorigami.com/images/igo_213_pineapple_tess.pdf

http://www.happyfolding.com/instructions-fujimoto-hydrangea<

https://origamiusa.org/thefold/article/tessellations-brief-theory-warping-paper

http://www.origamitessellations.com/