The ways of Tessellation

I attended a class at the Liverpool 2012 origami convention run by Tom Hull, and which featured his amazing Hexagonal Lace Tessellation. He was a great teacher and provided diagrams. By the end of the two hours, pretty much everyone who wanted to complete the model had done it. The only exceptions where a couple of folders who had actually gained a fair amount of experience creating tessellations. We'd even included some our examples in the exhibition. Where did we go wrong.

Well, it my own case the starting point was pure stupidity. I messed up a simple triangular grid which I'd casually folded the night before. Fair enough. However, I was also struggling conceptually, despite the fact that the Lace Tessellation is a relatively straightforward model.

The reason's for my problems were at least partly because of the mental toolkit that I was using to fold the designs.

There are essentially three ways of thinking about tessellation patterns.

The way of the pattern

The starting point for CP tessellators is to create a crease pattern, either by hand or using computers. They then either print or laser-cut this pattern on paper and fold it from that, or work out methods of folding the required reference lines to create the pre-creases. Once that's done, they can develop the sequence which will be used to collapse the model into its final form. It's an involved process, but it's capable of producing amazing results which would be impossible to create using any other method. A leading proponent of this technique is Robert J Lang.

The way of the molecule

Illan Garibi wrote a paper about this technique for Origami USA. He shows how you can build complex tessellations out of smaller units and provides guidelines for linking the units together to generate larger patterns. His pineapple tessellation is one of my all time favourites. Eric Gjerde has used similar techniques for some of his simpler models. Another proponent is a Polish folder called Halina Roscizwska-Norloch whose work I've discovered recently.

Molecular folding does have its limitations but it's accessible for beginners and is capable of generating beautiful models. And as a purist, you could argue that it's more in keeping with the original tessellation patterns first created by the Romans.

The way of the way

The procedural school starts with a base and expands that either outwards or inwards using a series of rules. Spread Hexagons is procedural as is the Roman Diaz's Fractal Flower. The leading proponent is Shuzu Fujomoto whose Hydrangea and Clover folds demonstrate the technique's power in a compelling way. You can fold the Hydrangea either by adding new levels inwards from the centre or folding a base and working outwards. The inwards or fractal method allows you to decide on the number of levels after you've started folding. The outwards method produces neater results allow you have take the number of levels into account during the pre-creasing stage.

The big advantage of this method is that once you've learned the rules, you can make use of them in a range of other models. You can take one of the patterns in Project F, add a simple tweak, and then fold it using the techniques using the classic Fujimoto star. Ultimately, that's how I was able to fold the Fujimoto star in the first place. I'd learned the rules from the Roman Diaz Fractal Flower so it was a logical step to apply them to the Star.

Spread Hexagons is procedural. Roman Diaz's Fractal Flowers is procedural.

In conclusion

So how does this affect my ability to fold the Lace Tessellation? Well, like I said, I was stupid. However, I'm a molecular folder who's devoted a lot of time to discovering the simplest of molecule-based tessellations. I'm also a huge fan of the work of Shuzu Fujimoto, so I've folded a number of procedural models as well. I haven't folded many CP-type models at all. This was also true of the other person who was struggling, although I'm sure he's mastered the model now.

The result was I spent two hours attempting to decompose the Lace tessellation into its component molecules.Try as I might, I couldn't find the logic behind the apparently arbitrary reference lines, although they're obvious enough now.

With the crease pattern and a little help from Tom Hull, I was able to complete a single molecule by the end of the session, so I left the session with a smile on my face. Years later, I've finally learned how to fold the model, albeit badly. If you've folded one of these models, perhaps at an origami class, try looking at the back. The pattern of linked triangles and hexagons is lovely.