In recent years there's been a great buzz around a new area of origami: tessellations! Essentially, these are patterns you fold and which you can repeat over and over, extending the design. In other words, it is a tiling.
But this article isn't so much about what tessellations are. If you'd like to know more about that, David Lister wrote some essays, which give great background information and an interesting insight into the history of tessellations: Part 1, Part 2, and Part 3. Ilan Garibi also wrote an article classifying tessellations, going into the creation process, as well as giving tips on useful tools: Tessellations: A Brief Theory of Warping Paper (TheFold Issue 2).
While designers such as Yoshihide Momotani and Shuzo Fujimoto explored tessellations early on, only recently did these folds become more main-stream. With it an explosion of stunning designs have emerged.
As is usually true for a new area of origami, great pictures started to appear, but folding these tessellations remained a bit of a secret for some time. Then crease patterns were shared, but these aren't well understood by everyone. The first big publication (in English) that demystified tessellations and explained how to fold several stunning projects was Origami Tessellations by Eric Gjerde (also available at The Source).
Still, the learning curve for folding tessellations is steep for many, sometimes too steep. Although Eric Gjerde's book is fantastic, it does require good basic origami skills and perhaps a love for working out the details yourself. This is why I decided to make a bunch of instructional videos, which help a bit with those details. For starters, I presented some origami tessellations step-by-step. It's perhaps the easiest way to get going if you already folded other non-tessellation models, be it from video or diagrams.
But the true magic of tessellations lies in creating your own designs and patterns and how easy it is compared to designing representational origami. You can start experimenting: fold a grid and then see how you can collapse the grid into shapes and continue on. At least at first, it's probably better to concentrate on structures that do fold flat, although tessellations with 3D components are definitely possible and have a beauty in themselves.
Or, if you aren't so much about experimenting in the wild, you can learn about some basic rules that will apply to all tessellations. Knowing these rules opens you a whole world of exploring tessellations. I myself was introduced to these rules by Ilan Garibi when I met him at the CDO Convention in 2010 (in Italy). I then made a video, lovingly and jokingly calling it a "Tessellesson". It demonstrates the technique on a model called "Bricks" by Ilan Garibi, which works on a square grid. But the rules can be applied to other grids and in particular also triangle grids.
(There is a longer version of this video here)
And once you know the rules, all you need is some "molecules" that you want to use and combine to construct your crease pattern to collapse. You can either - as before - experiment to come up with some of these. Or you can use techniques others have already explored. Some of the more common techniques I have presented in short videos. In particular, here's a playlist on techniques that work on a triangle grid, and which are also presented in Eric Gjerde's Origami Tessellations:
Once you master these techniques you can use them in two ways: either to create your own designs by combining them; or simply to understand how to collapse crease patterns of designs others have come up with and shared. For example, Eric Gjerde has posted some crease patterns on his website http://www.origamitessellations.com/, as have many others on their websites or Flickr albums (e.g. Andy Wilson, Christiane Bettens, Ilan Garibi, and Melina Hermsen, and Ralf Konrad). Anna Alekseeva put together a set of pictures classifying different tessellations according to their wallpaper group. Another great resource is Alex Bateman's website. It not only includes crease patterns, but also a program, with which you can create crease patterns for tessellations.
With that I wish you happy folding and exploring the world of origami tessellations!