What's a Tiling - and why does it matter?

I addressed this very question in this week's livestream, which you can see here:

Tilings are the roadmap for where pleats go and what they connect in a tessellation.

They're also the level of analysis that's most beneficial for folding and designing.

After all, if you know that a particular hexagon in a pattern is in a position of 6-fold rotational symmetry and you know one of the things that goes next to it, then you know all of the things that go next to it!

Thinking in terms of tilings and symmetries allows you to generalize and describe tessellations more succinctly too.

For example, the hexagons and triangles 6-fold tiling has a simple alternating symmetry available where you can choose one hexagon twist, one triangle twist, and the spacing between them and with those three choices describe the entire tessellation.

Now I don't know about you, but those three choices are much easier for me to remember than an entire crease pattern!

In fact, you may have been thinking in terms of tilings already and not realized it.

For example, if you know that square twists go together in loops of four twists or that triangle twists go together in loops of six, that's thinking in tilings.

The benefit of thinking in tilings explicitly instead of implicitly is that you can plan longer-range patterns and anticipate potential problems while folding.

Now that I've been aligning my pattern repeats to the edge of the paper in my display pieces, I've become very sensitive to whether my repeats are aligned with natural tiling breaks - places where you can stop folding without having pleat overlaps.

This alignment determines whether I'll need to fold all the way to the edge or if I can leave a clean border.

If I was only thinking in terms of what twists go next to each other, I wouldn't be able to see this broader pattern.

The kicker is that I've identified several tiling patterns, including one that's used frequently, that don't have natural tiling breaks at all!

Whether you're using tilings to better understand what you're folding or for coming up with new designs, I hope you keep looking for the deeper structures underlying origami tessellations and see if you can come up with a few of your own!

Happy folding,

Madonna