Articles and Tutorials

An Unusual Symmetry These hybrid hexagon twists (there's actually two kinds) are 3-fold rotationally symmetric. This is weird for something with 6 pleats - usually these kinds of twists have 6-fold rotational symmetry. But by putting the 3-fold six-pleated twist in the middle, I was able to push the remaining two 3-fold symmetries out to the next hexagons and use two different kinds of triangle twists. And so Hybrid Hex Weave uses three different kinds of hexagon twist and two different kinds of triangle twist while the simplest patterns in this tiling use only one of each type of twist. I've found...

It's easy to get sucked into the details of the exact placement of mountain and valley folds when you start looking at crease patterns for tessellations (and then think you need to precrease and collapse them!), but there's a simpler way to see these patterns.

A Whole Lot of 3   Sometimes it can be hard to tell what's 3 and what's 6. Now this may sound confusing - how can you not know the difference between 3 and 6? But when there's 6 things, with alternation between each one, is it 3 or is it 6? In terms of rotational symmetry, the answer is 3. In terms of twists in a loop, the answer is 6. And to make matters worse, you can't look for 3-fold symmetries around the point in question to decide - they're all going to be 3-fold in either case!...

Managing many pleats at once is one of the harder things about tessellations, but using these three-way intersections one at a time lets us minimize the difficulty and focus on the steps.

These three-way intersections can be put together in a huge variety of ways, from rotational to mirrored to combinations of both.

When we start folding tessellations we're presented with a couple grid options - all of which are aligned with the edge of the paper in some way.

This fact is invisible in the way we talk about grids too.

We don't say "square grid aligned with the edges of the paper" - it's just "a square grid".