Alignment - of what?
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".
But really a grid is an abstract concept and we have to make decisions about how we're going to apply that concept to the paper in front of us.
So, what if we want to align our grid to something else?
What if we want to align our grid to the pattern repeats?
That's where rotated grids come into play.
This alignment question comes into sharp relief with patterns like Woven Strips where the pattern repeats in stripes that are very clearly not oriented in alignment with the edges of the paper!
While I have done the math to align this kind of pattern on this kind of grid, I haven't tested it yet in practice - which is why this piece isn't aligned with the repeats.
We see these shifting horizontal bars in the pattern, crossed by slanted bars.
It's as if the trapezoid twists are pushing the slanted bars into place, like a rotating portion of a machine.
In this particular orientation it also looks like thrust faults and rolling waves of tsunamis.
In the vertical orientation it looks like a chain of buckets, raising and lowering some cargo.
This trapezoids and triangles tiling is very similar to the common tiling of hexagons and triangles with 6-fold rotational symmetry.
Each pair of trapezoids fits into one hexagon of the more common tiling and so, just like that tiling, this tessellation has six triangle twists around each pair of trapezoids.
The twist is that the two trapezoids in the pair are on opposite sides of the paper, and so you get the same number of each kind of twist on each side.
As far as I can tell, this tiling will have the same kinds of variations as the simple hexagons and triangles tiling.
The "hexagon" in this case is fixed as the isoarea pair of trapezoid twists - all that remains is a choice of closed or open triangle twist, on the same or opposite side as the closest twist.
Many variations available
Whenever I'm talking to people unfamiliar with tessellations, they have an intuition that the tessellations have all been discovered - that there's nothing new left to find.
That couldn't be farther from the truth!
In fact, there are thousands of tessellations waiting to be discovered.
Nearly every week I figure out some new system of tessellations - usually with at least a dozen possible variations.
Now, I'm only seeing three obvious variations on Woven Strips, but that doesn't include changing spacing or going to more open triangle twists.
It also doesn't include longer-range repeats, adding spacing between particular repeats, or adding spacing in particular directions.
It doesn't include keeping the trapezoids on the same side and adding mirror symmetries or seeing if I can use different triangle twists in the same tessellation.
The more you look, the more you find.