Hybrid Lattice Origami Tessellation

We don't usually think of tessellations as being changeable after they're folded, but this one is!

The hybrid square twists that it's made of can be folded in one of two ways, making this twist a binary pixel.

I chose to align all the twists on the same side in the same direction, but you could choose other patterns - stripes, checkerboards, maybe even letters - and you can have either the same or a different pattern on the other side!

I'm calling this design Hybrid Lattice and it could be folded in an even denser spacing to really highlight the binary patterns of the twists.

Symmetries

What symmetries do you see in this pattern?

Are there any 4-fold symmetries? Any 2-fold symmetries? (Hint - which way are the arrows pointing?)

How are the twists connected - are all the pleats perpendicular to each other?

This tessellation is an unusual example of the square tiling that has no rotational symmetry in its folded form, only translational symmetry.

That said, these hybrid square twists have two possible folded states from the initial pleat positions and you could fold a version of this tessellation with 2-fold rotational symmetry with a checkerboard pattern of the hybrid square squash directions on each side.

How would you choose to fold this - with or without rotational symmetry?

Deeper Structure

Hybrid Lattice is an isoarea tessellation using a square tiling.

Now, what does that mean?

It means that it looks the same on the front and back (up to rotation, mirroring, and translation) and all of the twists have four perpendicular pleats.

Going further, it means that half of the twists are on the front and half are on the back, and (in this case) each of the twists serves the same function in the broader pattern.

It's an example of one of the six symmetry patterns on square tilings that I'm teaching inside of Tessellations by Tiles - the alternating twist symmetry.

You can play with this symmetry yourself - just pick a square twist (closed/open, front), then pick another square twist for its neighbors (closed/open, front/back). Alternate between these two twists and see what you find!

Once you understand tilings and their symmetries, you can write generator functions like the last paragraph to make new tessellation designs - and the possibilities are endless!

So, what will you make with an alternating pattern?