Star Weave and Reverse Engineering

Oh the thinks you can think!

So, it turns out that we have more choices available to us than we might suspect with Hexagon and Triangle 6-fold tilings!

Instead of just having one choice for the next hexagon out from the middle, we actually get four hexagons to choose from - closed or open on either the front or the back.

And that's after we choose the central hexagon (2 choices) and the triangle twist (also 4 choices)!

Taking all of these options into account, we have 2*4*4=32 options for arranging hexagon and triangle twists in this extended symmetry and we haven't even talked about spacing yet.

A closer look

Looking from the "weave" side of the pattern, we can clearly see which hexagons have 6-fold or 3-fold rotational symmetry.

(If you squint it's even clearer)

What may not be clear is the location of the 2-fold rotational symmetry.

Whenever there's a 6-fold rotational symmetry in a repeating pattern there are also positions of 2- and 3-fold rotational symmetry.

In this case, our 2-fold symmetry is in the center of the loops with two closed hexagons.

You spin my head right round (right round)

When I'm designing new tessellations, I usually have a specific pattern in mind.

For Star Weave, that pattern was the extended symmetry on a hexagons and triangles 6-fold tiling.

But how did I know about that pattern?

I didn't know about that pattern until recently, even though I had used it in a design over 2 years ago.

I didn't know that pattern until I started focusing on the locations of symmetries that are possible in specific tilings - until I started trying to explain why different options are available to my Tessellations by Tiles students.

And let me tell you - this knowledge has rocked my world!

All of a sudden, many patterns that I thought were one-off coincidences are actually explainable, predictable, and systematic.

New variations in these patterns are intentional - like Star Weave.

I thought I had been paying close attention when I was focusing on tilings, but I was actually making simplifying assumptions left, right, and center.

Now I recognize that it's not only the tiling that matters, but how you apply symmetry to that tiling.

Folding without a Crease Pattern

When I was starting to fold tessellations in 2017, I quickly exhausted the supply of video tutorials and moved on to photo tutorials, then to reverse engineering.

Luckily, the tutorials I did find taught me the basic twists - closed and open square, triangle, and hexagon twists and also rhombus twists.

With that background and an awareness of some simple tilings (square, equilateral triangle, hexagons and triangles 6-fold) I was able to reverse engineer my next 100+ tessellations.

And so can you.

With an eye to which twists are being used, how they're connected, how far apart they are, and how they repeat you can reverse engineer any twist-based tessellation.

Watch the video above for more tips specific to various difficulty levels of tessellations, then follow along with the folding at the end to fold Star Weave for yourself!