There are two ways to look at twists:
- Where should the pleats go relative to the center of the twist?
- Where should the center of the twist go relative to these particular pleats?
These different perspectives let you see different options and I call them the designing and the folding perspective, respectively.
Using these perspectives we can see that different sizes of square twists create a line that extends to those same sizes on the opposite side of the page.
We also see a zone around that line where properties of two different sizes of square twist can be hybridized together.
In box pleating terms this creates a level shifter and in tessellations it moves the pleats around in unusual ways, affording different symmetry styles that can be codified as design equations.
This lecture is a preview of the paper I'll be presenting at the Bridges conference in July 2023.