(Read Part One Here) We can talk about a pleat as an overlap in the paper that lays flat on itself when folded. In order for that to occur, there must be pairs of mountain and valley folds. The simplest interpretation of that is a single mountain and a single valley fold. Those creases can be parallel or not, those are really the only two options. The simplest option is that they are parallel to each other, and so that is what we will start with. A pleat composed of exactly one mountain and one valley fold that are parallel to each other, we call a simple pleat. Our notation system is designed to describe the intersection of simple pleats that are parallel to the diagonals of a hexagon. To see this in action, take a hexagon and label each of the six corners through . We can define the location of a line on this grid (technically a ray) by its distance from one of these axes. The format looks like this: This defines the axis from which the ray is derived, then how many units away from that axis diagonal the ray is, with counterclockwise (CCW) around the origin being positive and clockwise (CW) being negative. Taking this to a pleat, we can use this to locate a crease on this grid; we locate the mountain first. The last part to getting a pleat on the plane is to define where the valley fold is in relation to the mountain. For that, we use a superscript and the format |mp where x is the axis from which the pleat is derived, m is the location of the mountain fold, and p is location of the valley fold with relation to the mountain fold (in other words, the shape of the pleat). A positive m lies CCW of its origin axis and a negative m is CW. A positive p means that mountain is hit first going CCW and a negative p means the valley is hit first. Here are few pleat examples.<div class="paragraph">Now that we have pleat notation, we can use that to define an entire pleat intersection. There are six axes, so we can set up a hextuple system that describes the pleats in relation to axis A1, , , , , , , with a dash as a placeholder for axes that don’t contribute to the intersection. As examples, a hex twist would be written as (0<sup>1</sup>, 01, 01, 01, 01, 01) and a triangle twist could be written as (01, -, 01, -, 01, -). The six twists described above would be notated as follows.</div> Now we have a system for describing pleat intersections. Every pleat intersection notation can have an infinite number of interpretations, but a pleat intersection can only be described using a single hextuple notation (along with all of its isomorphic permutations such as translations and rotations, of course). You can see more of these notations and the corresponding notation in my new website page, The Database.
There will be later blog posts exploring this system and its implications, but for now, see if you can use it to understand and fold the twists in the Database link. If you have twists of your own you’d like to have added, feel free to send them to me at brdparker@gmail.com, and I will offer you full accreditation. Maybe we can get a worldwide collection of twists for folders to play with!
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