(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 firstname.lastname@example.org, and I will offer you full accreditation. Maybe we can get a worldwide collection of twists for folders to play with!
The origami world is vast.
Just. Look. At. All. Of. These. Different. Origami. Styles. It amazes me daily that that breadth of expression can be made with a single uncut sheet of paper.
My own style is an extension of what other folders have been doing with origami tessellations, and I’ve put my own understanding into the design. I practice two primary types of origami styles: pleat patterns and corrugations. I’ll discuss corrugations in a later post, but for now I want to deal with pleat patterns. These were the subject of my first book, and I’m expanding quite a bit on that material in the forthcoming second edition (whole separate blog post). These patterns are composed of pleats that converge on an area to create (generally) flat twists. These twists can be connected with others and tiled as much as the folder desires so long as there is enough paper, and there are a massive number of possibilities. Here are some examples:
(most of my earlier experiments can be seen on my old flickr page here)
They’re often visually interesting and and pique viewers’ curiosity. “How many sheets of paper is that?” they ask (just the one). “Where do you buy paper with those triangles in it?” (You make those triangles by hand). “How do you make those twists? (a lot of practice)”. Even among the practitioners, a lot of the terminology is not set in stone, and it’s not uncommon to have a conversation with another origami artist and have to align your terminology with their own. It’s getting more streamlined, but there is a long way to go with regards to formalized terms. That said, to understand origami patterning, artists come up with different ways to approach the design of such foldings. They can look at the polygon the twist creates and try to figure out why it was that shape. They can try to look at the space between the twists and create pleats that will make those negative spaces. I like to study the pleats as driving over the paper until they intersect and then I figure out what the middle will be.
I describe six twists in my book, with photos and crease patterns that look like this (click one of the photos to view the slideshow):
I believe the most relevant information to be the approach of the pleats toward the center, rather than what actually happens at the center itself. Stripping that away, I call what remains a pleat pattern (as opposed to the standard crease pattern). In this, creases extend until they hit the mountain fold of another pleat, and that is all of the detailing required. This is a generic pleat intersection, and the same pleat assignment can be used for a vast number of interpretations of that specific intersection of pleats.
The pleat patterns for the six twists are as in the photos below:
Since the first edition of my book, I wanted to create a simple and effective way to describe these pleat intersections. I thought, if you can look at a pleat intersection and say “that’s a ____” then it’d be a lot easier to relate to other folders the way it intersected, and it would make documentation a lot easier as well. I put forth one method of describing which was… adequate, I suppose. Now that I’m writing the second edition, I’ve taken to revamping the system, this time with a couple of colleagues. A couple of colleagues and I were discussing some potential options, and it sort of took off. We have developed a system for understanding the way pleats intersect on a hexagonal grid. It is still very much a work in progress, and at the moment, is only being used to describe:
Despite these restrictions, there is a truly vast number of twists that can be created. You can view the database linked here for more information and continue to Part Two for the details of the system itself.
Well, all of my origami friends have blogs, so I should too, right? From Eric Gjerde’s Origami Tessellations, to Phillip Bell’s Fitful Frog, to Joel Cooper’s blog, OrigamiJoel (seriously, go check out these sites; they’re amazing), it’s become a sort of rite of passage. And I’m looking at my website thinking… man, this makes it look like it’s just a hobby. That won’t do.
I've been studying origami design seriously since 2007 and in the past 12 years, my writing ability has improved (I think), as has my ability to describe what I'm doing with the paper. So I feel like I can talk about what I feel is important to my style of origami. To boot, it occurs to me that over the past few years I’ve done a rather poor job of documenting my career in the art field, and I need to stop that. Since 2009 I've done several dozen art exhibits, a myriad of classes, collaborations, projects. I’ve attended quite a few origami conventions, worked in and/or ran three studios, sold works, and met fascinating people. There are stories there, and they should be written down somewhere.
Add in the fact that much of what I know about art is through trial and error or conversations. I never went to art school (I majored in French), and most of what I’ve learned has been through word of mouth and conversations. I've had people tell me to hide the struggles I go through for the sake of exploration because "the display of success begets success,” which I suppose is true to an extent. But in practice, I've found that way of thinking leads me to being more reserved about the process, and as a process-based artist, this is the most important part! This blog should be a place to show my results, as well as my experiments, my trials, and sometimes just to write my thoughts on the process of building an art career.
So then how often should I post? Weekly? Biweekly? I dunno, I’ll have to feel how much of a response there is and how quickly I can write. At the moment, I have a lot that’s been building up and I’ll probably write quite a few posts and schedule them out for the next few months, but after that… I honestly do not know. This will be a test of a sorts to see, well, how much I can blab, how much I have to say. I hope you enjoy the material!
Ben Parker Origami Artist
I fold paper a lot.
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