Articles and Tutorials — origami

My introduction to the possibility of a career in origami came during the Geometric Folding Algorithms class I took in my last semester at MIT.

This blog post features a list of academic articles I’ve has published since starting Gathering Folds.

Origami tessellations are infinitely repeating patterns folded from a single sheet of paper.

My designs are in the twist-based style and I fold my patterns one twist at a time whenever possible.

There are some patterns that when I see them, I know I have to fold them.

Usually I keep a queue of about 10 of these patterns, but sometimes one skips to the front of the line.

Siren's Call is one of those.

The ways that variations can be made are kinda endless.

You can change the spacing, switch a twist to the other side, change a position from closed to open or vice versa.

Or, in some cases, you can stick more twists in between the twists in your tessellation!

To start folding from crease patterns, we must first decide what part of the pattern to fold - what to put in the center, and how many repeats we want.

Next, we need to actually fold the grid.

Finally, we can start folding!

A Whole Lot of 3   Sometimes it can be hard to tell what's 3 and what's 6. Now this may sound confusing - how can you not know the difference between 3 and 6? But when there's 6 things, with alternation between each one, is it 3 or is it 6? In terms of rotational symmetry, the answer is 3. In terms of twists in a loop, the answer is 6. And to make matters worse, you can't look for 3-fold symmetries around the point in question to decide - they're all going to be 3-fold in either case!...

Managing many pleats at once is one of the harder things about tessellations, but using these three-way intersections one at a time lets us minimize the difficulty and focus on the steps.

These three-way intersections can be put together in a huge variety of ways, from rotational to mirrored to combinations of both.

When we start folding tessellations we're presented with a couple grid options - all of which are aligned with the edge of the paper in some way.

This fact is invisible in the way we talk about grids too.

We don't say "square grid aligned with the edges of the paper" - it's just "a square grid".

Many folders would like to start learning tessellations, but aren't sure where to start. In this video I introduced the four major grid types that every tessellation folder needs to know and clarified the distinction between twist-based tessellations, corrugations, and tessellations that require an all-at-once collapse. You see, there are easier and harder ways to fold tessellations and when many folders see a crease pattern for the first time they think that they need to precrease everything and then collapse all at once. That's actually the hardest way to learn tessellations - you have more work at the beginning, more...

What are twist blocks?   Whenever I look at tessellation crease patterns, I mentally  break them up into their component parts. With twist-based tessellations, those parts are twists and they're connected by pleats. The good news is that the same set of twists are used over and over again in different tessellations, which means that we can treat them like building blocks and drag and drop to snap them together! I made a set of digital twist blocks in both square and triangle grid for my Tessellations by Tiles students last fall and found that they really speed up the diagramming...

Patterns in the Tessellation   You can see many things in any tessellation - here I'm seeing a close mimic of the purl knitting stitch, and also an Easter bunny. I'm also seeing parallel mirror symmetry lines with a string of twists between them, a sequence of triangle-rhombus-triangle that I use in many tessellations, and an unusual arrangement of rhombi and triangles. Which aspect I choose to emphasize varies with the lighting, the level of zoom, and who I'm talking to.   Details As we look closely at this tessellation, we can see that there's the zigzagging rhombi and triangles,...

Tiling Surprise Believe it or not, Hybrid Closed Islands has the same tiling structure as Robin Scholz's Celtic Circle. Both of these tessellations feature twists with six pleats (hexagon twists) surrounded by twists with three pleats (triangle twists) and vice versa. However, when you look at Hybrid Closed Islands it's easy to see only triangles - because the six-pleated twists only have three-fold rotational symmetry.   Centering Choices In this tessellation, I have three equivalent choices of what to put in the center, and whatever I put in the center will be emphasized in the pattern. Each of these three...

The more I fold grids, the more I realize that the possibilities are endless.

For example, I used to think that triangle grids on hexagons could only be centered on grid intersections - that I had to start by folding the hexagon in half in one way or another.

If you're already familiar with isoarea tessellations, you might notice something unusual about this one - it uses hexagon twists!

Most of the time you'll see isoarea patterns in tessellations with all the same kind of twist - squares, equilateral triangles, right triangles, and occasionally rhombi.

While you may already be familiar with origami terms like mountain and valley folds, sinks, squashes, swivels, and rabbit ears, tessellations have a vocabulary of their own. Unfortunately, the vocabulary of tessellations is not standardized across all tessellation designers and each tessellation book uses different definitions - and different colors and symbols for their diagrams. This can make it hard to learn, since everyone is saying things in different ways. I try to stick to consistent definitions throughout my teaching, and to explain my terms in plain English - no math speak. Here's my list of the most important terms to...

So, you've seen the math behind rotated grids but you're still confused about how to *actually* use them for your patterns. You aren't alone and you're in the right place.