Articles and Tutorials

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...

Backlighting Surprise!   As the name implies, Knotted Web Tight is a close relative of Knotted Web - everything's just one spacing closer together. Interestingly, the donut shape that's somewhat visible in Knotted Web is much more pronounced in Knotted Web Tight and shows up as overlapping circles in the backlighting. This was a huge surprise to me when I first saw it while I was folding! No part of this pattern is actually curved, and yet the backlighting seems to show only curves! What's more, since this is an isoarea pattern the same backlighting is seen on both sides....

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...

Fully Flexible   Sometimes when you're thinking about tessellations, often when you're looking at what someone else has folded, you have a flash of insight and see new possibilities for patterns that you hadn't considered before. That's exactly what happened when I was looking at Miguel Gañan's book, Twist and Tess, when I first got it. I was looking through the crease patterns for interesting tilings, and came across a pattern that used the hexagon-rhombus-triangle-rhombus loop (which is usually very restrictive) in an expanded form that has specific places where the spacing is very variable! Immediately, I thought of Lens...

I addressed this very question in this week's livestream, which you can see here:   Tilings are the roadmap for where pleats go and what they connect in a tessellation. They're also the level of analysis that's most beneficial for folding and designing. After all, if you know that a particular hexagon in a pattern is in a position of 6-fold rotational symmetry and you know one of the things that goes next to it, then you know all of the things that go next to it! Thinking in terms of tilings and symmetries allows you to generalize and describe...

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,...

There's plenty of tutorials out there for folding triangle grids on squares, but the translation to folding these grids on rectangles in the orientation of your choice isn't well known. That's why I went live this week to talk about these grids - and why you might want to use them.   We often think of triangle grid tessellations with 6-fold symmetry that work well on hexagons, but there's also tessellations on triangle grids with dominant 3-fold or 2-fold symmetries too! In the 2-fold case or cases with one direction of mirror symmetry a rectangle is often the shape that...

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...

In the beginning, everyone learns origami from someone else.

As you become more advanced, you may start to rely more on recorded tutorials, books, and other diagrams.

At some point, you start creating your own designs - and then how do you learn more?

Serendipity There are some things that you can't know in advance. In the garden I can know what I planted and where, but not when it'll come up or how many plants will survive. In tessellations I can know my symmetry pattern, know what twists I'm using and in what spacing, but still not know what will emerge from the backlighting. In both cases, the end results are often more beautiful than we could have planned for. In both cases, I like to space things close together to see what happens. I'm often pleasantly surprised in the garden, just like...

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.