Articles and Tutorials

Many origami folders have been admiring tessellations from afar for years, intimidated by the apparent difficulty and the length of preparation needed just to get started. Others have dabbled in tessellations with live classes at in-person or online conventions and books, but come away frustrated because they get lost and can’t continue or the instructions are unclear or they aren’t able to recreate what they folded live, much less extend it to fold more repeats. Folders often look for the right paper or the right tools to make tessellations easy, but don’t realize that the way they’re attempting to fold...

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!

So many options! When you put triangles next to hexagons you give yourself many, many options of what to fold next. You can fold another hexagon (of a different type!), a rhombus in any size or side of the paper, and probably more options that I haven't explored yet. Open Triangle Linked Flowers shows a dense arrangement of closed hexagons, open triangles, and standard rhombi - all on the same side.   Details Matter Would you tuck your triangles under or over? The crease pattern won't tell you, but you must decide if you want your tessellation to lay flat....

A little tweak for big results As I'm folding I have a constant refrain of "wouldn't it be cool if ..." going through my head. This time that thought popped up as I was preparing for the square tiling lessons in the current round of Tessellations by Tiles and coming up with new designs that take advantage of extended symmetry patterns available in the tiling. I knew what would happen if I used the hybrid twists to add extra space to a closed square in a weave pattern - that's Square Knots Weave - but what would happen if the hybrid...

Let's talk rotated grids Rotated grids are a powerful tool for folding display-ready tessellations. Since I figured out the mathematical underpinnings of this tool last year, I've considered rotation for every single display-style tessellation that I've folded. But when I talk about rotated grids, many people's eyes glaze over. It's not obvious to non-tessellators whether a piece has been folded on a rotated grid or not - and even experienced tessellators can struggle to identify rotation if they're not familiar with the technique. The bottom line with rotation is that I use it to align repeating elements of the pattern...

Atypical Symmetry Pattern Most of the time when you see a tessellation with only hexagons and triangles, there's only one hexagon and only one triangle and they alternate back and forth. But that's not the only way! Using the Hexagons and Triangles 6-fold Islands symmetry gets you a choice of a second hexagon too. You can choose an open or closed central hexagon, a triangle of any type on any side, and then a hexagon of any type on any side too! I figured out that all of the options work by playing with my crease pattern building block templates,...

One Space Away Groovy Triangles and Emergent Hexagons are one space away, yet feel worlds apart. The dominant feature in backlighting of Groovy Triangles is a Star of David made with open triangle twists on alternating sides, while the dominant feature of Emergent Hexagons is a honeycomb structure of solid dark bars. This is something I've found with other closely related tessellations too. In equilateral triangle tiling tessellations with clusters of four twists, there are several spacing considerations: between clusters, within clusters, and global lines.  In Groovy Triangles, the spacing within clusters is as dense as possible, with spacing between...

What are skew lines? Skew lines are the lonliest pairs of lines - they don't intersect at a single point, and don't maintain a constant distance from each other like parallel lines. They just pass near each other then continue out to infinity. It's pretty rare for a tessellation to have pattern elements align this directly when the twists are on opposite sides of the paper - much less have lines in two different directions on the two sides! How is this possible? Skew Lines is made up of clusters of four closed square twists, which are then mirrored all...

Oh the thinks you can think! So, it turns out that we have more choices available to us than we might suspect with Hexagon and Triangle 6-fold tilings! Instead of just having one choice for the next hexagon out from the middle, we actually get four hexagons to choose from - closed or open on either the front or the back. And that's after we choose the central hexagon (2 choices) and the triangle twist (also 4 choices)! Taking all of these options into account, we have 2*4*4=32 options for arranging hexagon and triangle twists in this extended symmetry and...

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!

An Unusual Symmetry These hybrid hexagon twists (there's actually two kinds) are 3-fold rotationally symmetric. This is weird for something with 6 pleats - usually these kinds of twists have 6-fold rotational symmetry. But by putting the 3-fold six-pleated twist in the middle, I was able to push the remaining two 3-fold symmetries out to the next hexagons and use two different kinds of triangle twists. And so Hybrid Hex Weave uses three different kinds of hexagon twist and two different kinds of triangle twist while the simplest patterns in this tiling use only one of each type of twist. I've found...

It's easy to get sucked into the details of the exact placement of mountain and valley folds when you start looking at crease patterns for tessellations (and then think you need to precrease and collapse them!), but there's a simpler way to see these patterns.

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...

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...