Tiling-Based Origami Tessellation Design

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

There are several different styles within this niche craft, including corrugations, curved-crease tessellations, twist-based tessellations, flagstone tessellations, and pleat intersection-based tessellations.

Two folding methods are most popular: collapsing everything at once after pre-scoring all required fold lines or collapsing gradually, one piece of the pattern at a time.

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

I’ve designed over 500 patterns in the last five years, and there are infinitely many patterns waiting to be discovered by a folder like you.

This article will walk you through the framework I use to design patterns, which is also the most powerful mental tool I’ve found for folding tessellations one twist at a time.

What folds are used for these patterns?

Normal origami works by making one new fold at a time, but twists have lots of things going on at once.

Most of the folds are the pleats leaving the central shape, and these lines are already present in the paper through the background grid that’s folded in advance.

The outline of the central shape is rarely on grid lines, and so these folds are made when the twist is squashed flat.

First the pleats are set up and then the center is twisted around and flattened to create that central shape.

Twists can either be rotated right- or left-handedly, and each pleat will go between two twists with opposite rotation directions.

In my crease patterns, red solid lines show mountain folds and blue dashed lines show valley folds.

The twist above is a right-handed closed triangle twist, shown as a crease pattern and in the folded form, both front and back.

These twists can be on either side of the paper and most of my designs have some twists on one side and some twists on the other.

When you’re looking at a crease pattern like the one above, the twists where the central polygon is outlined in mountain folds are on the side you’re looking at, while the central polygons outlined in valley folds correspond to twists on the opposite side.

These crease patterns are usually shown on a background grid, so you can count the spaces between twists that are connected by a pleat.

Can you see that in this example there are hexagons on the front and triangles on the back?

Is there a simpler representation than a crease pattern?

Crease patterns can be overwhelming and most people think that they have to keep track of all the fold lines at once.

Thankfully, these patterns can be simplified to what twist shapes are present, which twist shapes are next to each other, and the specific identity of the twist for each shape.

In the crease pattern example above, each hexagon is connected to six triangles and each triangle is connected to three hexagons.

There is only one hexagon twist used in the whole pattern and also only one triangle twist used, and they always have the same spacing between them.

That matches with the pattern below, and you only need to make sure you’ve used the correct spacing once as you’re folding - for the rest of the twists you can check back with what you’ve already folded!

This representation of what shapes are used and how they’re connected is called a tiling, and there is one twist in the pattern for each tile (each shape) in the tiling.

Each tiling represents many different patterns, which can be found by making different choices for the contents of the tiles.

In the tiling of alternating hexagons and triangles above, you can pick any hexagon twist to fill the hexagon tile and any triangle twist on either side of the paper to fill the triangle tile.

There’s also multiple ways to make choices within a tiling and these ways correspond to different ways of coloring the tiling, as shown below.

Each color of tile represents a different choice, so you could have different contents for the dark blue and light blue hexagon tiles.

In fact, in order to preserve the symmetry of the coloring of the tiling you may be required to have distinct twists in different tiles.

The number of options here gets very large, very quickly, and you end up with infinitely many ways to generate patterns that themselves repeat infinitely.

What guarantees that the pattern is infinite?

All flat infinitely repeating patterns match one of the 17 wallpaper symmetry groups that have been known for hundreds or (in some cases) thousands of years.

The highest-symmetry cases repeat like a hexagon, like a square, or like a triangle, as shown below.

The different red shapes in each of these diagrams represent points of rotational symmetry - points where you can rotate the pattern at certain angles and see the same exact pattern again.

Each type of symmetry has a specific relationship between these points of symmetry and so when you find that relationship in a coloring of a tiling and in the crease pattern made by choices of twists matching that coloring of a tiling, you can be sure that your pattern will repeat infinitely.

That’s how I found the different colorings of the tiling in the first place - I picked different locations for the positions of symmetry and then followed the symmetry rules to fill in the rest of the colors!

Of the six colorings in the last section, three of them have hexagonal symmetry and three have triangular symmetry - can you see which ones are which?

How do you design new patterns?

First, I choose what tiling I want to work with.

I’ve worked with over 60 different tilings and each one has its own folding characteristics.

With some tilings you fold in rings around the center, while others fold best in stripes or in rings around the center that leave certain twists partially folded temporarily.

Then I check and see how many different choices are available to be made - how many different tiling positions there are and what their symmetric constraints are.

In the case of the tiling on the left (Hex-Tri-Rhomb-Tri), there is a choice of hexagon, a choice of triangle, and a choice of rhombus, and there will be no conflicts due to symmetry because there’s one symmetry point that’s between twists instead of inside twists, assuming we place the symmetry points as densely as possible.

In the case of the middle tiling (Split [Hexagons and Triangles 2-fold]), there are two distinct trapezoids (the colors show a distinction between left- and right-handed twists) and two distinct triangles, but I know from experience that these two choices per shape work well with copies of the same twist but on the opposite side of the paper.

We can further choose whether to use any of the possible mirror symmetry lines present in the tiling (the horizontal lines that go straight through), and use crease pattern template files to test which options work.

For the tiling on the right (Hexagon Double Rhombus Triangle), I see one hexagon tile, two distinct rhombus tiles, and one triangle tile.

All three positions of rotational symmetry are inside of a tile, so it’ll be easiest to choose the contents of the triangle and both rhombus tiles first before seeing what that gives us for the hexagon.

Just like with the middle tiling, I’d be playing with the twists in a template file to see what does and what does not connect.

Once you have your choices drawn with the spacing that you want, then it’s time to fold!

Here’s some examples of what you can fold from each of these tilings:

There’s tons of patterns out there just waiting to be discovered - maybe some of them will be discovered and folded for the first time by you!

The thrill of discovery is still keeping me searching for more and more patterns - even after discovering over 500 new designs.

Will you join me on the journey?

Each month in Tessellation Academy I focus on a specific topic in origami tessellation design such as alternating square twists, arrangements (colorings of the tiling) of equilateral triangle twists, or how to use only open hexagon and closed triangle twists in many different arrangements.

I release a beginner and an advanced folding tutorial video during the first two weeks of the month, then we have a live Zoom call to discuss the theory behind the topic and design a new pattern using that theory.

You can join at any time, with monthly or annual plans available, at training.gatheringfolds.com/academy and I hope to see you at our next live session!