Basic Twists for Origami Tessellations

Origami tessellations are repeating patterns folded from a single sheet of paper, but there’s not just one style or level of difficulty to this kind of origami.

There’s patterns with curved folds, patterns with 3D elements, patterns that need to be fully precreased before collapsing, and patterns that are folded one twist at a time.

The easiest way to get started folding tessellations is with grid-based tessellations using pleat intersections (like Spread Hexagons in Origami Tessellations by Eric Gjerde) and then practicing the basic twists to start folding tessellations one twist at a time.

The freedom in starting from a grid and folding one twist at a time is that any one of hundreds (or thousands) of patterns can be folded from that grid.

In the beginning this freedom is daunting and I encourage my students to draw the crease pattern (CP) on their grid and even precrease as needed.

Once you’ve mastered the basic twists, however, this freedom allows you to make different choices of spacing and twist to create variations on the fly.

So, what are these basic twists?

When Fujimoto first folded twist-based origami tessellations he used only the closed forms of the triangle, square, and hexagon twists.

By the time I started folding tessellations in 2017, open forms of all of these twist shapes were commonly used as well as rhombus and right triangle twists.

You would see more advanced twists such as stacked twists, isoarea twists, trapezoid twists, and mixed-depth rhombus twists used in Joel Cooper’s masks and the most advanced flagstone designs, but the bulk of the photos I used for reverse engineering used only those eight most basic twists.

Closed and open (equilateral) triangle twists

  

 

 

Using only these eight twists, hundreds if not thousands of origami tessellation designs are available to be folded.

Many of these are quite simple, using only left- and right-handed forms of the same twist or alternating between two different twists.

  

You can continue using these familiar twists in more advanced tessellations too if you increase the structural complexity.

The structure of an origami tessellation refers to what twist shapes are used and how these shapes are connected by pleats.

These structures are usually represented by a tiling, where any two shapes in the tiling that share an edge will have a pleat connecting them in the origami tessellation.

There’s two ways to increase this kind of complexity in an origami tessellation: increase the complexity of the tiling (from a 1-uniform to a 2- or 3-uniform tiling) or change how the symmetry points are arranged on a simple tiling (by creating different colorings of the tiling).

Neither of these options have theoretical limits on the number of designs that are possible.

The only real limit is how much time you’re willing to spend gridding and how big your paper is.

In fact, I’ve found over 40 tessellation designs that use only open hexagon and closed triangle twists - and that search was limited to patterns that fit three repeats inside of 80 grid divisions.

   

There are plenty of other twists too, and no theoretical limit on the number of twists either.

Sometimes I use more advanced twists to resolve symmetry constraints in a design or to create pattern elements that are impossible with the basic twists.

They’re good tools to be aware of, but not strictly necessary to learn to fold origami tessellations.

I’d recommend learning twists one at a time and practicing that one twist until you’ve internalized how it works.

This is my approach to teaching tessellation folding in Basic Twists Bootcamp and Tessellation Starter Sequence, my paid courses on starting to fold tessellations and learning to read crease patterns.

My free Tessellation Foundations resource introduces my top gridding techniques alongside basic skills like reordering pleat overlaps and it’s a good way to learn your first two twists - the closed square and closed triangle twists.