Edited by Jane Rosemarin

Editor’s Note: Scroll down for information on Madonna’s new free online course, Summer of Twists.

## The Anatomy of 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 are 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 is with grid-based tessellations using intersections of pleats (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 it’s possible to fold thousands of different patterns 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 precrease as needed. Once you’ve mastered the basic twists, you can make different choices of spacing and twist to create variations on the fly.

So, what are these basic twists? When Shuzo 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 were rhombus and right triangle twists. (See the closed and open twists in the photos below.) There are more advanced twists, such as stacked twists, iso-area twists, trapezoid twists and mixed-depth rhombus twists in Joel Cooper’s masks and the most advanced flagstone designs, but the bulk of the photos I used for reverse engineering were made of only the eight most basic twists.

Using these eight twists, hundreds — if not thousands — of origami tessellations can be folded. Many 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 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. There are two ways to add complexity: by increasing the number of vertices of the tiling (from a 1-uniform to a 2- or 3-uniform tiling) or by changing 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.

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. These twists are 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.

## A New Video Series

I’m excited to announce a new free challenge: Summer of Twists. Each of the 30 video tutorials in Summer of Twists will teach you a different twist, including all eight basic twists plus 22 more advanced options. Whether you fold all 30 or stick to the eight most basic, you’re sure to learn something new about folding origami tessellations. Each tutorial is paired with a longer video tutorial that teaches how to use the twist in the context of a tessellation pattern, so you could come out of the challenge with 30 finished pieces in addition to your 30 twist demonstrations. The crease patterns for these longer tutorials are available at The Source in the “Summer of Twists” ebook, which also includes a bonus crease pattern for each twist so you can test your new folding skills.

The Summer of twists tutorials are being released on my YouTube channel. They began on June 3 on a Monday-Wednesday-Friday schedule and will continue until August 9. You you can sign up for email reminders to make sure you don’t miss a single one.

Learning to fold twists doesn’t just make you a better tessellation folder, and it’s not just for folders who are already completing complex models from diagrams. My students often remark that they’ve seen all of their origami skills improve after studying tessellations — from predicting what the paper will look like after a fold is made to reading complex diagrams to folding precisely. Folding tessellations has improved it all. They also tell me that when something stressful is happening in their lives they just want to sit and grid, keeping their minds and hands occupied without trying to figure out something new. Whatever your motivation for folding tessellations, I’d advise starting with grids, pleats, and basic twists, folded one at a time. There’s a whole world of twists and tessellations out there — will you explore it with me?