Tessellations: A Brief Theory of Warping Paper

Tessellations are the new trend in the origami world. Definitions, which, by definition, try to draw definite border lines, can only do injustice to this field. There are Corrugations, Molecules, Curved Tessellations, and so many other subcategories. Here is my humble addition to this field.

First, let's define the types of tessellations. I am not familiar with any formal definition, even Wikipedia leaves this term unexplored.

Types of Tessellations

I like to divide the tessellation world into 4 categories:

Classic
Corrugation
Recursive
Back & Forth AKA Organic tessellation

Classic

"Red Flower" Tessellation designed and folded by Ilan Garibi, Tant

The Classic is, well, classic. Since Fujimoto, a Japanese origami master who published books that included origami tessellations in the 1960s till nowadays, this is the most common type. It's based on two major grid types - the Hexagon and the Square. It is made of Molecules that can be spread in all four (or six) directions, covering a continuous surface. By nature, the surface of the final model will have an odd number of layers (one, three, five, or even more) throughout. This is because whenever the paper is first folded this way, it must be folded the other way, too, to allow continuity. This change of number of layers gives the most amazing effect when you backlight your model. One layer is a bit transparent while three layers and more are dark. For that reason one tessellation can give you four models: first side, the other side, and both sides backlit.

Corrugation

"Diamond Corrugation" Tessellation designed and folded by Ilan Garibi, 40gsm Kraft paper

Borrowed from the English dictionary, a corrugation is: a wrinkle; fold; furrow; ridge. This type of tessellation has no triple or more layers. The entire original surface of the paper is visible to the eye, and the pattern is usually in the form of waves. There is no point in using backlighting on this type.

Recursive

"Hydrangea" designed by Shuzo Fujimoto, folded by Sara Adams, Dreamy paper

Recursive tessellations are unique. The concept here is to make the same fold in a smaller scale on one square in a repetitive manner. A classic recursive model is the Hydrangea by Fujimoto.

Back & Forth, or Organic tessellation

"Wave" Tessellation designed and folded by Ilan Garibi, Elephant Hide

Last in my list is the group of folds that are simply done this way: one row folded back and two rows are folded forth. Repeat.

Taking it Further

Of course, you can combine several types of tessellations in one project. For example, you may want to use a classic tessellation to form a border for a recursive central piece (left), or use a recursive tessellation as a molecule for a classic tessellation (right).

"Windows" Tessellation designed and folded by Ilan Garibi, Elephant Hide

High density tiling of the "Hydrangea" by Shuzo Fujimoto, folded by Sara Adams, Rainbow Kraft paper

More examples

This video shows further classic tessellations:

Creating a Tessellation

Tessellations are usually folded in three steps only: Grid, Precrease (all other folds than the grid) and collapse. Each step has some basic know-how and many tips, but before elaborating on that, we have to discuss the single molecule - the core of the model.

When I try to create a new classic tessellation, I follow the Way of the Molecule. A molecule is one unit of the tessellation: the repetitive part. I try to find an interesting molecule that follows two very simple rules:

Can I make a fold that has all of its sides (or, at least, every two opposite sides) to be totally the same?
Can I make sure all the original edges of the paper will stay on the edges of the folded molecule?

If this happens, you can tessellate this fold. To understand this concept let's take a look at the simplest fold - an unfolded square paper. All the original edges of the paper are on the edge of this molecule, and all four sides are identical. We can spread this molecule to all 4 directions, endlessly.

Moving to real molecules, let's start with only two square molecules. A trick for checking whether they can be tessellated is to glue-tape the edges together. If you can spread the folded molecules to their original flat state, align them and put a cellotape to hold them together, and then collapse the combined units, why cut the paper in the first place? Now start with a 1×2 rectangle of a paper, crease all needed creases on both squares, and collapse without the need for a cellotape.

For a better example, take the basic twist fold (the base of Kawasaki Rose). The pattern on all four sides is the same; hence you can put two molecules one by the other, unfold the common edges and tape them together. Now collapse both of them to get your first 1×2 molecules tessellation. Again, if we can tape two, why can't we start with two connected squares, in other words a rectangle of 1×2 proportions? Do all folds on both squares, and collapse.

Note that the twist direction had to be changed on the second molecule, to allow continuity.

These are simple rules, yet they are probably sufficient to make all classic tessellations.

Creation

To create a new molecule (for a Classic or a Recursive Tess) all that's needed is to follow two pieces of advice: Dare and Play. Dare to doodle with paper, grid it, mold it, and then try again. And again. It works for me.

A tip for this section - having a molecule that applies to the rules may not be enough. There is a third request - it must be collapsible. To make sure it is so, first try to collapse your molecule on a 2×2 tessellation, and then on a 3×3, which have the center molecule bordered by other molecule from all 4 sides. If those two tessellations can be collapsed, you can go to larger numbers.

Originality

Do not feel bad if you came up with this marvelous model, just to get a remark on your Flickr page, saying: "Oh, I did that a year ago, see it here...". This field of creation is very mathematical and many people discovered the same model independently. Enjoy your power of creation, even if you are not the first!

The Grid

Tessellations are usually made from grids. Divide the paper to 32 equal parts on both directions to get a 32×32 Grid. 16 grids may be good for beginners, and 64 grids will bring an amazing effect and finger aches. Common grids are the Square and the Hexagon (which includes the Triangle grid, since we all fold our hexagon by making six triangles). More rare ones are the Rectangle, usually 1×2 proportion, and the Diamond:

Making a grid may seem trivial, but the truth is far from that. My first tip is always doing your creases bi-directional. It is a double effort, but it is often very helpful in the collapse phase.

Now, to get a 2×2 grid, you need one horizontal crease and one perpendicular to it. That's easy to do - just divide into half in both directions. To get a 4×4 - divide all halves into quarters. Still easy - this is a cupboard fold, actually. Going to 8×8 starts to be problematic - getting the first ¹⁄₈ line is easy - fold the edge to the ¹⁄₄ line. But how do you get the ³⁄₈ line?

One solution immediately pops up - fold the edge to the ³⁄₄ line, which divides this length to ³⁄₈. This may be good for a 8×8 grid, but not that good for a 16×16, lousy for a 32×32, terrible for a 64×64 and disastrous for a 128×128 grid. The reason lies in the fact that every crease shrinks the paper a bit. When you fold the edge to the ⁴⁶⁄₆₄ line to get the ²³⁄₆₄ line, the distance of the 46th line is not ⁴⁶⁄₆₄ of the paper, but shorter than that. See this full sheet of Elephant Hide after folding a 128×128 grid on it. It's aligned on the bottom right corner with a full, virgin sheet (70×100cm):

It may seem negligible, 2.6% of shrinkage, but believe me - it does make a major change - squares are no longer squares, and a diagonal creases does not fall on the corners.

The right diagram shows how it should be done: in order to get the ³⁄₈, pinch the ¹⁄₄ line and fold it upon the ¹⁄₂ line. To get the ²³⁄₆₄ line, pinch the 22nd line upon the 24th line. And reverse that fold, so you will finish with all your horizontal creases as mountain folds.

The next tip regards the rotation of the paper. Since it is quite difficult to make a perpendicular crease against 64 mountain fold lines, do not crease 64 horizontal creases and then rotate the paper to complete the perpendicular ones. Instead divide a side in half, and rotate. Divide the new side into quarters, and rotate. Back to the first side, divided it into 8ths, and rotate. Divide into 16ths, and rotate. You get it, I hope.

The same logic applies to the Hexagon grid.

Some grids demand the diagonals to be folded, too. This brings up another tip - for grids up to 8×8, you can use the "fold the corner of a square to the opposite one" to fold line AB in one movement of the hand. For denser grids a more careful approach is needed. You can use the corner as a general guide, but when you mark the fold, refer to each single square as one - fold from the corner of it to the opposite one, and move to the next square. This is much tedious way, but far more accurate!

Precreasing

Unlike traditional origami, tessellations are mostly based on precreases. Some have a step by step way to get there, but for most you have to prepare all the folds in advance, and then collapse the model.

For most tessellations, this is the most time consuming phase. I think the most basic molecule needs at least 4 creases, and usually much more. Multiply your molecule number by that and understand how many short creases in all 4 directions you should fold. Luckily, there are short cuts for that. Many creases will align and can be folded in one stroke.

Notice that molecules 1, 2 and 3 in the image all contribute to one straight line (in red): the top-left fold line of molecule 2 (AB); the bottom-right line of molecule 1 (BC); and again the top-left fold line of molecule 3 (CD). So folding in one movement line AD gives you a major short cut.

Finding a shortcut is a bit harder with the inner squares that do not form a straight combined line. Here we move to another method: Counting. All 3 blue lines are aligned. Now start with the blue line of Molecule 1, and count 3 squares (marked in green) to get to the next square to which a diagonal fold needs to be added (in blue again). So this is the rhythm for that CP: crease 1, count 3, and repeat.

There is a cosmic law, saying that in every 32×32 grid and up tessellation you fold, you will make a mistake: dislocating a crease, jumping over a fold line in the grid, whatever. Do not feel bad. It's a cosmic law.

Collapse

Can't do it! Can't do it! Oh, maybe I can. (Deep breath). Oh, yes I can!

Yep, collapse is the stage that endangers your sanity. "It can't be done" comes to your mind again and again, and the cat jumps whenever a frustration cry is sounded. But, it really can be done. OK, enough with the psychology issues.

Collapsing a model can be done from center to edge, or from the corner, row by row. The rule is simple - with square grids I start at the corner, completing one molecule, folding the rest of the paper all the way to the edges, as if this is the only molecule to be folded. Then I add another one and so on, till the end of the row. The first row is simple. The second row starts easily, but the second molecule is surrounded from two sides with folded molecules and the other two sides are flat paper. This is where the difficulties lay. If you can fold that molecule, all the rest are just the same. For some tessellations, it is good to partially collapse all molecules in a row, before collapsing them one by one. This saves a bit of work in unfolding creases that will be in your way for the next row.

For a Hexagon tessellation, I always start from the center, (unless the center has no molecule, and then you start with the inner circle of 6 molecules). I do not really know why. This is how it is. Then you make the first circle around, with 6 molecules, when the last (the 6th in the first circle) can give you some troubles. This is mostly because you need to unfold some pleats you made to allow yourself to work on the paper. When opening the paper, you may have to be careful not to unfold the adjoining molecules - but it is necessary to complete that final molecule. If this proves too difficult, do try to fold the 5th and the 6th together instead!

The only tip I can give here is "do not give up". The folder is stronger than the paper.

When folding a tessellation, I sometimes found out that there is a rhythm, a repetitive method to do things correctly - write it down the moment you realize that sequence. It will help you tremendously the next time you try to fold it.

Tools

Paper

Elephant Hide.: Read Gadi's and my review to understand why: http://www.origamiusa.org/thefold001_paper
Tant: This is your second choice. The review was published earlier this issue.
Glassine: This is great for light effects.