The Mathematics, Laws and Theory Behind Crease Patterns

Did you know that paper folding developed in China, Korea, and Europe (in addition to Japan)?

Did you also know that the art of paper folding, most commonly associated with Japan, may have in fact been invented in China?

Cai Lun, a Chinese eunich and official of the Han Dynasty created paper. It was around this time that paper folding as an art form (or ‘zhe zhi,’ “folding paper” in Chinese) was created as well.

Paper folding played an integral role in Chinese ceremonial purposes, especially funerals. Golden nuggets, or yuan bao, were created from paper and then were used as a sacred ritual, lit during funerals.

Eventually, the Chinese may have introduced this art form to the Japanese.

In fact, it was Japanese Buddhist monks that further shared this art form with the Japanese people.

Originally known as “oritaka” (folded shapes in Japanese), paper folding was used in Shinto ceremonies and, because of the high cost of paper, was reserved for the elite.

In 1880 oritaka became origami, “fold paper” in Japanese—”ori” meaning “fold,” and “kami” meaning “paper.”

During the 20th century the “grandmaster of origami,” Akira Yoshizawa, helped make origami known to the world.

This, and the affordability of paper, made this art form accessible to the masses and helped propel origami forward.

Now that this art form is celebrated and shared internationally, we want to help you become a part of the origami world, and teach you how to read and fold origami crease patterns.

 

Origami is both a form of art and mathematics

In learning how to read and fold origami crease patterns, it’s important to first understand the mathematical principals pertaining to origami.

Think of it this way: You wouldn’t start reading sheet music if you didn’t know the theory fundamentals such as how many counts a quarter note gets, how many beats are in a meter, and what the time signature means.

Math and the mathematic laws governing paper folding are a large part of origami’s fundamentals.

Origami is both art and math, as it’s a pattern of creases.

Specifically in a TED talk, Robert Lang states “They [origami] have to obey four simple laws… The first law is two-colorability. You can color any crease pattern with just two colors without ever having the same color meeting.”

He goes on to discuss the second law, “The directions of the folds at any vertex —the number of mountain folds, the number of valley folds — always differs by two.”

This is essentially Maekawa’s Theorem.

Ok, let’s explain what Lang (and Maekawa’s Theorem) is saying.

A mountain fold (or mountain crease) is what it sounds like — a fold where the two ends of paper go down and the fold is pointed upwards. It looks like a mountain.

A valley fold (or valley crease) is the opposite. The fold is at the bottom, and the ends of paper are facing upwards, imitating a valley.

And, in case you don’t know, a vertex is pretty much a corner.

Or, if you want to get all fancy and impress a bunch of people, it’s “a point in a geometrical solid common to three or more sides.”

Basically, where two sides intersect.

So, in origami the vertex is where the mountain and valley folds meet.

And, if you subtract the mountain folds from the valley folds (or vice versa), the absolute difference would be two. (Remember to leave out negative numbers.)

For example, 5 mountain creases and 3 valley creases would be accurate, since 5-3=2.

However, 6 mountain creases and 2 valley creases wouldn’t work because 6-2=4.

This rule is essential when learning how to read and fold crease patterns, as it can save you a lot of time trying to fold 6 mountain creases and 5 valley creases since that’s an impossible feat.

 

The two other rules Lang talks about

The third law of origami is that no matter how many times you try to stack folds and sheets, a sheet can never penetrate a fold.

(This rule will prove to be essential when learning how to read and fold crease patterns.)

This goes with the Miura map fold.

Japanese Astrophysicist, Koryo Miura created this origami crease pattern where a single shape is repeated over and over…and over again.

For the map fold, the pattern is a parallelogram. There are no gaps, and no sheets striking through the folds.

So, if you come across any crease patterns that require this, know that they’re incorrect, and you may want to move on to another pattern.

And lastly, the fourth law states that every other angle around the vertices comes out to 180 degrees.

This aligns with the Kawasaki Theorem, which states that in a flat figure, every other angle adds up to 180 degrees.

So, when you fold your paper and it looks off, get out a protractor and measure those angles.

Remember, origami is also an art of geometry and precision.

 

Which brings us to reading origami crease patterns

So, let’s put the math into action.

How do we read origami crease patterns?

Simple. The blueprint is right in front of you.

Take an origami crane.

Unfold it.

The creases are the blueprint.

You can follow the creases, refolding the paper back into the crane.

That’s great if you already have an origami figure. But, what about wanting to create a figure from uncreased paper—how do you read a pattern then?

Robert Lang points out that, “Conventional origami diagrams describe a figure by a folding sequence — a linear step-by-step pattern of progression.”

He continues, “Crease patterns, by contrast, provide a one-step connection from the unfolded square to the folded form, compressing hundreds of creases, and sometimes hours of folding into a single diagram!”

So, there’s no simple, how-to step-by-step instructions when it comes to crease patterns.

Instead, there’s a single design, filled with different lines that represent mountain and valley creases.

Lang states that, as of now, there isn’t a set style or key for mountain and valley folds.

The step-by-step pattern uses a dash-dot-dot line to signify a mountain fold.

And a dashed line to illustrate a valley fold.

Crease patterns may follow these types of lines.

Some may include two different colours to differentiate mountain from valley creases.

Lang, himself, uses straight, dark lines for mountain folds and coloured, dash lines for valley folds.

Also, when reading an origami crease pattern, you’ll realize that the creator will not have inserted every mountain and valley fold.

Since many crease patterns, have hundreds of folds, showing every fold would make the design look busy.

And could overwhelm the reader.

Instead, the base of the design is created.

The origami maker can then follow this base pattern to create the rest of the folds for the design.

 

How to fold origami crease patterns

We mentioned this in the beginning of this article, but it’s worth repeating.

Origami crease patterns use mountain and valley folds.

The vertex of a mountain fold is pointing up.

And the vertex is pointing down with a valley fold.

Simply fold the paper in half to replicate this.

And remember the mathematical principals governing origami.

• If your creation looks uneven or off, go through the math rules to see what the mistake is and fix it.
• Unfold your paper figure, take out two crayons and colour the pattern. If the two colours meet, you know where you need to go back and fix it (two-colorability law).
• Count how many mountain and valley creases there are at the intersection (vertex). If the difference isn’t two, you know you have to adjust the folds. (Maekawa’s Theorem).
• Does the pattern call for a sheet to penetrate a fold? If so, this is an incorrect pattern (third law of origami).
• And if the angles don’t line up, get out your protractor and start measuring (Kawasaki Theorem).

Origami has changed the ways we’ve thought about art, math, and science. Thanks to origami, we now have inventions such as airbags and heart stents.

How has origami impacted your life? Has this helped?

Let us know by leaving a comment.

And be sure to check out our free origami instructions and diagrams, so you can practice.