The satisfying math of folding origami - Evan Zodl

As the space telescope prepares to snap a photo,

the light of the nearby star blocks its view.

But the telescope has a trick up its sleeve:

a massive shield to block the glare.

This starshade has a diameter of about 35 meters—

that folds down to just under 2.5 meters,

small enough to carry on the end of a rocket.

Its compact design is based on an ancient art form.

Origami, which literally translates to “folding paper,”

is a Japanese practice dating back to at least the 17th century.

In origami, the same simple concepts

yield everything from a paper crane with about 20 steps,

to this dragon with over 1,000 steps, to a starshade.

A single, traditionally square sheet of paper

can be transformed into almost any shape, purely by folding.

Unfold that sheet, and there’s a pattern of lines,

each of which represents a concave valley fold or a convex mountain fold.

Origami artists arrange these folds to create crease patterns,

which serve as blueprints for their designs.

Though most origami models are three dimensional,

their crease patterns are usually designed to fold flat

without introducing any new creases or cutting the paper.

The mathematical rules behind flat-foldable crease patterns

are much simpler than those behind 3D crease patterns—

it’s easier to create an abstract 2D design and then shape it into a 3D form.

There are four rules that any flat-foldable crease pattern must obey.

First, the crease pattern must be two-colorable—

meaning the areas between creases can be filled with two colors

so that areas of the same color never touch.

Add another crease here,

and the crease pattern no longer displays two-colorability.

Second, the number of mountain and valley folds

at any interior vertex must differ by exactly two—

like the three valley folds and one mountain fold that meet here.

Here’s a closer look at what happens when we make the folds at this vertex.

If we add a mountain fold at this vertex, there are three valleys and two mountains.

If it’s a valley, there are four valleys and one mountain.

Either way, the model doesn't fall flat.

The third rule is that if we number all the angles

at an interior vertex moving clockwise or counterclockwise,

the even-numbered angles must add up to 180 degrees,

as must the odd-numbered angles.

Looking closer at the folds, we can see why.

If we add a crease and number the new angles at this vertex,

the even and odd angles no longer add up to 180 degrees,

and the model doesn’t fold flat.

Finally, a layer cannot penetrate a fold.

A 2D, flat-foldable base is often an abstract representation

of a final 3D shape.

Understanding the relationship between crease patterns, 2D bases,

and the final 3D form allows origami artists

to design incredibly complex shapes.

Take this crease pattern by origami artist Robert J. Lang.

The crease pattern allocates areas for a creature's legs,

tail, and other appendages.

When we fold the crease pattern into this flat base,

each of these allocated areas becomes a separate flap.

By narrowing, bending, and sculpting these flaps,

the 2D base becomes a 3D scorpion.

Now, what if we wanted to fold 7 of these flowers from the same sheet of paper?

If we can duplicate the flower’s crease pattern

and connect each of them in such a way that all four laws are satisfied,

we can create a tessellation, or a repeating pattern of shapes

that covers a plane without any gaps or overlaps.

The ability to fold a large surface into a compact shape

has applications from the vastness of space

to the microscopic world of our cells.

Using principles of origami,

medical engineers have re-imagined the traditional stent graft,

a tube used to open and support damaged blood vessels.

Through tessellation, the rigid tubular structure folds into a compact sheet

about half its expanded size.

Origami principles have been used in airbags, solar arrays,

self-folding robots, and even DNA nanostructures—

who knows what possibilities will unfold next.