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 fold 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.
