# Welcome to the Fold

A natural question in the context of origami mathematics is: What if we make the paper infinitely large?

# Welcome to the Fold

Sara Chari
Saint Mary's College of Maryland

Bates College and the National Science Foundation

Origami—from the Japanese words for “fold” (oru) and “paper” (kami)—is the art of folding paper to make certain shapes, with the simple constraints that one is not allowed to cut or glue any part of the paper. Advances in origami design have yielded some amazing artistic, and even scientific results!

If you’ve ever tried your hand at origami, you will know that once you make a particular shape, you can unfold the paper to see all the creases (lines on the paper, which may or may not intersect). Crease patterns also tell you how you should fold paper to obtain certain shapes. Here are some images (made by a former Bates College student, Wuyue Zhou), of a paper crane with a few initial steps.

Wuyue Zhou

And just for fun, here is an origami seashell folded by Manda Riehl and its corresponding crease pattern:

Images and folds by Manda Riehl

Erik Demaine is one of the leading experts in so called “mathematical” origami, which he describes on his website Folding and Unfolding as coming in two flavors:

“Origami mathematics is a recent branch of mathematics (whose major study started circa 1980) that studies the properties of origami, such as what patterns you might get when you unfold a flat origami. Computational origami is an even more recent branch of computer science that explores algorithms for creating origami or solving paper-folding problems. This field began in North America with Robert Lang's work on TreeMaker (circa 1993).

A natural question in the context of origami mathematics is: What if we make the paper infinitely large? This leads to a variety of designs left by the folds and, more importantly, the points at which the folds intersect.

Joe Buhler, Steve Butler, Warwick De Launay, and Ron Graham, inspired by a question of Demaine’s, first came up with the idea of an “origami construction” to describe these ideas more formally, in their 2012 paper Origami Rings. In this column, we will describe the main ideas behind these constructions, and some of the natural questions that arise.

# Origami Constructions

Just like with traditional origami, we set up a few constraints for our mathematical origami: we are allowed two points on the plane to start, typically 0 and 1. These are called "seed points" and they will “grow” outward into our origami construction. We are also allowed a finite set of angles or directions at which we are allowed to "fold" the plane (read: draw a line). Each pair of folds intersects in a single point unless the corresponding lines are parallel. When a new point is created this way, it gets added to our collection of points, along with the seed points.

So, if you have the line with angle $\alpha$ going through the point $p$, and the line with angle $\beta$ going through the point $q$, their intersection point (below) gets added to our set of points.

Folds must be made only through two points that have already been added to the collection, so at first the options are limited. As the process continues, there are more and more points and therefore, more and more lines that may be drawn. Below is a sequence, starting with the seed points and the angles 0, $\pi/4$ and $\pi/2$.

This process continues infinitely, so it may feel strange to think about an end result, but it is often possible to describe the final set of points, and sometimes they even form a nice pattern (more on that later). From here on out, we will refer to this set of points as the origami set.

After many iterations of the origami process begun above, a pattern is obvious.

Once you start playing with the origami point construction algorithm, you might ask yourself many questions. For example, when do we get a nice “visual” pattern (as in the example above)? Do we get other “nice” properties? Since the plane can also be thought of as the complex numbers, you could also ask: when is this subset also a subring? Given a subring, is there a corresponding origami construction? What happens if we try this in higher dimensions?

# Lattices or not?

Here are some examples of the first few iterations of origami constructions using different numbers of angles (made by some summer research students of Adriana’s at Bates). What do you notice?

It seems like the only “nice” example (at least visually) is the first one. In fact, Buhler, Butler, de Launay and Graham already showed that when there are 4 or more allowed angles, the origami set will be dense in the complex plane, meaning that for any point in the whole plane you wish to pick, you can find a point in the origami set that is as close to your original point as you would like. Another way to view this is for any circle you draw in the complex plane, there will be infinitely many points from your origami set within the circle. That’s a lot of points to keep track of! (Notice the “zoomed in” picture above.)

Our first, “nice” example is actually a lattice. This is a special kind of discrete infinite subset of $\mathbb{R}^2$—or, more generally, $\mathbb{R}^n$. (If you want to dive deeper into the special properties of lattices, there have been some great recent Feature Columns on the subject.)

But do you always get a lattice when you use three angles? The answer is yes, as long as one of your angles is 0, and this was proved by Dmitri Nedrenco in 2015.

# Origami Rings

Fun fact: when you Google “origami ring” what you find are things like this, which are probably super fun to make (we haven’t tried) but not the kind of “ring” we are talking about.

Rose rings designed by Hyo Ahn and folded by Origami Sophy. (CC BY-SA 2.0)

Which origami constructions lead to subrings of the complex numbers? This was in fact the central question in the Buhler et al. paper, and the answer is pretty cool—if your angle set forms a subgroup of the unit circle, then your origami set is a ring. Algebraic structure yields algebraic structure!

A question they posed in the same paper was essentially the inverse—which rings can be “origami constructed”? Juergen Kritschgau, in his 2015 senior Bates thesis, proved that it is possible to obtain the ring of integers of an imaginary quadratic field through an origami construction (this was published in the INTEGERS journal in 2017).

Finally, Florian Moller definitively answered this question in 2018: he gives a set of criteria that determines precisely when the construction will be a ring.

Note: an origami construction can be both a lattice and a ring (like the very first example we showed, which ends up being the Gaussian integers), but these two “nice” features are independent.

# Higher Dimensions

Since we live in a three dimensional world, one may wish to do mathematical origami in higher dimensions. The rules are the same except that the allowed angles need not be contained in the complex plane. In the image shown, the standard basis elements are called 1, i, and j. To the reader who wishes to think about multiplication, these belong to the 4-dimensional Hamilton quaternion algebra. Sadly, all four dimensions are needed in order to have a closed multiplication structure.

One major difference in higher dimensions is that the intersection of two lines is not guaranteed anymore if the lines are skew. Luckily, this does not pose any problems; we simply do not record an intersection point if it does not exist. This does, however, allow for the inclusion of an arbitrary number of "irrelevant" allowed angles that either do not intersect anything at all besides the seed points and other points already in the set. There is no limit to the number of angles to obtain a lattice, but there is a minimal number of angles that guarantee a lattice: 2n+1, where n is the dimension of the space. If carefully chosen, just one additional angle can force a lattice to become dense which is quite remarkable!

Even though there is not a nice multiplication structure in 3 dimensions, 3-dimensional lattices are of interest to chemists alike crystallography. In particular, Bravais lattices have been classified and one may wish to study which of them can be obtained via mathematical origami.

And what about even higher dimensions? We thought about this question, together with then-Bates student Deveena Banerjee, and found many other cool results. For more, see our paper

# What’s next?

There really are so many options, since these ideas are so new. What we’ve described here is also not the only way to “mathematize” origami! For example, if you extract certain geometric “axioms” from origami, you can trisect an angle! We have also not discussed the area of computational origami, and all the other scientific applications of origami.

Let us end with an invitation: what is your favorite origami design? What does its crease pattern look like?

Image and folds by Jill M. Bean.