Part 1 of 3 Series: What Paper Knows
Part one of three
Chill and Quill Books & Making
What a Thousand-Page Math Book Taught Me About My Quilling Strips
Or: the surprisingly moving experience of reading academic proofs and recognizing your own hands in them
I did not expect to pick up a dense computer science textbook called Geometric Folding Algorithms and feel, somewhere around chapter three, like someone had quietly written a love letter to every paper crafter who ever lived.
Let me explain. And let me start somewhere you might not expect: eighteen-year-old me, standing at a crossroads I didn't fully understand yet.
The choice I made, and what it cost me
When I was choosing my undergraduate major, I wanted to study art. I loved it, I was good at it, and it was where my mind naturally went when I had free time. But I chose mathematics instead, because I thought it was the practical choice. Art felt like a luxury. Math felt like a future.
Here's the irony I live with now: I have a PhD in educational policy, and one of the biggest debates happening at the national level in that field right now is whether the cost of splitting students into "practical" and "creative" tracks is higher than anyone admitted at the time. I chose math because I was told it was the sensible thing to do. I have spent the years since accumulating evidence that the framing was wrong.
What it misses, completely and consequentially, is that math and art are not two separate things competing for the same students. I would argue they are so deeply intertwined that they cannot fully exist without each other. The history of both fields is full of moments where one advanced only because someone crossed the line the other way.
Consider:
The line we drew between art and math, between practical and impractical, between useful and beautiful, was always a fiction. This book just happens to prove it with theorems.
The book is by Erik Demaine and Joseph O'Rourke, two mathematicians who study what paper can and cannot do. Linkages. Origami. Polyhedra. Hundreds of theorems about folding and unfolding. It is not, on its surface, a crafting book. There is not a single quilling strip anywhere in its 486 pages.
And yet.
We are both asking the same first question
Before I start any quilling piece, I hold the idea in my mind and ask: can I actually make this? Can a tight coil become a convincing rose petal? Can this particular negative space be held by surrounding scrolls? Can a strip of paper become a hummingbird wing?
This is, word for word, the central question of the entire book. The authors call it a "foldability question." Given this shape, this crease pattern, this polygon, can it be folded into what you're imagining? Sometimes the answer is yes, and they give you the algorithm. Sometimes the answer is "we don't know yet," and they box it up as an open problem. Sometimes the answer is a surprising, counterintuitive no.
What struck me is that I've been doing this my whole quilling life, reaching for an intuition that mathematicians have been formalizing since the 1990s. We are working on the same problem from different ends of the same strip of paper. My math degree and my quilling practice were never in different rooms. I just wasn't allowed to see it that way.
"Foldability questions ask: can this type of object fold to some general class of folded states?" The authors spend 400 pages proving that the answer is deeply, surprisingly complicated.
Curved creases are quilling's whole entire thing
Chapter 20 covers "rigid origami and curved creases," and I had to put the book down for a moment. It describes how paper naturally wants to form curves when you fold along a curved line rather than a straight one, producing sculptural, three-dimensional forms that straight-line origami simply cannot achieve.
Quilling is this. Every quilling strip I roll takes on a curve because of the paper's grain and the pressure of my fingers. The teardrop, the marquise, the loose scroll: these are all expressions of what paper does when you encourage it to follow a curve instead of forcing a crease. Chapter 20 gives me the mathematical vocabulary for something I've been doing by feel for years.
The book also mentions artist David Huffman, who spent decades making sculptural pieces using curved folds, treating the mathematics of paper as an artistic medium. He didn't publish most of his work. He just made things, quietly, because he found it beautiful. He was doing what I wanted to do at eighteen: holding math and art in both hands at once, refusing to choose.
The TreeMaker algorithm is basically how I plan a piece
Chapter 16 describes a piece of software called TreeMaker, built by origami master Robert Lang. To use it, you sketch the "tree" of the figure you want: here is the head, here are the antlers, here are the legs, here is how long each part needs to be. The algorithm computes the exact crease pattern to produce all those parts from one square sheet of paper.
When I plan a complex quilling piece, I do something similar. I sketch the composition, mark where the focal coils go, trace the flow lines from the center outward, decide which shapes support which. I'm building a tree. I'm assigning lengths and weights to each branch. I don't have an algorithm, but I have twenty years of practice and a pencil, and the process is structurally the same thing.
My math degree lives in this. The spatial reasoning, the structural thinking, the habit of asking "what are the constraints?" before I make a single move: that is applied mathematics. It just happens to produce something beautiful rather than a proof. Which, by the way, a proof also is.
The rigidity chapter made me look at my coils differently
Chapter 4 is about when a structure holds its shape and when it collapses. A triangle is rigid. A square is floppy. Whether a structure resists movement depends on how its pieces are connected and how tension is distributed across the whole.
A quilled coil holds its shape because of glue at the end and the friction between paper layers. A loose coil, before it's glued, is floppy, it can become anything. The moment you fix that end, it becomes rigid. You've made a structural decision. The paper remembers it.
There's a concept in the book called a tensegrity: a structure where components balance each other in tension, where no two pieces actually touch but the whole thing holds together anyway. I've always thought of a finished quilling piece that way, as a conversation between coils, each one pushing slightly against its neighbors, held not by any single element but by the balance of all of them. The math has a name for this now.
One thing the book taught me to stop apologizing for
There is a result in Chapter 17 that I've been telling people about since I read it. The "one complete straight cut" theorem says: you can fold any flat polygon so that a single straight scissor cut produces that exact shape. A star. A letter. A silhouette of a bird. One fold sequence, one cut.
When I tell non-crafters about quilling, I sometimes soften it. "It's just paper rolling," I say, or "it's pretty simple, really." I've spent years doing this, and I think it comes from the same place as choosing math over art at eighteen: a quiet internalized belief that beautiful things need to justify themselves by being useful, or technical, or hard to explain. The same cultural logic that tells students to pick the practical degree tells makers to downplay what they make.
The geometry of what a quilling strip does when you roll, tease, and shape it is not simple. It is the application of paper's physical properties to produce curves and structures that have been the subject of serious mathematical study for decades. And then there's Chapter 13, which made me laugh out loud alone on my couch.
What does NP-hard actually mean?
Chapter 13 proves that checking whether a full crease pattern can fold flat is NP-hard. Here is what that means in plain language.
Imagine you're trying to find someone in a massive, unfamiliar city. Checking whether a specific address is correct takes two seconds: you look at the sign. But finding the right address from scratch, with no map and no directions, could take you the rest of your life, because the number of streets to search grows faster than time allows.
NP-hard problems work exactly this way. Verifying a solution is fast and easy. Finding that solution from scratch is brutally, exponentially hard. Double the size of the problem and you might multiply the work by millions. At a certain scale, not even the fastest supercomputer on Earth could solve it within the lifetime of the universe.
Checking whether one small region of a crease pattern folds correctly? Easy. Checking whether the whole sheet works globally, every fold interacting with every other fold? That's the city without a map.
So the next time someone tells you paper crafting is not a "real" skill, hand them Chapter 13 and ask them to verify whether a crease pattern has a valid flat folding. It's NP-hard. You have time.
I'm done apologizing for it.
The mathematics of paper is not decorative. It is genuinely, formally difficult. And it is also genuinely, formally beautiful. Which was exactly the point all along.
On slowness, open problems, and what we lose when we split things apart
The book is full of what the authors call "open problems": questions nobody has answered yet, boxed and labeled throughout the text. Can every convex polyhedron be unfolded flat without overlapping? Nobody knows. Can a map be refolded to its original state along its crease lines? We don't have a fast algorithm. These questions have been sitting open for years, some for decades.
Quilling is a slow practice. You roll a strip and it takes the time it takes. You can't rush a good coil. Art teaches you to be present. Math teaches you to be precise. Both teach you to sit with a problem long enough for something to emerge. Neither is impractical. Neither is optional. And neither is fully itself without the other.
I picked up Geometric Folding Algorithms as a curiosity. I put it down thinking about my paper strips differently, and about the eighteen-year-old who loved math and art and was told to pick one. She was right all along. They were never two things.
Now if you'll excuse me, I have a hummingbird to plan out.