Part 1 of 3 in the Series: What Paper Knows
Part one of three · What Paper Knows
Chill and Quill Books & Making
What a 500-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. But what surprised me more was realizing this wasn't just about folding paper. It was about everything I had been told to keep separate.
If you make anything with your hands (quilling, origami, sewing, woodworking), I suspect this book has a secret love letter for you, too.
Let me start where the split began: seventeen-year-old me, being told I couldn't major in art. My mother had always supported my art. She just didn't want me to go broke, and that fear made sense. She told me it wasn't practical, and I believed her in the way that first-generation college students tend to believe the people steering them toward something safer. Art was quietly set aside. I didn't immediately land on mathematics. But math is where I graduated, and by the time I did, I had accepted the split as simply the way things were. It took me twenty years and a book about origami to start pulling that apart.
The irony I live with now, as someone with a PhD in Curriculum, Teaching and Educational policy who studies exactly these kinds of choices, is that the split was never real. The line we drew between art and math, between practical and beautiful, between useful and meaningful, was a decision someone made. It was not a description of how knowledge actually works. History makes this plain.
Consider three examples I keep in my back pocket:
The fiction held anyway. It shaped my education, my career choices, and the apologetic way I describe my quilling to people who don't make things. All of that, it turns out, was preparation for reading a textbook about paper.
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, or a strip of paper become a hummingbird wing?
This is 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 able 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 native language
If foldability gave me language for the questions I've always asked, curved creases gave me language for the way my hands already move.
Chapter 20 covers "rigid origami and curved creases." 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. 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 are all expressions of what paper does when you encourage it to follow a curve. The chapter gave me the mathematical vocabulary for something I had 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 seventeen-year-old me wanted to do: holding math and art in both hands at once, refusing to choose.
My process is structural, whether I call it that or not
Chapter 16 describes a piece of software called TreeMaker, built by origami master Robert Lang. You sketch the "tree" of the figure you want: 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 structurally identical. I sketch the composition, mark where the focal coils go, trace the flow lines, decide which shapes support which. I'm building a tree. My math degree lives in this: the spatial reasoning, the habit of asking "what are the constraints?" before I make a single move. It just happens to produce something beautiful rather than a proof, which, by the way, a proof also is.
Chapter 4 deepened this further. It 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, 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 is also a concept in the book called a tensegrity: think of a sculpture where rods float in space, held only by cables in tension, nothing touching but the whole thing holding together anyway. I've always thought of a finished quilling piece that way, each coil pushing slightly against its neighbors, the whole thing held not by any single element but by the balance of all of them. The math has a name for it now.
This is where the book stopped being a mirror and became a defense
There's a theorem in the book that says you can fold paper so that one straight cut produces an exact shape: a star, a letter, a bird silhouette. One fold sequence, one cut.
When I tell non-crafters about quilling, I sometimes soften it. "It's just paper rolling," I say, before they can say it first. That habit comes from the same place as choosing math over art at seventeen: the internalized belief that beautiful things need to justify themselves by being useful or technical. The same logic that tells students to pick the practical degree tells makers to downplay what they make.
And then there is Chapter 13, which made me laugh out loud alone on my couch.
Paper folding is secretly very hard math
Chapter 13 proves that checking whether a crease pattern can fold flat is an NP-hard problem. Here's what that means without the jargon.
NP-hard problems are like searching for one right address in a city you don't know. Checking an address takes seconds; finding it from scratch, with no map, could take practically forever.
Paper folding lands in that category. Checking whether a small section of a crease pattern works is easy. Checking whether the entire sheet folds correctly, every fold interacting with every other, can be as hard as the toughest problems in computer science.
I'm done apologizing for it. So I've stopped calling it "just paper rolling." When people ask what I do, I tell them I solve hard geometry problems with my hands.
If you've ever called what you make "just" a hobby, I want you to borrow my language. You're doing real geometry, real engineering, real design. You just happen to call it art.
The mathematics of paper is not decorative. It is genuinely, formally difficult, and genuinely, formally beautiful. If you've ever been told to choose between what's useful and what you love, let this strange field of folding be your Exhibit A that the split was never real. The line between practical and beautiful was always a fiction; this book just happens to prove it with theorems.
On open problems, and sitting with what we don't know
The book is full of "open problems," questions nobody has solved yet, boxed and labeled throughout the text. Can every convex polyhedron be unfolded flat without overlapping? Can a map always be refolded to its original state? No one knows. These boxes of unanswered questions look a lot like my desk when I'm in the middle of a piece: half-finished ideas, waiting for time and attention.
Quilling is a slow practice. You roll a strip and it takes the time it takes. Art teaches you to be present. Math teaches you to be precise. Both teach you to sit with a question 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 the seventeen-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 huge trident to plan out. My seventeen-year-old self lives in this too.