How I Invented a New Origami Polyhedron
My quest to fold a shape with no folding instructions.
It was a sunny summer morning a few months before my final year of high school, and the world outside was gleaming with energy. It was obvious how I was going to spend a day like this. “Ooh,” I thought, “The lighting by my window is perfect for making origami polyhedra!”
I nestled into my chair and pulled out a stack of multicolored square origami paper. “Let’s start with an old favorite.” I folded three sheets into the shape of two triangular pyramids connected base-to-base. This triangular bipyramid (dubbed a “fox box” online) was the most elegant polyhedron in my repertoire. It only required twelve folds, and the resulting shape was composed of sturdy, right isosceles triangles. I twirled it around, admiring its symmetry.
A close relative of the triangular bipyramid is the square bipyramid, the best variety being the octahedron. As a member of the Platonic solids, the octahedron is even more symmetric than its triangular cousin, since its faces are congruent regular polygons and the same number of faces meet at each vertex. I didn’t know how to make one¹, but the octahedron is such a classic shape that I figured it had already been well-studied by origami aficionados. Indeed, a quick Google search revealed multiple methods for making them.
Having seen ways to make triangular and square bipyramids, the natural next question struck me. “What about a pentagonal bipyramid?” The number five always seemed a little odd². It’s a prime, refusing to be broken into factors. Regular pentagons, no matter how you arrange them, can’t tile the plane nicely–they always leave gaps. Would five be similarly elusive in the realm of origami bipyramids? Or would it yield to some beautifully simple sequence of folds?
The internet revealed some enticing photos of folded paper pentagonal bipyramids, but they either came with no instructions or (perhaps even worse) they were made by cutting out, folding, and gluing together the net of a pentagonal bipyramid. This approach didn’t interest me because it’s easy to make any shape you want if you allow cutting and gluing. In the same way that the ancient Greeks only acknowledged the existence of shapes that could be constructed with a compass and straightedge³, I was only interested in a pure origami solution — square paper, only folds. As the internet had no solutions for me, I decided to try inventing my own.
I began by imposing some reasonable restrictions on the problem so I could at least have a place to start. Obviously, any crease pattern that produced a pentagonal bipyramid must have isosceles triangles in it that make up the polyhedron’s ten faces. The faces technically could be anywhere on the paper, but I figured that it would be reasonable to expect them to lie edge-to-edge, so as to form a net. Furthermore, I thought it would be best to make each face an equilateral triangle, because a polyhedral net with equilateral triangles could be embedded in a triangular grid, which was easy to make.
The next step was to figure out how to turn a flat grid into a three-dimensional shape. I naturally gravitated towards designs with symmetry, eventually settling on a rotationally symmetric clamshell structure that’s secured shut by inserting flaps into pockets. I had no idea if I could actually make such a thing, but it seemed to be the most promising goal.
Armed with this plan, I pulled out a fresh sheet of paper and got to work.
Two hours later, my desk was cluttered with countless crumpled-up, failed clamshell attempts. I tried triangular grids of all types, but none of them seemed to work. Grids that were too coarse didn’t give me enough material to form sturdy flaps and pockets, while grids that were too fine gave me a bunch of extra paper that I couldn’t tuck away anywhere.
I leaned back and exhaled in frustration. The answer to “Is there an elegant way to make an origami pentagonal bipyramid?” seemed to be a deeply unsatisfying “Nope!” I had spent my whole morning searching for a rare origami flower in a vast paper forest, and now I was utterly lost amongst the foliage. Incomplete thoughts swirled in my mind.
Pentagonal bipyramids consumed six days of my life that summer. By the end, I had arrived at a jank solution that used a strange parallelogram-shaped piece of paper. At the time I was pleased with this result. I even featured it in my college admissions essay, which was about the joy of creating things even if they are imperfect. The essay provided closure, in some sense, but every now and then I would get caught up in tackling the pentagonal bipyramid once again.
Fast-forward to spring, a few months before graduation. I was eating lunch in the cafeteria while experimenting with a triangular grid that had only three rows. With so few triangles, the possibilities were limited. To start, I found a place on the grid where six triangles met to form a hexagon and tucked one of the triangles away. This move gave the paper a whiff of pentagonality, but most of it was still flat.
Naively and without any plan in mind, I smooshed and prodded the flat part in hopes that I could mold it into the right shape. Unsurprisingly, there wasn’t enough material to form a full pentagonal bipyramid.⁴ What I had was about half a pentagonal bipyramid.
“Hmm…” My head cocked to one side. As a cat plays with yarn, I picked up the half I had made and studied it close to my face, feeling its worn folds. “…what if I made two halves?” The idea sounded revolutionary, yet so obvious in retrospect. I speedily folded up another paper the same way, hoping that I could somehow piece the two halves together into a whole. Alas, this was not the case. At every place where I needed a flap and a pocket, I instead had a flap and a flap. It was like trying to assemble Legos like this.
Still, I was convinced that the two-halves approach was the way to go. What I needed was not two identical halves, but two complementary halves. I went back to the drawing (folding?) board with a freshly creased grid and tried to design a shape that had pockets where my existing shape had flaps, and vice versa. With the help of some fortuitous smooshing, I arrived at what I hoped would be the answer.
Then came the moment of truth. Holding one paper in each shaky hand, I began coaxing the halves together, piece by piece, fold by fold, and…
It worked!!
I freaked out in a way that my lunch buddies probably thought was really weird.
“Dude, Matt, can you chill? What’s with the diamond thing you just made?”
“No you don’t understand, Eric, I’ve been working on this thing for like, a year!!” It was more like half a year, really. But after countless sessions where I thought until my brain hurt and folded until my fingertips were raw, I found it appropriate to be a little dramatic.
This pentagonal bipyramid was mine, something entirely original. In the inexplicable way that an artist relates to their art, I became emotionally attached to the weird little shape I held in my hands. It was somehow an extension of myself. It was my own little contribution to humanity’s collective origami knowledge.
That whole story took place three years ago, but the design is still in my mind with perfect clarity. I folded the pentagonal bipyramid that you see in this story as if I had come up with it yesterday. They say that people remember stuff better when that stuff has high emotional content. Every time I fold a pentagonal bipyramid, I’m reminded of the emotions that went into its creation: curiosity, frustration, and the joy of discovery.
This story contains bits and pieces lifted straight out of my Common App essay. #CiteYoSelf
- But I did learn how to make one just now so I could make you these sweet, sweet GIFs.
- Pun intended ;)
- Check out this much longer article about a tricky compass-and-straightedge problem that eluded mathematicians for thousands of years!
- This became obvious to me while writing this story because I realized the grid I was working with didn’t even have ten full triangles on it!
