Edited by Jane Rosemarin

For years, I looked at the modular world of kusudamas from afar. It takes patience and courage to fold 30 units. So, I admired these spherical wonders from a distance and wondered about the mysterious ways in which the units interlocked. Decades passed. Every so often, I would see a photo of a good one and feel a jolt of envy. Pale and kusudamaless, I would answer, “One of these days, soon, I will fold one. I promise.” As if someone were listening.

Turning the Star Box into a module. All photos and folding by Roman Gorelik.

Then salvation came. Or so I thought, because in my mind’s eye, I had imagined how to make my own kusudama design with just six units. For the module I planned to use a traditional Japanese origami star box, which looks like a container with four triangular petals. I modified it by inverting a petal so that it became a pocket. Now petals would fit into pockets. The future seemed bright.

A “just okay” kusudama with a distraction.

With the number six in mind, I started to fold the individual unit-boxes one by one. But the repetition quickly wore me out: It took me several weeks to fold just eight boxes! Likewise, the assembly was a fiasco, because nothing held. Petals fell out of pockets. I had to use small clothespins to hold modules together, and I had to solicit help from my family. At long last, my first kusudama was finished. After these Herculean efforts, I was exhausted. Eventually, I understood why: It was geometrically impossible.

From an aesthetic point of view, the six-piece kusudama looked just okay. In fact, the more I looked at it, the less comely it seemed. To appease my crying inner child, I put an adorable animal next to it and said: “Look: It isn’t so bad! Just look at that cute little critter nearby. It is so playful!”

The next iteration.

But with each passing day, my impatience with the kusudama grew. I was tempted to throw it out, but the memory of all that work stood in the way. To rid myself of the inner conflict, I searched for and found a new solution: I stacked the boxes in one dimension and the resulting “train” looped on itself, becoming an eight-piece wheel. All this transpired without extra force, strenuous effort or family members. It looked like a fun Ferris wheel. It was lovely and unique, but somehow, I could not derive any pleasure from looking at it.

However, by this time I had already invested too much of everything to get away from the boxes. After some thinking, the next box-related design became laughably apparent — I decided to add a new wheel to the existing one orthogonally. They would intersect at two points, like rings of a gyroscope toy. I did not have enough patience for a full new wheel. So, I made only half of a wheel. Suddenly, between the arches of the hemi-kusudama I discovered two spaces, each of which was perfect for adding a box. The result was interesting — a cactus-like spacecraft with tight bundles of petals protruding from triangular niches. It looked like the Rubik’s Snake toy from my childhood, a rhombicuboctahedron.

The hemi-kusudama was spiky and lonely. It sat on the shelf for days and looked at me. I looked back at it, too, and tried to summon some kindness towards it. “Maybe I can do something else with you?” I thought and disassembled the hemi-kusudama. I peered into one individual box.

“What else can I make from you?”
“A rose.”
“A rose?”
“Yes. The petals will become sepals. And the container will become petals. Just twist it!”

The yellow rose has four petals, the red three. See the video tutorial at the end of this article.

Grateful, I took the box’s advice and made a rose. It took some curved folds and twisting to get it done. When the rose emerged, I almost fell in love with it. Almost, because it had a flaw: Instead of tapering to a point at the bottom, it widened. After some troubleshooting, I found a solution. Namely, I folded one of the sepals in half lengthwise and hid it inside the rose, which narrowed the flower at the bottom just enough to appease the inner critic. Now I had permission to fall in love with abandon. It was easy to do because of all the curves and color changes. The rose became so dear to me that I was suddenly afraid of forgetting how to fold it. Which is why I recorded a video tutorial.

The disassembled hemi-kusudama provided enough boxes to experiment with the roses. I will not bore the reader with even half of the details but will state just the most essential ones. My original rose had four petals, but later I made a three-petaled version, which was just as elegant, if not more. Finally, I figured out how to make a five-petaled rose, but, contrary to my expectations, it was not better than its predecessors. I felt happy and posted some of these roses to Instagram.

After this string of folding experiences, I felt tired and perplexed. Was there a lesson to be learned from this? All I perceived were surface-level banalities, such as “things got better with time” and “the traditional box design holds a plethora of possibilities.” However, I felt that a deeper truth was hiding from me. This notion saddened, angered and excited me all at once.

Thankfully, I remembered how the box spoke to me about the roses. “Be brave,” I said to myself. “Ask it again!” And I did:

“Thanks for the roses, Box!”
“You bet.”
“Can I ask you something again?”
“We both know that I am missing the big secret. What is it, Box? ... please!”
“Look beyond the square.”
“Beyond the square? What do you mean?”

The traditional Star Box folded from various starting shapes.

But the Box was silent. I beseeched it continuously, even with tears in my eyes, but heard only silence. Angry, I unfolded the box to the initial square. “There!” I said, “you deserve it! This is your punishment for speaking in riddles! And also for the silent treatment!”

But the box did not mind being a square again. It just lay there peacefully and reflected happy light from the ridges of its mountain folds. I wiped away the tears and inspected the ridges. Two ridges extended from each corner of the square and bisected a 45-degree angle, such that the entire corner was divided into four equal angles. The ridges all terminated at vertices of a smaller square, which corresponded to the bottom of the container. The small square was rotated 45 degrees relative to the big square. I drew the crease pattern and suddenly knew what the box meant when it said to look beyond the square: This crease pattern could be applied to any regular polygon.

Crease patterns for the boxes above.

Since I am not a mathematician, I did not know how to prove it. So, I decided to check it empirically. First, I applied this crease pattern to a pentagon, and it worked: the resulting pentagonal box was cute and very short. I turned it into a vase that resembled a hybrid between a pomegranate and a persimmon. Then, I applied the crease pattern to an equilateral triangle and out came a small triangular box with very long petals.

I put the three boxes next to each other and smiled. Then, I danced a happy dance and took a photograph. I sighed a sigh of relief and felt content. “Let the mathematicians prove it, if they want,” I said. “It is good enough for me as it is.”

How to Make a Rose From a Traditional Star Box