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.
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.
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!”
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!”
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?”
“Sure.”
“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?”
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.
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.”