Hexaflexagon 8191

by Cecilia Wennerström

I am a mathematician. All of my life and all of my ambition is about mathematics. Everyone knows me and respects me at the university, where I have done research for many years. There’s no simple way to explain what my research is about. Let us be content to say, that it’s about surfaces. In a vast workroom, I have my computers and my surfaces. My virtual surfaces sway and shimmer above the floor, carried by the embrace of the digital network. Like silk cloth, infinitely thin, colours sparkling, the mathematical formulas float in the air. The cloth can be cut, folded, rotated, stretched out and flattened to different shapes, all of them expressing a special mathematical idea.

My virtual cloth can be used to make hexaflexagons.

Anyone can take a strip of paper, consisting of ten equilateral triangles, and wind them into a hexagon. The paper has two sides, thus you have twenty triangular areas at your disposal. Two superfluous areas are pasted. This paper toy hasn’t just a front side and a reverse side, but also a third, inner side or page. Theoretically, a hexaflexagon can have an infinite number of pages; however, the thickness of the paper sets a limit for the game, when played with paper, scissors, glue and pencil.

For my floating, virtual silk cloth of equations, there are no limits. It’s infinitely thin, yet it has an upper side and an under side, just like a piece of paper.

“The difficult task,” I said one day to Irina Clemens, “is to create and simulate a virtual, two-dimensional area with the right shape. This is already done for many numbers, in fact all numbers up to 8190. Strange enough, no one has yet been able to construct the shape of the hexaflexagon surface that results in exactly 8191 hexagon pages.”

“I’m certain that I could do it,” she said.

“I guessed that you would accept the challenge,” I said. She is the best of the new, young doctoral candidates.

Shortly after, she came by, cheeks red with excitement, saying that she wanted to try her program for the 8191-sided hexaflexagon in my computer room. We decided to meet the following evening, and perform the experiment. She showed me her calculations and her outline of the surface. I was overwhelmed. It was evident that she had hit the target.

Full of excitement, we ran her calculations together. The basic surface itself was impressive — a monstrous, chaos-like polygon, almost filling the room. Now it was to be folded in the right way. Irina wrote her commands on the screen, until the giant surface had been folded together, being reduced to the simple strip of ten triangles. Fingers dancing on the keyboard, she flipped the strip to a standing position, folded back its upper part, wound it round itself, and simulated pasting the two last, superfluous pages. The final result, the hexagon, glistened in the air, like a big, two-dimensional diamond, many-thousand-paged, still infinitely thin. Next, a search program would entice the pages from their infolded hide-aways, counting thousands of foldings per second.

She had no luck the first time, as one of the virtual triangles came loose from the grip of the hexagon, flapping an unruly corner in the air.

“I have to refold it,” Irina said. “I think I know what’s wrong.”

Once more her fingers flew across the keyboard. The hexagon changed its colour; it vibrated so that you could imagine hearing a humming drone; a thousand and yet another thousand pages were shown in repetitive patterns. The speed was high; still it took hours, until the eight thousandth page showed up. The figure 8000 flickered by, and then the humdrum counting restarted from page 1.

“Do you think there’s still some error?” she asked.

“No,” I said. “This is right. I’m sure you have succeeded. You will be able to publish an article about this later. No one has succeeded before.” In the corner of my eye, I saw her trembling with pride.

Evening turned into night, as the sides of the hexagon replaced one another with incredible speed. We were close now. The page number 8190 flickered by and the counting restarted again from the figure 1. The excitement made both of us breath quickly and irregularly. Our eyes were locked stiff on the geometric shape in the air. The counting started from the beginning once more, every tedious series, every page subtly changing colour, an extra highlight Irina had created in her program code. She was a master. Her skill moved me.

And then came the magic moment, suddenly, when the hexagon showed the page number 8191. The counting came to a halt, the figure flickered, then disappeared, whereafter the surface went pitch black and completely still.

“How strange,” Irina said. “I didn’t code that. The number 8191 should have flickered by, just like the others, only with a tiny delay, which I put there to make the digits show clearly. The program seems to be hanging.”

She moved towards the keyboard, but I stopped her.

“No, let’s have a look at it. It’s an interesting phenomenon.” I took a couple of steps towards the coal-black hexagon, that hung unmoving in the air in front of us.

“Have a look at this,” I said. She came closer.

“You really must see this,” I added, heartily, meaning to break the uncanny feeling created by the stiff blackness of the virtual hexaflexagon.

“Do you see the little red dot at the upper right corner of the hexaflexagon? You have to move a bit closer. Why, isn’t the surface oddly dark? Move a little closer, and you’ll see.”

When the black surface spread out, folding its soft, dark corners around Irina’s body, I let her see my real body for a second. As usual, the eyes opened wide in utter horror before the body was hauled into the dark, hungry hole.

I waited a couple of seconds before I closed the gate to my belly. I don’t know if belly is the right word. It’s more like a place. Maybe it should have a more ceremonial name. I pondered this as I left the building, whistling. I think whistling is fun. It is strange to be able to form lips like these, and strange and funny to be able to whistle.

A month later, a new, young scientist came to our department. I made friends with him, and we talked for a long while about mathematics and mathematical games. His intelligence made my mouth water.

Not that I really have a mouth.

And the viscous fluids, that sometimes pleasurably trickle out of my body, have no points of similarity with water.

“I suppose,” I finally said to him, “that you at some time have heard about hexaflexagons?”

The story originally appeared in Swedish in Mitrania (January 2004). Translation by the author.

Editor’s note: an informative, illustrated page on hexaflexagons can be found at Hexaflexagons. It is an auxiliary page to one devoted to Martin Gardner, who was the mathematical games expert at Scientific American from 1956-86.