A mathematical corollary tale inspired by Edwin Abbott's Flatland.



As a lone hexagon, I remember when grandfather was executed for believing in the third dimension myth. In our flat world, all polygons thought he was insane: none of us could understand his visions of seeing above and below our land.

My grandfather was a square of much knowledge; he knew about the different configurations and coordinates of the inhabitants of 2-D World. He was a great luminary, though most mistook him for a lunatic. Being just a little hexagon at the time, I loved Grandfather. We would take long walks, vertex in vertex, and I'd listen to his stories about fantastic worlds beyond our own. "Grandson," he would say, "you're the only one with the courage to think outside the lines." I didn't know exactly what he was talking about, but it sounded like a geometric compliment, and I loved all 360 interior degrees of him for it. I remember his perfect right angles as we would walk through the forest and he would say to me, "Grandson, you have the courage to see the larger world, to venture to the next land." I never would have guessed, at the time, that being able to imagine new dimensions and believe in something others could not would eventually cost him everything.

In the time squares following his death, my family despaired losing him. His "prophecies" about Spaceland, the special "guide" he found there, and Spaceland's interdigitation with 0-D World, 1-D World and 2-D World are silenced now. But on the last day of the millennium, I thought of him, unaware that I was about to become a vector of Grandpa's ideas.

I was asleep when it happened. I was in the middle of a dream about walking through the grid forest that Grandpa and I used to love so well, and I stopped to pick a flower with beautiful coordinates. Suddenly, a large figure in the mist came rushing toward me. I screamed, but it did not stop, intersecting my regular sides, and I was deflected backward into the angled darkness of my dream. Half awake, or some fraction into sleep, I heard a mysterious voice outside the house. I went to investigate and there I found a newcomer, cloaked in shimmering light; a rounded sort of shape without a face. My first thought was to run away, but some remainder of curiosity made me stay. In a low voice, it began to speak.

"Now is the time. You must head there - immediately."

"Where?" My voice quivered with fear as the shape seemed to grow in size. "Who are you, and what do you want with me?"

"You must go to the origin. Head straight to the origin without pursuing tangents. Begin at the Great Planes and proceed integer-wise. Once at the origin, I will reveal the way, for I am the King of Spheres."

"Wait!" I pleaded in vain, for the King of Spheres began to shrink and fade until he vanished. "Why should I go to the origin? What way will be revealed? Why do you call yourself a sphere instead of a circle? And what has all this to do with me?" My questions echoed into squares of thin air.

I was vexed, but intrigued all the same. From my present coordinates I headed southwest toward this origin, wondering what would be there when I arrived. Grandpa once said that mysterious jet streams always flowed at the places that he called axes, although in 2-D World we always knew there were just two of them.

I traveled westward for hours, hurrying to catch up to the king. All of my edges were aching from the journey. Then, after traveling for what seemed like infinity, I caught sight of a line which flowed without end, directly in front of my edge. I was thrilled to realize that this was the famous jet stream Grandpa had told me about. I headed southward, and after a while I found the point at which the axes met. I also found myself, once again, before the King of Spheres.

In a quiet voice, what appeared to be a great circle said, "Without speaking a word, you will stand on the origin, the point of ultimate intersection."

Although I wanted to question this command, I complied in silence, just like a perfect square. Or rather, regular hexagon. As I stepped on the point, I felt a strange force coming over and under me. It was as if all the winds came together at that single point in space, lifting me upward. I gasped, part thrilled, part horrified, as I looked around my levitating self, seeing the ground disappear beneath me. Once I rose above the plane, I saw the guide that my grandfather had described to me. And I saw the King of Spheres. They were one and the same.

With a shaky voice, I asked, "Are you the guide that my grandfather saw?"

He said nothing, but nodded. Then he reached out and gave me a gentle shove, pushing me on my side and tipping me to a new angle. I was amazed at what I was able to see. This was Spaceland, the third dimension Grandpa had said existed. And here I was, seeing it for the first time, spreading before me in all its three-dimensional reality. Memories of Grandpa flooded my mind, and tears filled my eyes. I could hear his words, clear as a crystalgon, "Just let yourself imagine a shape going upward, like a square going Northward and it will form a cube with six sides, or a circle going upward, and it will form a sphere."

"Now you know the truth."

"Why are you telling me this truth, King of Spheres?" I asked. But there was no answer. I closed my eyes, dizzy from these strange events, this new dimension, and opened them to find myself back in my bed. Had this been just a dream? Had this just been my own imagination playing tricks on me? Could it really have been true? I looked at the calendar beside my bed and remembered that today was the start of the new millennium, the beginning of new truths. I wondered what else there was beyond our world or even their world, extending well beyond the limits of our own thinking.