Teaching My Grandson Three Dimensions: My Attempt and Its Results

I awoke rejoicing, and began to reflect on the glorious career before me. I would go forth, methought, at once, and evangelize the whole of Flatland. Even to Women and Soldiers should the Gospel of Three Dimensions be proclaimed. I would begin with my Wife.

I woke full of joy and began to think about the glorious future ahead of me. I would go out at once, I thought, and spread the Gospel throughout Flatland. Even Women and Soldiers would hear the Gospel of Three Dimensions. I would begin with my Wife.

Just as I had decided on the plan of my operations, I heard the sound of many voices in the street commanding silence. Then followed a louder voice. It was a herald's proclamation. Listening attentively, I recognized the words of the Resolution of the Council, enjoining the arrest, imprisonment, or execution of any one who should pervert the minds of the people by delusions, and by professing to have received revelations from another World.

Just as I decided on my plan, I heard many voices in the street calling for silence. Then a louder voice announced a herald's proclamation. Listening closely, I recognized the words of the Council's Resolution. It ordered the arrest, imprisonment, or execution of anyone who corrupted people's minds with delusions and claims of revelations from another World.

I reflected. This danger was not to be trifled with. It would be better to avoid it by omitting all mention of my Revelation, and by proceeding on the path of Demonstration—which after all, seemed so simple and so conclusive that nothing would be lost by discarding the former means. "Upward, not Northward"—was the clue to the whole proof. It had seemed to me fairly clear before I fell asleep; and when I first awoke, fresh from my dream, it had appeared as patent as Arithmetic; but somehow it did not seem to me quite so obvious now. Though my Wife entered the room opportunely just at that moment, I decided, after we had exchanged a few words of commonplace conversation, not to begin with her.

I thought it over. This danger had to be taken seriously. It would be better to avoid it by leaving out all mention of my Revelation and relying on Demonstration instead. After all, Demonstration seemed so simple and convincing that nothing would be lost by abandoning Revelation. "Upward, not Northward" was the key to the whole proof. The proof had seemed fairly clear to me before I fell asleep. When I first woke, fresh from my dream, it had seemed as obvious as Arithmetic. But somehow the proof did not seem quite so obvious now. Although my Wife entered the room at just the right moment, I decided not to start with her. I made that decision after we had exchanged a few words of everyday conversation.

My Pentagonal Sons were men of character and standing, and physicians of no mean reputation, but not great in mathematics, and, in that respect, unfit for my purpose. But it occurred to me that a young and docile Hexagon, with a mathematical turn, would be a most suitable pupil. Why therefore not make my first experiment with my little precocious Grandson, whose casual remarks on the meaning of 33 had met with the approval of the Sphere? Discussing the matter with him, a mere boy, I should be in perfect safety; for he would know nothing of the Proclamation of the Council; whereas I could not feel sure that my Sons—so greatly did their patriotism and reverence for the Circles predominate over mere blind affection—might not feel compelled to hand me over to the Prefect, if they found me seriously maintaining the seditious heresy of the Third Dimension.

My Pentagonal Sons were men of character and standing, and physicians with strong reputations. But they were not skilled in mathematics, and so were unsuitable for my purpose. Then I thought a young Hexagon, willing to learn and with a talent for mathematics, would be an ideal pupil. So why not try my first experiment with my precocious little Grandson? His casual remarks about the meaning of 33 had won the Sphere’s approval. I would be perfectly safe discussing the matter with him, since he was only a boy. He would know nothing of the Council’s Proclamation. But I could not be sure my Sons would not feel compelled to hand me over to the Prefect. Their patriotism and respect for the Circles far outweighed mere blind affection. They might report me if they found me seriously defending the rebellious heresy of the Third Dimension.

But the first thing to be done was to satisfy in some way the curiosity of my Wife, who naturally wished to know something of the reasons for which the Circle had desired that mysterious interview, and of the means by which he had entered the house. Without entering into the details of the elaborate account I gave her,—an account, I fear, not quite so consistent with truth as my Readers in Spaceland might desire,—I must be content with saying that I succeeded at last in persuading her to return quietly to her household duties without eliciting from me any reference to the World of Three Dimensions. This done, I immediately sent for my Grandson; for, to confess the truth, I felt that all that I had seen and heard was in some strange way slipping away from me, like the image of a half-grasped, tantalizing dream, and I longed to essay my skill in making a first disciple.

But first I had to find some way to satisfy my Wife’s curiosity. Naturally, she wanted to know why the Circle had requested that mysterious meeting and how he had entered the house. I will not go into the details of the elaborate story I told her. I fear my account was not quite as truthful as my Readers in Spaceland might wish. I must settle for saying that I finally persuaded her to return quietly to her household duties. She did so without drawing any mention of the World of Three Dimensions from me. Once that was done, I immediately sent for my Grandson. For, to tell the truth, I felt all I had seen and heard was somehow slipping away from me. It was like the image of a tantalizing dream I could not quite grasp. And I longed to try my skill at winning my first disciple.

When my Grandson entered the room I carefully secured the door. Then, sitting down by his side and taking our mathematical tablets,—or, as you would call them, Lines—I told him we would resume the lesson of yesterday. I taught him once more how a Point by motion in One Dimension produces a Line, and how a straight Line in Two Dimensions produces a Square. After this, forcing a laugh, I said, "And now, you scamp, you wanted to make me believe that a Square may in the same way by motion 'Upward, not Northward' produce another figure, a sort of extra Square in Three Dimensions. Say that again, you young rascal."

When my Grandson entered the room, I carefully secured the door. Then I sat beside him and took out our mathematical tablets. You would call them Lines. I told him we would continue yesterday's lesson. I taught him again how a Point moving in One Dimension produces a Line. Then I showed him how a straight Line moving in Two Dimensions produces a Square. Afterward, I forced a laugh and said, "And now, you scamp, you wanted to make me believe a Square may produce another figure the same way. It would move 'Upward, not Northward' and produce a sort of extra Square in Three Dimensions. Say that again, you young rascal."

At this moment we heard once more the herald's "O yes! O yes!" outside in the street proclaiming the Resolution of the Council. Young though he was, my Grandson—who was unusually intelligent for his age, and bred up in perfect reverence for the authority of the Circles—took in the situation with an acuteness for which I was quite unprepared. He remained silent till the last words of the Proclamation had died away, and then, bursting into tears, "Dear Grandpapa," he said, "that was only my fun, and of course I meant nothing at all by it; and we did not know anything then about the new Law; and I don't think I said anything about the Third Dimension; and I am sure I did not say one word about 'Upward, not Northward', for that would be such nonsense, you know. How could a thing move Upward, and not Northward? Upward and not Northward! Even if I were a baby, I could not be so absurd as that. How silly it is! Ha! ha! ha!"

At that moment, we again heard the herald’s "Hear ye! Hear ye!" outside in the street, announcing the Council’s Resolution. Young though he was, my Grandson understood the situation with a sharpness I had not expected. He was unusually intelligent for his age and had been raised to respect the Circles’ authority completely. He stayed silent until the last words of the Proclamation had faded away. Then he burst into tears. "Dear Grandpapa," he said, "I was only joking, and of course I meant nothing at all by it. And we didn’t know anything then about the new Law. And I don’t think I said anything about the Third Dimension. And I’m sure I didn’t say one word about 'Upward, not Northward', because that would be such nonsense, you know. How could a thing move Upward, and not Northward? Upward and not Northward! Even if I were a baby, I couldn’t be as absurd as that. How silly it is! Ha! ha! ha!"

"Not at all silly," said I, losing my temper; "here for example, I take this Square," and, at the word, I grasped a moveable Square, which was lying at hand—"and I move it, you see, not Northward but—yes, I move it Upward—that is to say, not Northward, but I move it somewhere—not exactly like this, but somehow—" Here I brought my sentence to an inane conclusion, shaking the Square about in a purposeless manner, much to the amusement of my Grandson, who burst out laughing louder than ever, and declared that I was not teaching him, but joking with him; and so saying he unlocked the door and ran out of the room. Thus ended my first attempt to convert a pupil to the Gospel of Three Dimensions.

"Not silly at all," I said, losing my temper. "For example, I take this Square." As I spoke, I picked up a movable Square lying nearby. "And I move it, you see, not Northward but, yes, I move it Upward. That is, not Northward, but I move it somewhere. Not exactly like this, but somehow." Here my sentence trailed off into nonsense as I shook the Square around aimlessly. My Grandson was greatly amused and burst out laughing louder than ever. He declared that I was not teaching him but joking with him. With that, he unlocked the door and ran out of the room. So ended my first attempt to convert a pupil to the Gospel of Three Dimensions.