How I Tried and Failed to Explain Flatland

Thinking that it was time to bring down the Monarch from his raptures to the level of common sense, I determined to endeavour to open up to him some glimpses of the truth, that is to say of the nature of things in Flatland. So I began thus: "How does your Royal Highness distinguish the shapes and positions of his subjects? I for my part noticed by the sense of sight, before I entered your Kingdom, that some of your people are Lines and others Points, and that some of the Lines are larger—" "You speak of an impossibility," interrupted the King; "you must have seen a vision; for to detect the difference between a Line and a Point by the sense of sight is, as every one knows, in the nature of things, impossible; but it can be detected by the sense of hearing, and by the same means my shape can be exactly ascertained. Behold me—I am a Line, the longest in Lineland, over six inches of Space—" "Of Length", I ventured to suggest. "Fool," said he, "Space is Length. Interrupt me again, and I have done."

I thought it was time to bring the Monarch down from his ecstasy to the level of common sense. I decided to give him a few glimpses of the truth about the nature of things in Flatland. So I began: "How does your Royal Highness distinguish the shapes and positions of your subjects? Before I entered your Kingdom, I could see that some of your people are Lines and others Points. I also saw that some of the Lines are larger..." "You speak of something impossible," interrupted the King. "You must have seen a vision. As everyone knows, the very nature of things makes it impossible to distinguish a Line from a Point by sight. But hearing can detect the difference, and hearing can also reveal my exact shape. Look at me. I am a Line, the longest in Lineland, over six inches of Space..." "Of Length," I ventured to suggest. "Fool," he said, "Space is Length. Interrupt me again, and I will say no more."

I apologized; but he continued scornfully, "Since you are impervious to argument, you shall hear with your ears how by means of my two voices I reveal my shape to my Wives, who are at this moment six thousand miles seventy yards two feet eight inches away, the one to the North, the other to the South. Listen, I call to them."

I apologized, but he continued scornfully, "Since arguments cannot convince you, you shall hear how my two voices reveal my shape to my Wives. They are now six thousand miles, seventy yards, two feet, and eight inches away. One is to the North, the other to the South. Listen as I call to them."

He chirruped, and then complacently continued: "My wives at this moment receiving the sound of one of my voices, closely followed by the other, and perceiving that the latter reaches them after an interval in which sound can traverse 6.457 inches, infer that one of my mouths is 6.457 inches further from them than the other, and accordingly know my shape to be 6.457 inches. But you will of course understand that my wives do not make this calculation every time they hear my two voices. They made it, once for all, before we were married. But they COULD make it at any time. And in the same way I can estimate the shape of any of my Male subjects by the sense of sound."

He chirped, then continued with smug satisfaction: "My wives are now hearing one of my voices, closely followed by the other. They notice a gap between the voices, during which sound can travel 6.457 inches. So they conclude that one of my mouths is 6.457 inches farther from them than the other. From this, they know that my shape measures 6.457 inches. But of course you understand that my wives do not make this calculation every time they hear my two voices. They made the calculation once and for all before we were married. But they COULD make it at any time. And in the same way, I can estimate the shape of any of my Male subjects by sound."

"But how," said I, "if a Man feigns a Woman's voice with one of his two voices, or so disguises his Southern voice that it cannot be recognized as the echo of the Northern? May not such deceptions cause great inconvenience? And have you no means of checking frauds of this kind by commanding your neighbouring subjects to feel one another?" This of course was a very stupid question, for feeling could not have answered the purpose; but I asked with the view of irritating the Monarch, and I succeeded perfectly.

"But what," I said, "if a Man uses one of his two voices to imitate a Woman's voice? Or if he disguises his Southern voice so no one recognizes it as the echo of his Northern voice? Might such tricks not cause serious problems? And can't you prevent such fraud by ordering neighbouring subjects to feel one another?" This was, of course, a very stupid question, because feeling could not have served that purpose. But I asked to irritate the Monarch, and I succeeded perfectly.

"What!" cried he in horror, "explain your meaning." "Feel, touch, come into contact," I replied. "If you mean by FEELING," said the King, "approaching so close as to leave no space between two individuals, know, Stranger, that this offence is punishable in my dominions by death. And the reason is obvious. The frail form of a Woman, being liable to be shattered by such an approximation, must be preserved by the State; but since Women cannot be distinguished by the sense of sight from Men, the Law ordains universally that neither Man nor Woman shall be approached so closely as to destroy the interval between the approximator and the approximated.

"What!" he cried in horror. "Explain what you mean." "Feel, touch, make contact," I replied. "If by FEELING," said the King, "you mean coming so close that no space remains between two individuals, know this, Stranger. In my kingdom, that offense is punishable by death. And the reason is obvious. A Woman's fragile body could be shattered by such close contact, so the State must protect her. But since Women cannot be distinguished from Men by sight, the Law applies to everyone. No one may approach either a Man or a Woman so closely that no gap remains between them.

"And indeed what possible purpose would be served by this illegal and unnatural excess of approximation which you call TOUCHING, when all the ends of so brutal and coarse a process are attained at once more easily and more exactly by the sense of hearing? As to your suggested danger of deception, it is non-existent: for the Voice, being the essence of one's Being, cannot be thus changed at will. But come, suppose that I had the power of passing through solid things, so that I could penetrate my subjects, one after another, even to the number of a billion, verifying the size and distance of each by the sense of FEELING: how much time and energy would be wasted in this clumsy and inaccurate method! Whereas now, in one moment of audition, I take as it were the census and statistics, local, corporeal, mental and spiritual, of every living being in Lineland. Hark, only hark!"

"And really, what possible purpose could this illegal and unnatural closeness you call TOUCHING serve? Hearing achieves all the aims of such a brutal, crude process at once, more easily and more precisely. As for the risk of deception you suggest, there is none. The Voice is the essence of one's Being and cannot be changed at will. But come, suppose I could pass through solid objects. I could then pass through my subjects, one after another, even a billion of them. I could check each subject's size and distance by FEELING. How much time and energy would be wasted on this clumsy, inaccurate method! Yet now, in a single moment of listening, I can take what amounts to a census of every living being in Lineland. I can gather statistics on each one's location, body, mind, and spirit. Listen, just listen!"

So saying he paused and listened, as if in an ecstasy, to a sound which seemed to me no better than a tiny chirping from an innumerable multitude of lilliputian grasshoppers.

After saying this, he paused and listened to a sound, as if overcome with joy. To me, the sound seemed no better than the faint chirping of countless tiny grasshoppers.

"Truly," replied I, "your sense of hearing serves you in good stead, and fills up many of your deficiencies. But permit me to point out that your life in Lineland must be deplorably dull. To see nothing but a Point! Not even to be able to contemplate a Straight Line! Nay, not even to know what a Straight Line is! To see, yet be cut off from those Linear prospects which are vouchsafed to us in Flatland! Better surely to have no sense of sight at all than to see so little! I grant you I have not your discriminative faculty of hearing; for the concert of all Lineland which gives you such intense pleasure, is to me no better than a multitudinous twittering or chirping. But at least I can discern, by sight, a Line from a Point. And let me prove it. Just before I came into your kingdom, I saw you dancing from left to right, and then from right to left, with Seven Men and a Woman in your immediate proximity on the left, and eight Men and two Women on your right. Is not this correct?"

"Truly," I replied, "your hearing serves you well and makes up for many of your limitations. But let me point out that your life in Lineland must be terribly dull. To see nothing but a Point! Not even to be able to look at a Straight Line! No, not even to know what a Straight Line is! To see, yet be denied the views of Lines that we enjoy in Flatland! Surely it would be better to have no sight at all than to see so little! I admit I cannot distinguish sounds as well as you can. To you, the music of all Lineland brings intense pleasure. But to me, that music is no more than a mass of twittering or chirping sounds. At least, though, I can tell a Line from a Point by sight. And let me prove it. Just before I entered your kingdom, I saw you dancing from left to right, then from right to left. Seven Men and a Woman were immediately to your left, and eight Men and two Women were to your right. Is that not correct?"

"It is correct," said the King, "so far as the numbers and sexes are concerned, though I know not what you mean by 'right' and 'left'. But I deny that you saw these things. For how could you see the Line, that is to say the inside, of any Man? But you must have heard these things, and then dreamed that you saw them. And let me ask what you mean by those words 'left' and 'right'. I suppose it is your way of saying Northward and Southward."

"That is correct," said the King, "as far as the numbers and sexes go, though I don't know what you mean by 'right' and 'left'. But I deny that you saw these things. How could you see the Line, meaning the inside, of any Man? You must have heard these things and then dreamed that you saw them. And let me ask what you mean by the words 'left' and 'right'. I suppose they are your way of saying Northward and Southward."

"Not so," replied I; "besides your motion of Northward and Southward, there is another motion which I call from right to left."

"That's not true," I replied. "Besides moving north and south, you can move another way, which I call right to left."

KING. Exhibit to me, if you please, this motion from left to right.

KING. Please show me this movement from left to right.

I. Nay, that I cannot do, unless you could step out of your Line altogether.

I. No, I can't do that unless you could step outside your Line entirely.

KING. Out of my Line? Do you mean out of the world? Out of Space?

KING. Outside my Line? Do you mean outside the world? Outside Space?

I. Well, yes. Out of YOUR World. Out of YOUR Space. For your Space is not the true Space. True Space is a Plane; but your Space is only a Line.

I. Well, yes. Out of YOUR World. Out of YOUR Space. Because your Space is not true Space. True Space is a Plane, but your Space is only a Line.

KING. If you cannot indicate this motion from left to right by yourself moving in it, then I beg you to describe it to me in words.

KING. If you cannot show me this left-to-right motion by moving that way yourself, then please describe the motion in words.

I. If you cannot tell your right side from your left, I fear that no words of mine can make my meaning clear to you. But surely you cannot be ignorant of so simple a distinction.

I. If you cannot tell your right side from your left, I fear no words of mine can help you understand. But surely you know such a simple difference.

KING. I do not in the least understand you.

KING. I don't understand you at all.

I. Alas! How shall I make it clear? When you move straight on, does it not sometimes occur to you that you COULD move in some other way, turning your eye round so as to look in the direction towards which your side is now fronting? In other words, instead of always moving in the direction of one of your extremities, do you never feel a desire to move in the direction, so to speak, of your side?

I. Oh dear! How can I make this clear? When you move straight ahead, do you sometimes think you COULD move another way? Could you turn your eye to look in the direction your side now faces? In other words, instead of always moving toward one of your ends, do you ever want to move sideways? Toward your side, so to speak?

KING. Never. And what do you mean? How can a man's inside "front" in any direction? Or how can a man move in the direction of his inside?

KING. Never. And what do you mean? How can a man's inside "face" any direction? Or how can a man move toward his own inside?

I. Well then, since words cannot explain the matter, I will try deeds, and will move gradually out of Lineland in the direction which I desire to indicate to you.

I. Well then, since words cannot explain the matter, I will try actions instead. I will move gradually out of Lineland in the direction I want to show you.

At the word I began to move my body out of Lineland. As long as any part of me remained in his dominion and in his view, the King kept exclaiming, "I see you, I see you still; you are not moving." But when I had at last moved myself out of his Line, he cried in his shrillest voice, "She is vanished; she is dead." "I am not dead," replied I; "I am simply out of Lineland, that is to say, out of the Straight Line which you call Space, and in the true Space, where I can see things as they are. And at this moment I can see your Line, or side—or inside as you are pleased to call it; and I can see also the Men and Women on the North and South of you, whom I will now enumerate, describing their order, their size, and the interval between each."

At those words, I began to move my body out of Lineland. As long as any part of me remained in his kingdom and within his sight, the King kept exclaiming, "I see you, I still see you. You are not moving." But when I had finally moved completely out of his Line, he cried in his shrillest voice, "She has vanished. She is dead." "I am not dead," I replied. "I am simply outside Lineland, outside the Straight Line you call Space. I am in true Space, where I can see things as they are. And right now I can see your Line, or side, or inside, as you prefer to call it. I can also see the Men and Women to your North and South. I will now list them, describing their order, their size, and the distance between each."

When I had done this at great length, I cried triumphantly, "Does that at last convince you?" And, with that, I once more entered Lineland, taking up the same position as before.

After doing this at great length, I cried triumphantly, "Does that finally convince you?" With that, I entered Lineland again and took the same position as before.

But the Monarch replied, "If you were a Man of sense—though, as you appear to have only one voice I have little doubt you are not a Man but a Woman—but, if you had a particle of sense, you would listen to reason. You ask me to believe that there is another Line besides that which my senses indicate, and another motion besides that of which I am daily conscious. I, in return, ask you to describe in words or indicate by motion that other Line of which you speak. Instead of moving, you merely exercise some magic art of vanishing and returning to sight; and instead of any lucid description of your new World, you simply tell me the numbers and sizes of some forty of my retinue, facts known to any child in my capital. Can anything be more irrational or audacious? Acknowledge your folly or depart from my dominions."

But the Monarch replied, "If you were a sensible Man, you would listen to reason. Though you seem to have only one voice, so I have little doubt you are a Woman, not a Man. But if you had even a little sense, you would listen to reason. You ask me to believe there is another Line besides the one my senses detect. You also ask me to believe there is another kind of movement besides the one I experience daily. In return, I ask you to describe that other Line in words or show it through movement. Instead of moving, you merely use some magic trick to disappear and reappear. And instead of clearly describing your new World, you simply give me the numbers and sizes of some forty of my attendants. Any child in my capital knows these facts. Could anything be more unreasonable or more brazen? Admit your foolishness or leave my kingdom."

Furious at his perversity, and especially indignant that he professed to be ignorant of my sex, I retorted in no measured terms, "Besotted Being! You think yourself the perfection of existence, while you are in reality the most imperfect and imbecile. You profess to see, whereas you can see nothing but a Point! You plume yourself on inferring the existence of a Straight Line; but I CAN SEE Straight Lines, and infer the existence of Angles, Triangles, Squares, Pentagons, Hexagons, and even Circles. Why waste more words? Suffice it that I am the completion of your incomplete self. You are a Line, but I am a Line of Lines, called in my country a Square: and even I, infinitely superior though I am to you, am of little account among the great nobles of Flatland, whence I have come to visit you, in the hope of enlightening your ignorance."

Furious at his stubbornness, I was especially outraged that he claimed not to know my sex. I replied harshly, "You fool! You think you are the most perfect being, when you are actually the most imperfect and stupid. You claim to see, but you can see nothing but a Point! You pride yourself on deducing that a Straight Line exists. But I CAN SEE Straight Lines and deduce the existence of Angles, Triangles, Squares, Pentagons, Hexagons, and even Circles. Why waste more words? It is enough to say that I am the complete version of your incomplete self. You are a Line, but I am a Line of Lines, called a Square in my country. And although I am infinitely superior to you, even I count for little among the great nobles of Flatland. I have come from Flatland to visit you, hoping to free you from your ignorance."

Hearing these words the King advanced towards me with a menacing cry as if to pierce me through the diagonal; and in that same moment there arose from myriads of his subjects a multitudinous war-cry, increasing in vehemence till at last methought it rivalled the roar of an army of a hundred thousand Isosceles, and the artillery of a thousand Pentagons. Spell-bound and motionless, I could neither speak nor move to avert the impending destruction; and still the noise grew louder, and the King came closer, when I awoke to find the breakfast-bell recalling me to the realities of Flatland.

Hearing these words, the King advanced toward me with a threatening cry, as if to pierce me through the diagonal. At that same moment, countless thousands of his subjects raised a mighty war cry. It grew so fierce that I thought it rivaled the roar of an army of a hundred thousand Isosceles. It also seemed to rival the artillery of a thousand Pentagons. Frozen and helpless, I could neither speak nor move to prevent my approaching destruction. Still the noise grew louder, and the King came closer. Then I awoke to find the breakfast bell calling me back to the realities of Flatland.