The Sphere’s Further Spaceland Mysteries, My Desire for Still More, and What Followed

When I saw my poor brother led away to imprisonment, I attempted to leap down into the Council Chamber, desiring to intercede on his behalf, or at least bid him farewell. But I found that I had no motion of my own. I absolutely depended on the volition of my Guide, who said in gloomy tones, "Heed not thy brother; haply thou shalt have ample time hereafter to condole with him. Follow me."

When I saw my poor brother led away to prison, I tried to leap down into the Council Chamber. I wanted to plead on his behalf, or at least say goodbye. But I found I could not move on my own. I depended entirely on the will of my Guide, who said in a gloomy voice, "Pay no attention to your brother. Perhaps you will have plenty of time later to comfort him. Follow me."

Once more we ascended into space. "Hitherto," said the Sphere, "I have shewn you naught save Plane Figures and their interiors. Now I must introduce you to Solids, and reveal to you the plan upon which they are constructed. Behold this multitude of moveable square cards. See, I put one on another, not, as you supposed, Northward of the other, but ON the other. Now a second, now a third. See, I am building up a Solid by a multitude of Squares parallel to one another. Now the Solid is complete, being as high as it is long and broad, and we call it a Cube."

Once again we rose into space. "So far," said the Sphere, "I have shown you only Plane Figures and their interiors. Now I must introduce you to Solids and show you how they are constructed. Look at these many movable square cards. Watch as I place one on another, not Northward of the other, as you supposed, but ON the other. Now a second, now a third. See, I am building a Solid from many Squares parallel to one another. Now the Solid is complete. It is as high as it is long and broad, and we call it a Cube."

"Pardon me, my Lord," replied I; "but to my eye the appearance is as of an Irregular Figure whose inside is laid open to the view; in other words, methinks I see no Solid, but a Plane such as we infer in Flatland; only of an Irregularity which betokens some monstrous criminal, so that the very sight of it is painful to my eyes."

"Forgive me, my Lord," I replied. "But to my eye, it looks like an Irregular Figure with its inside exposed. In other words, I think I see no Solid, but a Plane like those we infer in Flatland. Only its Irregularity suggests some monstrous criminal, so that the very sight of it hurts my eyes."

"True," said the Sphere, "it appears to you a Plane, because you are not accustomed to light and shade and perspective; just as in Flatland a Hexagon would appear a Straight Line to one who has not the Art of Sight Recognition. But in reality it is a Solid, as you shall learn by the sense of Feeling."

"True," said the Sphere, "it looks like a Plane to you because you are not used to light, shade, and perspective. In just the same way, a Hexagon in Flatland would look like a Straight Line to someone without the Art of Sight Recognition. But it is actually a Solid, as you will learn by touch."

He then introduced me to the Cube, and I found that this marvellous Being was indeed no Plane, but a Solid; and that he was endowed with six plane sides and eight terminal points called solid angles; and I remembered the saying of the Sphere that just such a Creature as this would be formed by a Square moving, in Space, parallel to himself: and I rejoiced to think that so insignificant a Creature as I could in some sense be called the Progenitor of so illustrious an offspring.

He then introduced me to the Cube, and I discovered that this marvelous being was indeed a Solid, not a Plane. He had six flat faces and eight corner points called solid angles. I remembered the Sphere saying that a Square moving through Space, parallel to himself, would form just such a creature. And I rejoiced to think that someone as insignificant as I could, in a sense, father so distinguished an offspring.

But still I could not fully understand the meaning of what my Teacher had told me concerning "light" and "shade" and "perspective"; and I did not hesitate to put my difficulties before him.

But I still did not fully understand what my Teacher had told me about "light" and "shade" and "perspective". And I did not hesitate to tell him what I found confusing.

Were I to give the Sphere's explanation of these matters, succinct and clear though it was, it would be tedious to an inhabitant of Space, who knows these things already. Suffice it, that by his lucid statements, and by changing the position of objects and lights, and by allowing me to feel the several objects and even his own sacred Person, he at last made all things clear to me, so that I could now readily distinguish between a Circle and a Sphere, a Plane Figure and a Solid.

If I gave the Sphere's explanation of these matters, it would bore an inhabitant of Space. Although his explanation was brief and clear, anyone living in Space already knows these things. It is enough to say that he finally made everything clear to me. He did so through clear explanations, by moving objects and lights, and by letting me touch the objects. He even let me touch his own sacred Person. As a result, I could now easily tell a Circle from a Sphere, and a Plane Figure from a Solid.

This was the Climax, the Paradise, of my strange eventful History. Henceforth I have to relate the story of my miserable Fall:—most miserable, yet surely most undeserved! For why should the thirst for knowledge be aroused, only to be disappointed and punished? My volition shrinks from the painful task of recalling my humiliation; yet, like a second Prometheus, I will endure this and worse, if by any means I may arouse in the interiors of Plane and Solid Humanity a spirit of rebellion against the Conceit which would limit our Dimensions to Two or Three or any number short of Infinity. Away then with all personal considerations! Let me continue to the end, as I began, without further digressions or anticipations, pursuing the plain path of dispassionate History. The exact facts, the exact words,—and they are burnt in upon my brain,—shall be set down without alteration of an iota; and let my Readers judge between me and Destiny.

This was the high point, the Paradise, of my strange and eventful story. From here on, I must tell the story of my miserable Fall. Most miserable, yet surely most undeserved! For why should a thirst for knowledge be awakened, only to be disappointed and punished? I shrink from the painful task of recalling my humiliation. Yet, like a second Prometheus, I will endure this pain and worse for the chance to inspire rebellion. I hope to awaken that spirit in the hearts of Plane and Solid Humanity. Let them rebel against the arrogant belief that our Dimensions must stop at Two, Three, or any number below Infinity. Away, then, with all personal concerns! Let me continue to the end as I began, without further digressions or glimpses ahead. I will follow the straightforward path of impartial History. I will record the exact facts and the exact words, without changing the smallest detail. Those facts and words are burned into my brain. Let my Readers judge between me and Destiny.

The Sphere would willingly have continued his lessons by indoctrinating me in the conformation of all regular Solids, Cylinders, Cones, Pyramids, Pentahedrons, Hexahedrons, Dodecahedrons, and Spheres: but I ventured to interrupt him. Not that I was wearied of knowledge. On the contrary, I thirsted for yet deeper and fuller draughts than he was offering to me.

The Sphere would gladly have continued teaching me about the shapes of all regular Solids: Cylinders, Cones, Pyramids, Pentahedrons, Hexahedrons, Dodecahedrons, and Spheres. But I dared to interrupt him. Not because I was tired of learning. On the contrary, I thirsted for knowledge even deeper and richer than he was offering me.

"Pardon me," said I, "O Thou Whom I must no longer address as the Perfection of all Beauty; but let me beg thee to vouchsafe thy servant a sight of thine interior."

"Forgive me," I said. "I must no longer call you the Perfection of all Beauty. But I beg you to let your servant see inside you."

SPHERE. My what?

SPHERE. My what?

I. Thine interior: thy stomach, thy intestines.

I. Your insides: your stomach, your intestines.

SPHERE. Whence this ill-timed impertinent request? And what mean you by saying that I am no longer the Perfection of all Beauty?

SPHERE. Why make such a rude request at such a bad time? And what do you mean by saying I am no longer the Perfection of all Beauty?

I. My Lord, your own wisdom has taught me to aspire to One even more great, more beautiful, and more closely approximate to Perfection than yourself. As you yourself, superior to all Flatland forms, combine many Circles in One, so doubtless there is One above you who combines many Spheres in One Supreme Existence, surpassing even the Solids of Spaceland. And even as we, who are now in Space, look down on Flatland and see the insides of all things, so of a certainty there is yet above us some higher, purer region, whither thou dost surely purpose to lead me—O Thou Whom I shall always call, everywhere and in all Dimensions, my Priest, Philosopher, and Friend—some yet more spacious Space, some more dimensionable Dimensionality, from the vantage-ground of which we shall look down together upon the revealed insides of Solid things, and where thine own intestines, and those of thy kindred Spheres, will lie exposed to the view of the poor wandering exile from Flatland, to whom so much has already been vouchsafed.

I. My Lord, your own wisdom has taught me to aspire to One even greater and more beautiful than yourself. This One comes even closer to Perfection. You, who are superior to all Flatland forms, combine many Circles in One. So doubtless there is One above you who combines many Spheres in One Supreme Existence. This Supreme Existence surpasses even the Solids of Spaceland. And just as we, now in Space, look down on Flatland and see the insides of all things, so there is certainly some higher, purer region above us. Surely you intend to lead me there. Wherever I am, in all Dimensions, I shall always call you my Priest, Philosopher, and Friend. That higher region must be some still more spacious Space, some Dimensionality with still more dimensions. From that vantage point, we shall look down together and see the exposed insides of Solid things. There, your own intestines, and those of your fellow Spheres, will be visible to this poor, wandering exile from Flatland. So much has already been granted to me.

SPHERE. Pooh! Stuff! Enough of this trifling! The time is short, and much remains to be done before you are fit to proclaim the Gospel of Three Dimensions to your blind benighted countrymen in Flatland.

SPHERE. Nonsense! Rubbish! Enough of this fooling around! Time is short, and much remains to be done before you are ready. Then you can proclaim the Gospel of Three Dimensions to your blind, ignorant countrymen in Flatland.

I. Nay, gracious Teacher, deny me not what I know it is in thy power to perform. Grant me but one glimpse of thine interior, and I am satisfied for ever, remaining henceforth thy docile pupil, thy unemancipable slave, ready to receive all thy teachings and to feed upon the words that fall from thy lips.

I. No, gracious Teacher, do not refuse me what I know you have the power to do. Give me just one glimpse of your interior, and I will be satisfied forever. From then on, I will remain your obedient pupil, your slave who can never be freed. I will be ready to accept all your teachings and feed on every word from your lips.

SPHERE. Well, then, to content and silence you, let me say at once, I would shew you what you wish if I could; but I cannot. Would you have me turn my stomach inside out to oblige you?

SPHERE. Well, then, to satisfy you and keep you quiet, let me say this now. I would show you what you want if I could. But I cannot. Do you expect me to turn my stomach inside out just to please you?

I. But my Lord has shewn me the intestines of all my countrymen in the Land of Two Dimensions by taking me with him into the Land of Three. What therefore more easy than now to take his servant on a second journey into the blessed region of the Fourth Dimension, where I shall look down with him once more upon this land of Three Dimensions, and see the inside of every three-dimensioned house, the secrets of the solid earth, the treasures of the mines in Spaceland, and the intestines of every solid living creature, even of the noble and adorable Spheres.

I. But my Lord has shown me the intestines of all my countrymen in the Land of Two Dimensions. He did so by taking me with him into the Land of Three. So what could be easier than taking his servant on a second journey, into the blessed region of the Fourth Dimension? From there, I shall once more look down with him upon this land of Three Dimensions. I shall see inside every three-dimensional house, the secrets of the solid earth, and the treasures of Spaceland's mines. I shall also see the intestines of every solid living creature, even the noble and adorable Spheres.

SPHERE. But where is this land of Four Dimensions?

SPHERE. But where is this land of Four Dimensions?

I. I know not: but doubtless my Teacher knows.

I. I don't know, but my Teacher undoubtedly knows.

SPHERE. Not I. There is no such land. The very idea of it is utterly inconceivable.

SPHERE. Not me. No such land exists. The very idea of such a land is completely unimaginable.

I. Not inconceivable, my Lord, to me, and therefore still less inconceivable to my Master. Nay, I despair not that, even here, in this region of Three Dimensions, your Lordship's art may make the Fourth Dimension visible to me; just as in the Land of Two Dimensions my Teacher's skill would fain have opened the eyes of his blind servant to the invisible presence of a Third Dimension, though I saw it not.

I. I can imagine it, my Lord, so my Master must find it even easier to imagine. Indeed, I still hope your Lordship’s skill may make the Fourth Dimension visible to me, even here in Three Dimensions. Just as, in the Land of Two Dimensions, my Teacher tried through his skill to open his blind servant’s eyes. He wanted me to see the invisible presence of a Third Dimension, though I did not see it.

Let me recall the past. Was I not taught below that when I saw a Line and inferred a Plane, I in reality saw a Third unrecognized Dimension, not the same as brightness, called "height"? And does it not now follow that, in this region, when I see a Plane and infer a Solid, I really see a Fourth unrecognized Dimension, not the same as colour, but existent, though infinitesimal and incapable of measurement?

Let me think back. Wasn't I taught below that when I saw a Line and inferred a Plane, I actually saw an unrecognized Third Dimension? Wasn't that Dimension different from brightness and called "height"? And doesn't it now follow that the same is true in this region when I see a Plane and infer a Solid? Am I not actually seeing an unrecognized Fourth Dimension? Isn't that Dimension different from colour, yet real, though infinitesimally small and impossible to measure?

And besides this, there is the Argument from Analogy of Figures.

And besides this, there is the argument based on similarities between shapes.

SPHERE. Analogy! Nonsense: what analogy?

SPHERE. Analogy! Nonsense! What analogy?

I. Your Lordship tempts his servant to see whether he remembers the revelations imparted to him. Trifle not with me, my Lord; I crave, I thirst, for more knowledge. Doubtless we cannot SEE that other higher Spaceland now, because we we have no eye in our stomachs. But, just as there WAS the realm of Flatland, though that poor puny Lineland Monarch could neither turn to left nor right to discern it, and just as there WAS close at hand, and touching my frame, the land of Three Dimensions, though I, blind senseless wretch, had no power to touch it, no eye in my interior to discern it, so of a surety there is a Fourth Dimension, which my Lord perceives with the inner eye of thought. And that it must exist my Lord himself has taught me. Or can he have forgotten what he himself imparted to his servant?

I. Your Lordship is testing his servant to see whether he remembers the revelations shared with him. Do not toy with me, my Lord. I crave, I thirst for more knowledge. Surely we cannot SEE that other, higher Spaceland now because we have no eyes in our stomachs. But Flatland WAS there, though that poor, puny Lineland Monarch could turn neither left nor right to see it. Just as surely, the land of Three Dimensions WAS close at hand, touching my body. Yet I, a blind, senseless wretch, could not touch it and had no eye inside me to see it. So, just as surely, a Fourth Dimension exists, which my Lord sees with the inner eye of thought. My Lord himself has taught me that the Fourth Dimension must exist. Or has he forgotten what he himself taught his servant?

In One Dimension, did not a moving Point produce a Line with TWO terminal points?

In One Dimension, didn't a moving Point create a Line with TWO endpoints?

In Two Dimensions, did not a moving Line produce a Square with FOUR terminal points?

In Two Dimensions, didn't a moving Line create a Square with FOUR endpoints?

In Three Dimensions, did not a moving Square produce—did not this eye of mine behold it—that blessed Being, a Cube, with EIGHT terminal points?

In Three Dimensions, didn't a moving Square produce that blessed Being, a Cube, with EIGHT corners? Didn't I see that Cube with my own eyes?

And in Four Dimensions shall not a moving Cube—alas, for Analogy, and alas for the Progress of Truth, if it be not so—shall not, I say, the motion of a divine Cube result in a still more divine Organization with SIXTEEN terminal points?

And in Four Dimensions, will a divine Cube’s motion not produce an even more divine structure with SIXTEEN endpoints? Alas for Analogy, and alas for the Progress of Truth, if the answer is no!

Behold the infallible confirmation of the Series, 2, 4, 8, 16: is not this a Geometrical Progression? Is not this—if I might quote my Lord's own words—"strictly according to Analogy"?

Look at the undeniable confirmation of the series, 2, 4, 8, 16. Isn't this series a geometric progression? And isn't this progression, if I may quote my Lord's own words, "strictly according to Analogy"?

Again, was I not taught by my Lord that as in a Line there are TWO bounding Points, and in a Square there are FOUR bounding Lines, so in a Cube there must be SIX bounding Squares? Behold once more the confirming Series, 2, 4, 6: is not this an Arithmetical Progression? And consequently does it not of necessity follow that the more divine offspring of the divine Cube in the Land of Four Dimensions, must have 8 bounding Cubes: and is not this also, as my Lord has taught me to believe, "strictly according to Analogy"?

Again, did my Lord not teach me that a Line has TWO boundary Points, and a Square has FOUR boundary Lines? So must a Cube not have SIX boundary Squares? Look again at the supporting sequence, 2, 4, 6: is it not an arithmetic progression? So must the divine Cube’s more divine offspring in the Land of Four Dimensions not have 8 boundary Cubes? And is having 8 boundary Cubes not also, as my Lord taught me to believe, "strictly according to Analogy"?

O, my Lord, my Lord, behold, I cast myself in faith upon conjecture, not knowing the facts; and I appeal to your Lordship to confirm or deny my logical anticipations. If I am wrong, I yield, and will no longer demand a fourth Dimension; but, if I am right, my Lord will listen to reason.

Oh, my Lord, my Lord, I put my faith in a guess, without knowing the facts. I ask you to confirm or deny what logic leads me to expect. If I am wrong, I give in and will no longer demand a fourth Dimension. But if I am right, my Lord will listen to reason.

I ask therefore, is it, or is it not, the fact, that ere now your countrymen also have witnessed the descent of Beings of a higher order than their own, entering closed rooms, even as your Lordship entered mine, without the opening of doors or windows, and appearing and vanishing at will? On the reply to this question I am ready to stake everything. Deny it, and I am henceforth silent. Only vouchsafe an answer.

So I ask: have your countrymen also witnessed Beings of a higher order than themselves descending into their world? Have these Beings entered closed rooms without doors or windows being opened, just as your Lordship entered mine? Have they appeared and vanished at will? I am ready to stake everything on your answer to these questions. Deny that this has happened, and I will say no more. Just give me an answer.

SPHERE. (AFTER A PAUSE). It is reported so. But men are divided in opinion as to the facts. And even granting the facts, they explain them in different ways. And in any case, however great may be the number of different explanations, no one has adopted or suggested the theory of a Fourth Dimension. Therefore, pray have done with this trifling, and let us return to business.

SPHERE. (AFTER A PAUSE). That is what people report. But men disagree about the facts. And even if they accept the facts, they explain them in different ways. And in any case, no matter how many explanations there are, no one has accepted or suggested a Fourth Dimension. So please stop this nonsense, and let us get back to business.

I. I was certain of it. I was certain that my anticipations would be fulfilled. And now have patience with me and answer me yet one more question, best of Teachers! Those who have thus appeared—no one knows whence—and have returned—no one knows whither—have they also contracted their sections and vanished somehow into that more Spacious Space, whither I now entreat you to conduct me?

I. I was certain of it. I was certain that what I expected would come true. Now be patient with me and answer just one more question, best of Teachers! Those people appeared from somewhere unknown and returned to somewhere unknown. Have they also shrunk their sections and somehow vanished into that more Spacious Space? I now beg you to take me there.

SPHERE (MOODILY). They have vanished, certainly—if they ever appeared. But most people say that these visions arose from the thought—you will not understand me—from the brain; from the perturbed angularity of the Seer.

SPHERE (MOODILY). They have certainly vanished, if they ever appeared. But most people say these visions came from thought. You will not understand what I mean. The visions came from the brain, from the Seer’s disturbed angularity.

I. Say they so? Oh, believe them not. Or if it indeed be so, that this other Space is really Thoughtland, then take me to that blessed Region where I in Thought shall see the insides of all solid things. There, before my ravished eye, a Cube, moving in some altogether new direction, but strictly according to Analogy, so as to make every particle of his interior pass through a new kind of Space, with a wake of its own—shall create a still more perfect perfection than himself, with sixteen terminal Extra-solid angles, and Eight solid Cubes for his Perimeter. And once there, shall we stay our upward course? In that blessed region of Four Dimensions, shall we linger on the threshold of the Fifth, and not enter therein? Ah, no! Let us rather resolve that our ambition shall soar with our corporal ascent. Then, yielding to our intellectual onset, the gates of the Sixth Dimension shall fly open; after that a Seventh, and then an Eighth— How long I should have continued I know not. In vain did the Sphere, in his voice of thunder, reiterate his command of silence, and threaten me with the direst penalties if I persisted. Nothing could stem the flood of my ecstatic aspirations. Perhaps I was to blame; but indeed I was intoxicated with the recent draughts of Truth to which he himself had introduced me. However, the end was not long in coming. My words were cut short by a crash outside, and a simultaneous crash inside me, which impelled me through space with a velocity that precluded speech. Down! down! down! I was rapidly descending; and I knew that return to Flatland was my doom. One glimpse, one last and never-to-be-forgotten glimpse I had of that dull level wilderness—which was now to become my Universe again—spread out before my eye. Then a darkness. Then a final, all-consummating thunder-peal; and, when I came to myself, I was once more a common creeping Square, in my Study at home, listening to the Peace-Cry of my approaching Wife.

I. Is that what they say? Oh, do not believe them. But if this other Space really is Thoughtland, then take me to that blessed Region. There, in Thought, I shall see the insides of all solid things. Before my delighted eye, a Cube shall move in some entirely new direction, but strictly according to Analogy. Every particle inside him shall pass through a new kind of Space, leaving a trail of its own. In this way, he shall create something even more perfect than himself. It shall have sixteen terminal Extra-solid angles and Eight solid Cubes forming its Perimeter. And once there, shall we stop our upward journey? In that blessed region of Four Dimensions, shall we linger at the entrance to the Fifth without entering? Ah, no! Let us instead resolve that our ambition shall rise as our bodies rise. Then the gates of the Sixth Dimension shall fly open before our intellectual assault. After that, a Seventh, and then an Eighth. How long I would have continued, I do not know. In vain, the Sphere repeated his command for silence in his thunderous voice. He threatened me with the harshest penalties if I continued. Nothing could stop the flood of my ecstatic hopes. Perhaps I was to blame. But I was truly drunk on the Truth he himself had so recently given me to drink. Still, the end soon came. A crash outside and another inside me cut off my words at the same instant. The force hurled me through space so fast that I could not speak. Down! down! down! I was falling rapidly, and I knew I was doomed to return to Flatland. I caught one last, unforgettable glimpse of that dull, flat wilderness spread out before my eye. It was now to become my Universe again. Then darkness. Then a final thunderclap that ended everything. When I came to my senses, I was once more an ordinary crawling Square in my Study at home. I was listening to the Peace-Cry of my approaching Wife.