A Romance of Many Dimensions, by Edwin A. Abbot

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I. And how many solids or sides will appertain to this Being whom
I am to generate by the motion of my inside in an "upward" direction,
and whom you call a Cube?

SPHERE. How can you ask? And you a mathematician!
The side of anything is always, if I may so say, one Dimension behind
the thing. Consequently, as there is no Dimension behind a Point,
a Point has 0 sides; a Line, if I may say, has 2 sides
(for the Points of a Line may be called by courtesy, its sides);
a Square has 4 sides; 0, 2, 4; what Progression do you call that?

I. Arithmetical.

SPHERE. And what is the next number?

I. Six.

SPHERE. Exactly. Then you see you have answered your own question.
The Cube which you will generate will be bounded by six sides,
that is to say, six of your insides. You see it all now, eh?

"Monster," I shrieked, "be thou juggler, enchanter, dream, or devil,
no more will I endure thy mockeries. Either thou or I must perish."
And saying these words I precipitated myself upon him.

Section 17. How the Sphere, having in vain tried words,
resorted to deeds

It was in vain. I brought my hardest right angle into violent
collision with the Stranger, pressing on him with a force sufficient
to have destroyed any ordinary Circle: but I could feel him
slowly and unarrestably slipping from my contact; no edging to
the right nor to the left, but moving somehow out of the world,
and vanishing to nothing. Soon there was a blank. But still I heard
the Intruder's voice.

SPHERE. Why will you refuse to listen to reason?
I had hoped to find in you -- as being a man of sense
and an accomplished mathematician -- a fit apostle for the Gospel
of the Three Dimensions, which I am allowed to preach once only
in a thousand years: but now I know not how to convince you.
Stay, I have it. Deeds, and not words, shall proclaim the truth.
Listen, my friend.

I have told you I can see from my position in Space the inside
of all things that you consider closed. For example,
I see in yonder cupboard near which you are standing,
several of what you call boxes (but like everything else in Flatland,
they have no tops nor bottoms) full of money; I see also
two tablets of accounts. I am about to descend into that cupboard
and to bring you one of those tablets. I saw you lock the cupboard
half an hour ago, and I know you have the key in your possession.
But I descend from Space; the doors, you see, remain unmoved.
Now I am in the cupboard and am taking the tablet. Now I have it.
Now I ascend with it.

I rushed to the closet and dashed the door open. One of the tablets
was gone. With a mocking laugh, the Stranger appeared
in the other corner of the room, and at the same time the tablet
appeared upon the floor. I took it up. There could be no doubt --
it was the missing tablet.

I groaned with horror, doubting whether I was not out of my senses;
but the Stranger continued: "Surely you must now see
that my explanation, and no other, suits the phenomena. What you call
Solid things are really superficial; what you call Space is really
nothing but a great Plane. I am in Space, and look down upon
the insides of the things of which you only see the outsides.
You could leave this Plane yourself, if you could but summon up
the necessary volition. A slight upward or downward motion
would enable you to see all that I can see.

"The higher I mount, and the further I go from your Plane,
the more I can see, though of course I see it on a smaller scale.
For example, I am ascending; now I can see your neighbour the Hexagon
and his family in their several apartments; now I see
the inside of the Theatre, ten doors off, from which the audience
is only just departing; and on the other side a Circle in his study,
sitting at his books. Now I shall come back to you.
And, as a crowning proof, what do you say to my giving you a touch,
just the least touch, in your stomach? It will not seriously
injure you, and the slight pain you may suffer cannot be compared with
the mental benefit you will receive."

Before I could utter a word of remonstrance, I felt a shooting pain
in my inside, and a demoniacal laugh seemed to issue from within me.
A moment afterwards the sharp agony had ceased, leaving nothing but
a dull ache behind, and the Stranger began to reappear, saying,
as he gradually increased in size, "There, I have not hurt you much,
have I? If you are not convinced now, I don't know what will
convince you. What say you?"

My resolution was taken. It seemed intolerable that I should endure
existence subject to the arbitrary visitations of a Magician who could
thus play tricks with one's very stomach. If only I could in any way
manage to pin him against the wall till help came!

Once more I dashed my hardest angle against him, at the same time
alarming the whole household by my cries for aid. I believe,
at the moment of my onset, the Stranger had sunk below our Plane,
and really found difficulty in rising. In any case
he remained motionless, while I, hearing, as I thought,
the sound of some help approaching, pressed against him
with redoubled vigour, and continued to shout for assistance.

A convulsive shudder ran through the Sphere. "This must not be,"
I thought I heard him say: "either he must listen to reason,
or I must have recourse to the last resource of civilization."
Then, addressing me in a louder tone, he hurriedly exclaimed,
"Listen: no stranger must witness what you have witnessed.
Send your Wife back at once, before she enters the apartment.
The Gospel of Three Dimensions must not be thus frustrated.
Not thus must the fruits of one thousand years of waiting
be thrown away. I hear her coming. Back! back! Away from me,
or you must go with me -- whither you know not -- into the Land
of Three Dimensions!"

"Fool! Madman! Irregular!" I exclaimed; "never will I release thee;
thou shalt pay the penalty of thine impostures."

"Ha! Is it come to this?" thundered the Stranger: "then meet
your fate: out of your Plane you go. Once, twice, thrice!
'Tis done!"

Section 18. How I came to Spaceland, and what I saw there

An unspeakable horror seized me. There was a darkness;
then a dizzy, sickening sensation of sight that was not like seeing;
I saw a Line that was no Line; Space that was not Space:
I was myself, and not myself. When I could find voice,
I shrieked aloud in agony, "Either this is madness or it is Hell."
"It is neither," calmly replied the voice of the Sphere,
"it is Knowledge; it is Three Dimensions: open your eye once again
and try to look steadily."

I looked, and, behold, a new world! There stood before me,
visibly incorporate, all that I had before inferred, conjectured,
dreamed, of perfect Circular beauty. What seemed the centre
of the Stranger's form lay open to my view: yet I could see no heart,
nor lungs, nor arteries, only a beautiful harmonious Something --
for which I had no words; but you, my Readers in Spaceland,
would call it the surface of the Sphere.

Prostrating myself mentally before my Guide, I cried, "How is it,
O divine ideal of consummate loveliness and wisdom that I see
thy inside, and yet cannot discern thy heart, thy lungs, thy arteries,
thy liver?" "What you think you see, you see not," he replied;
"it is not given to you, nor to any other Being to behold
my internal parts. I am of a different order of Beings from those
in Flatland. Were I a Circle, you could discern my intestines,
but I am a Being, composed as I told you before, of many Circles,
the Many in the One, called in this country a Sphere. And,
just as the outside of a Cube is a Square, so the outside of a Sphere
presents the appearance of a Circle."

Bewildered though I was by my Teacher's enigmatic utterance,
I no longer chafed against it, but worshipped him in silent adoration.
He continued, with more mildness in his voice. "Distress not yourself
if you cannot at first understand the deeper mysteries of Spaceland.
By degrees they will dawn upon you. Let us begin by casting back
a glance at the region whence you came. Return with me a while
to the plains of Flatland, and I will shew you that which
you have often reasoned and thought about, but never seen
with the sense of sight -- a visible angle." "Impossible!" I cried;
but, the Sphere leading the way, I followed as if in a dream,
till once more his voice arrested me: "Look yonder,
and behold your own Pentagonal house, and all its inmates."

I looked below, and saw with my physical eye all that
domestic individuality which I had hitherto merely inferred
with the understanding. And how poor and shadowy was the inferred
conjecture in comparison with the reality which I now beheld!
My four Sons calmly asleep in the North-Western rooms,
my two orphan Grandsons to the South; the Servants, the Butler,
my Daughter, all in their several apartments. Only my
affectionate Wife, alarmed by my continued absence, had quitted
her room and was roving up and down in the Hall, anxiously awaiting
my return. Also the Page, aroused by my cries, had left his room,
and under pretext of ascertaining whether I had fallen
somewhere in a faint, was prying into the cabinet in my study.
All this I could now SEE, not merely infer; and as we came
nearer and nearer, I could discern even the contents of my cabinet,
and the two chests of gold, and the tablets of which the Sphere
had made mention.

<>

<>

/\
/ |My \
/ <> |Study \
/______ | ___ \
/ <> My Sons\ \|The \
/______/ \ Page / \
N / <> \ / My \
^ /______/ THE HALL \ Bedroom \
| \ <> My\ /
| \____| /\Wife's /
W--+--E \ My Wife / Apartment/
| ------- /\ --- \ WOMEN'S DOOR
| MEN'S DOOR \My Daughter
| /\ --== \ / The Scullion
S \ My Grandsons \ -==# \/ The Footman
\___ ___ _ _/ \-=#|/ The Butler
\ <> | <> | |THE CELLAR \ /
\____|____|_|____________/

###===--- ---===###
Policeman Policeman

Touched by my Wife's distress, I would have sprung downward
to reassure her, but I found myself incapable of motion.
"Trouble not yourself about your Wife," said my Guide:
"she will not be long left in anxiety; meantime, let us take
a survey of Flatland."

Once more I felt myself rising through space. It was even as
the Sphere had said. The further we receded from the object
we beheld, the larger became the field of vision. My native city,
with the interior of every house and every creature therein,
lay open to my view in miniature. We mounted higher, and lo,
the secrets of the earth, the depths of mines and inmost caverns
of the hills, were bared before me.

Awestruck at the sight of the mysteries of the earth,
thus unveiled before my unworthy eye, I said to my Companion,
"Behold, I am become as a God. For the wise men in our country say
that to see all things, or as they express it, OMNIVIDENCE,
is the attribute of God alone." There was something of scorn
in the voice of my Teacher as he made answer: "Is it so indeed?
Then the very pick-pockets and cut-throats of my country
are to be worshipped by your wise men as being Gods:
for there is not one of them that does not see as much as you see now.
But trust me, your wise men are wrong."

I. Then is omnividence the attribute of others besides Gods?

SPHERE. I do not know. But, if a pick-pocket or a cut-throat
of our country can see everything that is in your country,
surely that is no reason why the pick-pocket or cut-throat should be
accepted by you as a God. This omnividence, as you call it --
it is not a common word in Spaceland -- does it make you more just,
more merciful, less selfish, more loving? Not in the least.
Then how does it make you more divine?

I. "More merciful, more loving!" But these are the qualities
of women! And we know that a Circle is a higher Being
than a Straight Line, in so far as knowledge and wisdom
are more to be esteemed than mere affection.

SPHERE. It is not for me to classify human faculties according
to merit. Yet many of the best and wisest in Spaceland think more
of the affections than of the understanding, more of your despised
Straight Lines than of your belauded Circles. But enough of this.
Look yonder. Do you know that building?

I looked, and afar off I saw an immense Polygonal structure, in which
I recognized the General Assembly Hall of the States of Flatland,
surrounded by dense lines of Pentagonal buildings at right angles
to each other, which I knew to be streets; and I perceived that
I was approaching the great Metropolis.

"Here we descend," said my Guide. It was now morning,
the first hour of the first day of the two thousandth year of our era.
Acting, as was their wont, in strict accordance with precedent,
the highest Circles of the realm were meeting in solemn conclave,
as they had met on the first hour of the first day of the year 1000,
and also on the first hour of the first day of the year 0.

The minutes of the previous meetings were now read by one whom I
at once recognized as my brother, a perfectly Symmetrical Square,
and the Chief Clerk of the High Council. It was found recorded
on each occasion that: "Whereas the States had been troubled
by divers ill-intentioned persons pretending to have received
revelations from another World, and professing to produce
demonstrations whereby they had instigated to frenzy both themselves
and others, it had been for this cause unanimously resolved
by the Grand Council that on the first day of each millenary,
special injunctions be sent to the Prefects in the several districts
of Flatland, to make strict search for such misguided persons,
and without formality of mathematical examination, to destroy all such
as were Isosceles of any degree, to scourge and imprison
any regular Triangle, to cause any Square or Pentagon to be sent
to the district Asylum, and to arrest any one of higher rank,
sending him straightway to the Capital to be examined and judged
by the Council."

"You hear your fate," said the Sphere to me, while the Council
was passing for the third time the formal resolution.
"Death or imprisonment awaits the Apostle of the Gospel
of Three Dimensions." "Not so," replied I, "the matter is now
so clear to me, the nature of real space so palpable, that methinks
I could make a child understand it. Permit me but to descend
at this moment and enlighten them." "Not yet," said my Guide,
"the time will come for that. Meantime I must perform my mission.
Stay thou there in thy place." Saying these words,
he leaped with great dexterity into the sea (if I may so call it)
of Flatland, right in the midst of the ring of Counsellors. "I come,"
cried he, "to proclaim that there is a land of Three Dimensions."

I could see many of the younger Counsellors start back
in manifest horror, as the Sphere's circular section widened
before them. But on a sign from the presiding Circle
-- who shewed not the slightest alarm or surprise -- six Isosceles
of a low type from six different quarters rushed upon the Sphere.
"We have him," they cried; "No; yes; we have him still! he's going!
he's gone!"

"My Lords," said the President to the Junior Circles of the Council,
"there is not the slightest need for surprise; the secret archives,
to which I alone have access, tell me that a similar occurrence
happened on the last two millennial commencements. You will,
of course, say nothing of these trifles outside the Cabinet."

Raising his voice, he now summoned the guards. "Arrest the policemen;
gag them. You know your duty." After he had consigned to their fate
the wretched policemen -- ill-fated and unwilling witnesses
of a State-secret which they were not to be permitted to reveal --
he again addressed the Counsellors. "My Lords, the business
of the Council being concluded, I have only to wish you
a happy New Year." Before departing, he expressed, at some length,
to the Clerk, my excellent but most unfortunate brother,
his sincere regret that, in accordance with precedent and for the sake
of secrecy, he must condemn him to perpetual imprisonment,
but added his satisfaction that, unless some mention were made by him
of that day's incident, his life would be spared.

Section 19. How, though the Sphere shewed me other mysteries
of Spaceland, I still desired more; and what came of it

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."

<>

<>

(1) (2)
__________ __________
|\ |\ | \
| \ | \ | \
| \ ____|____\ | \
| | | | | |
|_____|____| | | |
\ | \ | \ |
\ | \ | \ |
\|_________\| \ __________|

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."

"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."

"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."

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.

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.

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.

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.

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.

"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."

SPHERE. My what?

I. Thine interior: thy stomach, thy intestines.

SPHERE. Whence this ill-timed impertinent request? And what
mean you by saying that 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.

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.

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.

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?

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.

SPHERE. But where is this land of Four Dimensions?

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

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

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.

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?

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