A Romance of Many Dimensions, by Edwin A. Abbot

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My body just before I disappeared
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Lineland ----> |\ \ \ \ \| The King
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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.

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."

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."

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.

Section 15. Concerning a Stranger from Spaceland

From dreams I proceed to facts.

It was the last day of the 1999th year of our era.
The pattering of the rain had long ago announced nightfall;
and I was sitting in the company of my wife, musing on the events
of the past and the prospects of the coming year, the coming century,
the coming Millennium.

[Note: When I say "sitting", of course I do not mean
any change of attitude such as you in Spaceland signify by that word;
for as we have no feet, we can no more "sit" nor "stand"
(in your sense of the word) than one of your soles or flounders.

Nevertheless, we perfectly well recognize the different mental states
of volition implied in "lying", "sitting", and "standing",
which are to some extent indicated to a beholder by a slight
increase of lustre corresponding to the increase of volition.

But on this, and a thousand other kindred subjects, time forbids me
to dwell.]

My four Sons and two orphan Grandchildren had retired
to their several apartments; and my wife alone remained with me
to see the old Millennium out and the new one in.

I was rapt in thought, pondering in my mind some words that had
casually issued from the mouth of my youngest Grandson,
a most promising young Hexagon of unusual brilliancy
and perfect angularity. His uncles and I had been giving him
his usual practical lesson in Sight Recognition, turning ourselves
upon our centres, now rapidly, now more slowly, and questioning him
as to our positions; and his answers had been so satisfactory
that I had been induced to reward him by giving him a few hints
on Arithmetic, as applied to Geometry.

Taking nine Squares, each an inch every way, I had put them together
so as to make one large Square, with a side of three inches,
and I had hence proved to my little Grandson that -- though it was
impossible for us to SEE the inside of the Square --
yet we might ascertain the number of square inches in a Square
by simply squaring the number of inches in the side: "and thus,"
said I, "we know that 3^2, or 9, represents the number
of square inches in a Square whose side is 3 inches long."

The little Hexagon meditated on this a while and then said to me;
"But you have been teaching me to raise numbers to the third power:
I suppose 3^3 must mean something in Geometry; what does it mean?"
"Nothing at all," replied I, "not at least in Geometry;
for Geometry has only Two Dimensions." And then I began
to shew the boy how a Point by moving through a length of three inches
makes a Line of three inches, which may be represented by 3;
and how a Line of three inches, moving parallel to itself through
a length of three inches, makes a Square of three inches every way,
which may be represented by 3^2.

Upon this, my Grandson, again returning to his former suggestion,
took me up rather suddenly and exclaimed, "Well, then,
if a Point by moving three inches, makes a Line of three inches
represented by 3; and if a straight Line of three inches,
moving parallel to itself, makes a Square of three inches every way,
represented by 3^2; it must be that a Square of three inches
every way, moving somehow parallel to itself (but I don't see how)
must make Something else (but I don't see what) of three inches
every way -- and this must be represented by 3^3."

"Go to bed," said I, a little ruffled by this interruption:
"if you would talk less nonsense, you would remember more sense."

So my Grandson had disappeared in disgrace; and there I sat
by my Wife's side, endeavouring to form a retrospect of the year 1999
and of the possibilities of the year 2000, but not quite able
to shake off the thoughts suggested by the prattle of my bright
little Hexagon. Only a few sands now remained in the half-hour glass.
Rousing myself from my reverie I turned the glass Northward
for the last time in the old Millennium; and in the act,
I exclaimed aloud, "The boy is a fool."

Straightway I became conscious of a Presence in the room,
and a chilling breath thrilled through my very being.
"He is no such thing," cried my Wife, "and you are breaking
the Commandments in thus dishonouring your own Grandson."
But I took no notice of her. Looking round in every direction
I could see nothing; yet still I FELT a Presence, and shivered
as the cold whisper came again. I started up. "What is the matter?"
said my Wife, "there is no draught; what are you looking for?
There is nothing." There was nothing; and I resumed my seat,
again exclaiming, "The boy is a fool, I say; 3^3 can have no meaning
in Geometry." At once there came a distinctly audible reply,
"The boy is not a fool; and 3^3 has an obvious Geometrical meaning."

My Wife as well as myself heard the words, although she did not
understand their meaning, and both of us sprang forward
in the direction of the sound. What was our horror when we saw
before us a Figure! At the first glance it appeared to be a Woman,
seen sideways; but a moment's observation shewed me that
the extremities passed into dimness too rapidly to represent
one of the Female Sex; and I should have thought it a Circle,
only that it seemed to change its size in a manner impossible
for a Circle or for any regular Figure of which I had had experience.

But my Wife had not my experience, nor the coolness necessary to note
these characteristics. With the usual hastiness and unreasoning
jealousy of her Sex, she flew at once to the conclusion
that a Woman had entered the house through some small aperture.
"How comes this person here?" she exclaimed, "you promised me,
my dear, that there should be no ventilators in our new house."
"Nor are there any," said I; "but what makes you think that
the stranger is a Woman? I see by my power of Sight Recognition ----"
"Oh, I have no patience with your Sight Recognition," replied she,
"'Feeling is believing' and 'A Straight Line to the touch is worth
a Circle to the sight'" -- two Proverbs, very common
with the Frailer Sex in Flatland.

"Well," said I, for I was afraid of irritating her, "if it must be so,
demand an introduction." Assuming her most gracious manner,
my Wife advanced towards the Stranger, "Permit me, Madam,
to feel and be felt by ----" then, suddenly recoiling, "Oh!
it is not a Woman, and there are no angles either, not a trace of one.
Can it be that I have so misbehaved to a perfect Circle?"

"I am indeed, in a certain sense a Circle," replied the Voice,
"and a more perfect Circle than any in Flatland; but to speak
more accurately, I am many Circles in one." Then he added
more mildly, "I have a message, dear Madam, to your husband,
which I must not deliver in your presence; and, if you would suffer us
to retire for a few minutes ----" But my Wife would not listen
to the proposal that our august Visitor should so incommode himself,
and assuring the Circle that the hour of her own retirement
had long passed, with many reiterated apologies for her
recent indiscretion, she at last retreated to her apartment.

I glanced at the half-hour glass. The last sands had fallen.
The third Millennium had begun.

Section 16. How the Stranger vainly endeavoured to reveal to me
in words the mysteries of Spaceland

As soon as the sound of the Peace-cry of my departing Wife
had died away, I began to approach the Stranger with the intention
of taking a nearer view and of bidding him be seated:
but his appearance struck me dumb and motionless with astonishment.
Without the slightest symptoms of angularity he nevertheless varied
every instant with gradations of size and brightness scarcely possible
for any Figure within the scope of my experience. The thought
flashed across me that I might have before me a burglar or cut-throat,
some monstrous Irregular Isosceles, who, by feigning the voice
of a Circle, had obtained admission somehow into the house,
and was now preparing to stab me with his acute angle.

In a sitting-room, the absence of Fog (and the season happened
to be remarkably dry), made it difficult for me to trust to
Sight Recognition, especially at the short distance at which
I was standing. Desperate with fear, I rushed forward
with an unceremonious, "You must permit me, Sir --" and felt him.
My Wife was right. There was not the trace of an angle,
not the slightest roughness or inequality: never in my life had I met
with a more perfect Circle. He remained motionless while I walked
round him, beginning from his eye and returning to it again.
Circular he was throughout, a perfectly satisfactory Circle;
there could not be a doubt of it. Then followed a dialogue,
which I will endeavour to set down as near as I can recollect it,
omitting only some of my profuse apologies -- for I was covered
with shame and humiliation that I, a Square, should have been guilty
of the impertinence of feeling a Circle. It was commenced
by the Stranger with some impatience at the lengthiness
of my introductory process.

STRANGER. Have you felt me enough by this time? Are you not
introduced to me yet?

I. Most illustrious Sir, excuse my awkwardness, which arises not
from ignorance of the usages of polite society, but from a little
surprise and nervousness, consequent on this somewhat
unexpected visit. And I beseech you to reveal my indiscretion
to no one, and especially not to my Wife. But before your Lordship
enters into further communications, would he deign to satisfy
the curiosity of one who would gladly know whence his Visitor came?

STRANGER. From Space, from Space, Sir: whence else?

I. Pardon me, my Lord, but is not your Lordship already in Space,
your Lordship and his humble servant, even at this moment?

STRANGER. Pooh! what do you know of Space? Define Space.

I. Space, my Lord, is height and breadth indefinitely prolonged.

STRANGER. Exactly: you see you do not even know what Space is.
You think it is of Two Dimensions only; but I have come
to announce to you a Third -- height, breadth, and length.

I. Your Lordship is pleased to be merry. We also speak
of length and height, or breadth and thickness, thus denoting
Two Dimensions by four names.

STRANGER. But I mean not only three names, but Three Dimensions.

I. Would your Lordship indicate or explain to me in what direction
is the Third Dimension, unknown to me?

STRANGER. I came from it. It is up above and down below.

I. My Lord means seemingly that it is Northward and Southward.

STRANGER. I mean nothing of the kind. I mean a direction in which
you cannot look, because you have no eye in your side.

I. Pardon me, my Lord, a moment's inspection will convince
your Lordship that I have a perfect luminary at the juncture of two
of my sides.

STRANGER. Yes: but in order to see into Space you ought to have
an eye, not on your Perimeter, but on your side, that is,
on what you would probably call your inside; but we in Spaceland
should call it your side.

I. An eye in my inside! An eye in my stomach! Your Lordship jests.

STRANGER. I am in no jesting humour. I tell you that
I come from Space, or, since you will not understand what Space means,
from the Land of Three Dimensions whence I but lately looked down
upon your Plane which you call Space forsooth. From that position
of advantage I discerned all that you speak of as SOLID
(by which you mean "enclosed on four sides"), your houses,
your churches, your very chests and safes, yes even your insides
and stomachs, all lying open and exposed to my view.

I. Such assertions are easily made, my Lord.

STRANGER. But not easily proved, you mean. But I mean to prove mine.

When I descended here, I saw your four Sons, the Pentagons,
each in his apartment, and your two Grandsons the Hexagons;
I saw your youngest Hexagon remain a while with you and then
retire to his room, leaving you and your Wife alone.
I saw your Isosceles servants, three in number, in the kitchen
at supper, and the little Page in the scullery. Then I came here,
and how do you think I came?

I. Through the roof, I suppose.

STRANGER. Not so. Your roof, as you know very well,
has been recently repaired, and has no aperture by which even a Woman
could penetrate. I tell you I come from Space. Are you not convinced
by what I have told you of your children and household?

I. Your Lordship must be aware that such facts touching
the belongings of his humble servant might be easily ascertained
by any one in the neighbourhood possessing your Lordship's
ample means of obtaining information.

STRANGER. (TO HIMSELF.) What must I do? Stay; one more argument
suggests itself to me. When you see a Straight Line -- your wife,
for example -- how many Dimensions do you attribute to her?

I. Your Lordship would treat me as if I were one of the vulgar who,
being ignorant of Mathematics, suppose that a Woman is really
a Straight Line, and only of One Dimension. No, no, my Lord;
we Squares are better advised, and are as well aware as your Lordship
that a Woman, though popularly called a Straight Line, is,
really and scientifically, a very thin Parallelogram,
possessing Two Dimensions, like the rest of us, viz.,
length and breadth (or thickness).

STRANGER. But the very fact that a Line is visible implies
that it possesses yet another Dimension.

I. My Lord, I have just acknowledged that a Woman is broad
as well as long. We see her length, we infer her breadth;
which, though very slight, is capable of measurement.

STRANGER. You do not understand me. I mean that when you see
a Woman, you ought -- besides inferring her breadth --
to see her length, and to SEE what we call her HEIGHT;
although that last Dimension is infinitesimal in your country.
If a Line were mere length without "height", it would cease to
occupy Space and would become invisible. Surely you must
recognize this?

I. I must indeed confess that I do not in the least
understand your Lordship. When we in Flatland see a Line,
we see length and BRIGHTNESS. If the brightness disappears,
the Line is extinguished, and, as you say, ceases to occupy Space.
But am I to suppose that your Lordship gives to brightness the title
of a Dimension, and that what we call "bright" you call "high"?

STRANGER. No, indeed. By "height" I mean a Dimension like
your length: only, with you, "height" is not so easily perceptible,
being extremely small.

I. My Lord, your assertion is easily put to the test.
You say I have a Third Dimension, which you call "height".
Now, Dimension implies direction and measurement. Do but measure
my "height", or merely indicate to me the direction in which
my "height" extends, and I will become your convert. Otherwise,
your Lordship's own understanding must hold me excused.

STRANGER. (TO HIMSELF.) I can do neither. How shall I
convince him? Surely a plain statement of facts followed by
ocular demonstration ought to suffice. -- Now, Sir; listen to me.

You are living on a Plane. What you style Flatland is
the vast level surface of what I may call a fluid, on, or in,
the top of which you and your countrymen move about,
without rising above it or falling below it.

I am not a plane Figure, but a Solid. You call me a Circle;
but in reality I am not a Circle, but an infinite number of Circles,
of size varying from a Point to a Circle of thirteen inches
in diameter, one placed on the top of the other. When I cut through
your plane as I am now doing, I make in your plane a section
which you, very rightly, call a Circle. For even a Sphere --
which is my proper name in my own country -- if he manifest himself
at all to an inhabitant of Flatland -- must needs manifest himself
as a Circle.

Do you not remember -- for I, who see all things, discerned last night
the phantasmal vision of Lineland written upon your brain --
do you not remember, I say, how, when you entered the realm
of Lineland, you were compelled to manifest yourself to the King,
not as a Square, but as a Line, because that Linear Realm had not
Dimensions enough to represent the whole of you, but only a slice
or section of you? In precisely the same way, your country
of Two Dimensions is not spacious enough to represent me,
a being of Three, but can only exhibit a slice or section of me,
which is what you call a Circle.

The diminished brightness of your eye indicates incredulity. But now
prepare to receive proof positive of the truth of my assertions.
You cannot indeed see more than one of my sections, or Circles,
at a time; for you have no power to raise your eye out of the plane
of Flatland; but you can at least see that, as I rise in Space,
so my sections become smaller. See now, I will rise; and the effect
upon your eye will be that my Circle will become smaller and smaller
till it dwindles to a point and finally vanishes.

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The Sphere on the
point of vanishing
(2) __-----__
The Sphere with The Sphere rising / \ (3)
his section __-----__ / \
at full size / \ | |
__-----__ / \ | |
/ \ | | | |
/ __ - __ \ | | \ / My
| -- -- | | __ --- __ | \ __ __ / Eye
--|-----------------|----\--__-------__--/------------===----------+(>
| -- __ __ -- | \ __ --- __ /
\ - / -----
\ __ __ /
-----

There was no "rising" that I could see; but he diminished
and finally vanished. I winked once or twice to make sure
that I was not dreaming. But it was no dream. For from the depths
of nowhere came forth a hollow voice -- close to my heart it seemed --
"Am I quite gone? Are you convinced now? Well, now I will
gradually return to Flatland and you shall see my section become
larger and larger."

Every reader in Spaceland will easily understand that
my mysterious Guest was speaking the language of truth
and even of simplicity. But to me, proficient though I was
in Flatland Mathematics, it was by no means a simple matter.
The rough diagram given above will make it clear to any
Spaceland child that the Sphere, ascending in the three positions
indicated there, must needs have manifested himself to me,
or to any Flatlander, as a Circle, at first of full size, then small,
and at last very small indeed, approaching to a Point. But to me,
although I saw the facts before me, the causes were as dark as ever.
All that I could comprehend was, that the Circle had made himself
smaller and vanished, and that he had now reappeared and was rapidly
making himself larger.

When he regained his original size, he heaved a deep sigh;
for he perceived by my silence that I had altogether failed
to comprehend him. And indeed I was now inclining to the belief
that he must be no Circle at all, but some extremely clever juggler;
or else that the old wives' tales were true, and that after all
there were such people as Enchanters and Magicians.

After a long pause he muttered to himself, "One resource alone remains,
if I am not to resort to action. I must try the method of Analogy."
Then followed a still longer silence, after which he continued
our dialogue.

SPHERE. Tell me, Mr. Mathematician; if a Point moves Northward,
and leaves a luminous wake, what name would you give to the wake?

I. A straight Line.

SPHERE. And a straight Line has how many extremities?

I. Two.

SPHERE. Now conceive the Northward straight Line moving parallel
to itself, East and West, so that every point in it leaves behind it
the wake of a straight Line. What name will you give to the Figure
thereby formed? We will suppose that it moves through a distance
equal to the original straight Line. -- What name, I say?

I. A Square.

SPHERE. And how many sides has a Square? How many angles?

I. Four sides and four angles.

SPHERE. Now stretch your imagination a little, and conceive
a Square in Flatland, moving parallel to itself upward.

I. What? Northward?

SPHERE. No, not Northward; upward; out of Flatland altogether.

If it moved Northward, the Southern points in the Square would have to
move through the positions previously occupied by the Northern points.
But that is not my meaning.

I mean that every Point in you -- for you are a Square and will serve
the purpose of my illustration -- every Point in you, that is to say
in what you call your inside, is to pass upwards through Space
in such a way that no Point shall pass through the position
previously occupied by any other Point; but each Point shall describe
a straight Line of its own. This is all in accordance with Analogy;
surely it must be clear to you.

Restraining my impatience -- for I was now under a strong temptation
to rush blindly at my Visitor and to precipitate him into Space,
or out of Flatland, anywhere, so that I could get rid of him --
I replied: --

"And what may be the nature of the Figure which I am to shape out
by this motion which you are pleased to denote by the word 'upward'?
I presume it is describable in the language of Flatland."

SPHERE. Oh, certainly. It is all plain and simple,
and in strict accordance with Analogy -- only, by the way,
you must not speak of the result as being a Figure, but as a Solid.
But I will describe it to you. Or rather not I, but Analogy.

We began with a single Point, which of course -- being itself a Point
-- has only ONE terminal Point.

One Point produces a Line with TWO terminal Points.

One Line produces a Square with FOUR terminal Points.

Now you can give yourself the answer to your own question: 1, 2, 4,
are evidently in Geometrical Progression. What is the next number?

I. Eight.

SPHERE. Exactly. The one Square produces a SOMETHING-WHICH-
YOU-DO-NOT-AS-YET-KNOW-A-NAME-FOR-BUT-WHICH-WE-CALL-A-CUBE
with EIGHT terminal Points. Now are you convinced?

I. And has this Creature sides, as well as angles or what you call
"terminal Points"?

SPHERE. Of course; and all according to Analogy. But, by the way,
not what YOU call sides, but what WE call sides.
You would call them SOLIDS.

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