Flatland: A Romance of Many Dimensions
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.
[ASCII approximation follows]
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.
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