Flatland: A Romance of Many Dimensions
I call our world Flatland, not because we call it so, but to make its
nature clearer to you, my happy readers, who are privileged to live in
Space.
Imagine a vast sheet of paper on which straight Lines, Triangles,
Squares, Pentagons, Hexagons, and other figures, instead of remaining
fixed in their places, move freely about, on or in the surface, but
without the power of rising above or sinking below it, very much like
shadows--only hard and with luminous edges--and you will then have a
pretty correct notion of my country and countrymen. Alas, a few years
ago, I should have said "my universe": but now my mind has been opened
to higher views of things.
In such a country, you will perceive at once that it is impossible that
there should be anything of what you call a "solid" kind; but I dare
say you will suppose that we could at least distinguish by sight the
Triangles, Squares, and other figures, moving about as I have described
them. On the contrary, we could see nothing of the kind, not at least
so as to distinguish one figure from another. Nothing was visible, nor
could be visible, to us, except Straight Lines; and the necessity of
this I will speedily demonstrate.
Place a penny on the middle of one of your tables in Space; and leaning
over it, look down upon it. It will appear a circle.
But now, drawing back to the edge of the table, gradually lower your
eye (thus bringing yourself more and more into the condition of the
inhabitants of Flatland), and you will find the penny becoming more and
more oval to your view, and at last when you have placed your eye
exactly on the edge of the table (so that you are, as it were, actually
a Flatlander) the penny will then have ceased to appear oval at all,
and will have become, so far as you can see, a straight line.
The same thing would happen if you were to treat in the same way a
Triangle, or Square, or any other figure cut out of pasteboard. As
soon as you look at it with your eye on the edge on the table, you will
find that it ceases to appear to you a figure, and that it becomes in
appearance a straight line. Take for example an equilateral
Triangle--who represents with us a Tradesman of the respectable class.
Fig. 1 represents the Tradesman as you would see him while you were
bending over him from above; figs. 2 and 3 represent the Tradesman, as
you would see him if your eye were close to the level, or all but on
the level of the table; and if your eye were quite on the level of the
table (and that is how we see him in Flatland) you would see nothing
but a straight line.
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When I was in Spaceland I heard that your sailors have very similar
experiences while they traverse your seas and discern some distant
island or coast lying on the horizon. The far-off land may have bays,
forelands, angles in and out to any number and extent; yet at a
distance you see none of these (unless indeed your sun shines bright
upon them revealing the projections and retirements by means of light
and shade), nothing but a grey unbroken line upon the water.
Well, that is just what we see when one of our triangular or other
acquaintances comes toward us in Flatland. As there is neither sun
with us, nor any light of such a kind as to make shadows, we have none
of the helps to the sight that you have in Spaceland. If our friend
comes closer to us we see his line becomes larger; if he leaves us it
becomes smaller: but still he looks like a straight line; be he a
Triangle, Square, Pentagon, Hexagon, Circle, what you will--a straight
Line he looks and nothing else.
You may perhaps ask how under these disadvantageous circumstances we
are able to distinguish our friends from one another: but the answer to
this very natural question will be more fitly and easily given when I
come to describe the inhabitants of Flatland. For the present let me
defer this subject, and say a word or two about the climate and houses
in our country.
Section 2. Of the Climate and Houses in Flatland
As with you, so also with us, there are four points of the compass
North, South, East, and West.
There being no sun nor other heavenly bodies, it is impossible for us
to determine the North in the usual way; but we have a method of our
own. By a Law of Nature with us, there is a constant attraction to the
South; and, although in temperate climates this is very slight--so that
even a Woman in reasonable health can journey several furlongs
northward without much difficulty--yet the hampering effect of the
southward attraction is quite sufficient to serve as a compass in most
parts of our earth. Moreover, the rain (which falls at stated
intervals) coming always from the North, is an additional assistance;
and in the towns we have the guidance of the houses, which of course
have their side-walls running for the most part North and South, so
that the roofs may keep off the rain from the North. In the country,
where there are no houses, the trunks of the trees serve as some sort
of guide. Altogether, we have not so much difficulty as might be
expected in determining our bearings.
Yet in our more temperate regions, in which the southward attraction is
hardly felt, walking sometimes in a perfectly desolate plain where
there have been no houses nor trees to guide me, I have been
occasionally compelled to remain stationary for hours together, waiting
till the rain came before continuing my journey. On the weak and aged,
and especially on delicate Females, the force of attraction tells much
more heavily than on the robust of the Male Sex, so that it is a point
of breeding, if you meet a Lady in the street, always to give her the
North side of the way--by no means an easy thing to do always at short
notice when you are in rude health and in a climate where it is
difficult to tell your North from your South.
Windows there are none in our houses: for the light comes to us alike
in our homes and out of them, by day and by night, equally at all times
and in all places, whence we know not. It was in old days, with our
learned men, an interesting and oft-investigated question, "What is the
origin of light?" and the solution of it has been repeatedly attempted,
with no other result than to crowd our lunatic asylums with the
would-be solvers. Hence, after fruitless attempts to suppress such
investigations indirectly by making them liable to a heavy tax, the
Legislature, in comparatively recent times, absolutely prohibited them.
I--alas, I alone in Flatland--know now only too well the true solution
of this mysterious problem; but my knowledge cannot be made
intelligible to a single one of my countrymen; and I am mocked at--I,
the sole possessor of the truths of Space and of the theory of the
introduction of Light from the world of three Dimensions--as if I were
the maddest of the mad! But a truce to these painful digressions: let
me return to our houses.
The most common form for the construction of a house is five-sided or
pentagonal, as in the annexed figure. The two Northern sides RO, OF,
constitute the roof, and for the most part have no doors; on the East
is a small door for the Women; on the West a much larger one for the
Men; the South side or floor is usually doorless.
Square and triangular houses are not allowed, and for this reason. The
angles of a Square (and still more those of an equilateral Triangle),
being much more pointed than those of a Pentagon, and the lines of
inanimate objects (such as houses) being dimmer than the lines of Men
and Women, it follows that there is no little danger lest the points of
a square or triangular house residence might do serious injury to an
inconsiderate or perhaps absent-minded traveller suddenly therefore,
running against them: and as early as the eleventh century of our era,
triangular houses were universally forbidden by Law, the only
exceptions being fortifications, powder-magazines, barracks, and other
state buildings, which it is not desirable that the general public
should approach without circumspection.
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At this period, square houses were still everywhere permitted, though
discouraged by a special tax. But, about three centuries afterwards,
the Law decided that in all towns containing a population above ten
thousand, the angle of a Pentagon was the smallest house-angle that
could be allowed consistently with the public safety. The good sense
of the community has seconded the efforts of the Legislature; and now,
even in the country, the pentagonal construction has superseded every
other. It is only now and then in some very remote and backward
agricultural district that an antiquarian may still discover a square
house.
Section 3. Concerning the Inhabitants of Flatland
The greatest length or breadth of a full grown inhabitant of Flatland
may be estimated at about eleven of your inches. Twelve inches may be
regarded as a maximum.
Our Women are Straight Lines.
Our Soldiers and Lowest Classes of Workmen are Triangles with two equal
sides, each about eleven inches long, and a base or third side so short
(often not exceeding half an inch) that they form at their vertices a
very sharp and formidable angle. Indeed when their bases are of the
most degraded type (not more than the eighth part of an inch in size),
they can hardly be distinguished from Straight Lines or Women; so
extremely pointed are their vertices. With us, as with you, these
Triangles are distinguished from others by being called Isosceles; and
by this name I shall refer to them in the following pages.
Our Middle Class consists of Equilateral or Equal-Sided Triangles.
Our Professional Men and Gentlemen are Squares (to which class I myself
belong) and Five-Sided Figures or Pentagons.
Next above these come the Nobility, of whom there are several degrees,
beginning at Six-Sided Figures, or Hexagons, and from thence rising in
the number of their sides till they receive the honourable title of
Polygonal, or many-sided. Finally when the number of the sides becomes
so numerous, and the sides themselves so small, that the figure cannot
be distinguished from a circle, he is included in the Circular or
Priestly order; and this is the highest class of all.
It is a Law of Nature with us that a male child shall have one more
side than his father, so that each generation shall rise (as a rule)
one step in the scale of development and nobility. Thus the son of a
Square is a Pentagon; the son of a Pentagon, a Hexagon; and so on.
But this rule applies not always to the Tradesmen, and still less often
to the Soldiers, and to the Workmen; who indeed can hardly be said to
deserve the name of human Figures, since they have not all their sides
equal. With them therefore the Law of Nature does not hold; and the
son of an Isosceles (i.e. a Triangle with two sides equal) remains
Isosceles still. Nevertheless, all hope is not shut out, even from the
Isosceles, that his posterity may ultimately rise above his degraded