Flatland (fifth edition)

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Flatland (fifth edition) (1963)
by Edwin Abbott Abbott
3974888Flatland (fifth edition)1963William Garnett


A Romance of Many Dimensions

With Illustrations
by the Author,
With Introduction by

[Fifth Edition, Revised]

"Fie, fie how franticly I square my talk!"



New York, Hagerstown, San Francisco, London

Published in the United States
in 1963
by Barnes & Noble, Inc.
by special arrangement with
Basil Blackwell
Oxford, England

L. C. Catalogue Card Number: 63-12454

Hardbound Edition SBN 0389 01531 8
Paperback Edition SBN 0389 00245 3

76 77 78 79 8012 11 10 9 8 7 6 5

Printed in the United States of America


The Inhabitants of Space in General

And H. C. in Particular

This Work is Dedicated

By a Humble Native of Flatland

In the Hope that

Even as he was Initiated into the Mysteries

Of Three Dimensions

Having been previously conversant

With Only Two

So the Citizens of that Celestial Region

May aspire yet higher and higher

To the Secrets of Four Five or even Six Dimensions

Thereby contributing

To the Enlargement of the Imagination

And the possible Development

Of that most rare and excellent Gift of Modesty

Among the Superior Races

Of Solid Humanity


In an address to the Committee of the Cayley Portrait Fund in 1874 Clerk Maxwell, after referring in humorous terms to the work of Arthur Cayley in higher algebra and algebraical geometry, concluded his eulogium with the lines—

March on, symbolic host! with step sublime,
Up to the flaming bounds of Space and Time!
There pause, until by Dickenson depicted,
In two dimensions, we the form may trace
Of him whose soul, too large for vulgar space
In n dimensions flourished unrestricted.

In those days any conception of "dimensions" beyond length, breadth and height was confined to advanced mathematicians; and even among them, with very few exceptions, the fourth and higher dimensions afforded only a field for the practice of algebraical analysis with four or more variables instead of the three which sufficiently describe the space to which our foot-rules are applicable. Any geometrical conclusions reached were regarded only as analogies to the corresponding results in geometry of three dimensions and not as having any bearing on the system of Nature. As an illustration, reference may be made to the "more divine offspring of the divine Cube in the Land of Four Dimensions" mentioned on p. 94 infra which has for its faces eight three-dimensional cubes and possesses sixteen four-dimensional angular points or comers.

During the present century the work of Einstein, Lorentz, Larmor, Whitehead and others has shewn that at least four dimensions of space-time are necessary to account for the observed phenomena of nature, and there are some suggestions of the necessity for more than four. It is only when dealing with very high velocities, such as are comparable with the velocity of light, that the unity of time with space thrusts itself upon the notice of physicists, for even with such a velocity as that of the planet Mercury in its orbit it is only after the lapse of centuries that any divergence from the motion strictly calculated on the basis of Euclidean Geometry and Newton's laws of gravitation and of motion has become apparent. The observed behaviour of electrons, moving in high vacua with velocities comparable with the velocity of light, has confirmed some of Einstein's conclusions and necessitated a revision of our fundamental notions of kinematics and the laws of motion when these high velocities are concerned. But the whole subject of Relativity has strongly appealed to popular interest through the brilliant confirmation of Einstein's theory of gravitation by the bending of light in passing close to the sun's surface and the consequent apparent displacement of stars which are very close to the sun from their true relative position when photographed during a solar eclipse. The best popular exposition of the whole subject of relativity and gravitation is to be found in Professor Eddington's Space, Time, and Gravitation.

But when a great truth comes to light it is generally found that there have already been prophets crying in the wilderness and preparing the way for the reception of the Revelation when the full time has come. In an anonymous letter published in Nature on February 12th, 1920, entitled "Euclid, Newton, and Einstein," attention was called to such a prophet in the following words:—

"Some thirty or more years ago, a little jeu d'esprit was written by Dr. Edwin Abbott, entitled 'Flatland.' At the time of its publication it did not attract as much attention as it deserved. Dr. Abbott pictures intelligent beings whose whole experience is confined to a plane, or other space of two dimensions, who have no faculties by which they can become conscious of anything outside that space and no means of moving off the surface on which they live. He then asks the reader, who has the consciousness of the third dimension, to imagine a sphere descending upon the plane of Flatland and passing through it. How will the inhabitants regard this phenomenon? They will not see the approaching sphere and will have no conception of its solidity. They will only be conscious of the circle in which it cuts their plane. This circle, at first a point, will gradually increase in diameter, driving the inhabitants of Flatlands outwards from its circumference, and this will go on until half the sphere has passed through the plane, when the circle will gradually contract to a point and then vanish, leaving the Flatlanders in undisturbed possession of their country. . . . Their experience will be that of a circular obstacle gradually expanding or growing, and then contracting, and they will attribute to growth in time what the external observer in three dimensions assigns to motion in the third dimension. Transfer this analogy to a movement of the fourth dimension through three-dimensional space. Assume the past and future of the universe to be all depicted in four-dimensional space and visible to any being who has consciousness of the fourth dimension. If there is motion of our three-dimensional space relative to the fourth dimension, all the changes we experience and assign to the flow of time will be due simply to this movement, the whole of the future as well as the past always existing in the fourth dimension."

It will be noticed that in the presentation of the Sphere to the Flatlander the third dimension involves time through the motion of the Sphere. In the Space-Time Continuum of the Theory of Relativity the fourth dimension is a time function, and the simplest element is an "event." One set of parallel sections of the four-dimensional continuum present the universe as it exists in three-dimensional space at the instants corresponding to the sections. Sections in all other directions involve the time element and represent the universe as it appears to an observer in motion.

There are some mathematical minds which are completely satisfied by the results expressed in algebraical symbols of the analysis of a continuum of four dimensions; but there are others which crave for the visualization of these results which, in their symbolic forms, they do not question. To many, perhaps to the great majority, of these, Dr. Abbott's sphere penetrating Flatland points the way to the clearest imagery of the fourth dimension to which they are likely to attain.



EDITION, 1884.


If my poor Flatland friend retained the vigour of mind which he enjoyed when he began to compose these Memoirs, I should not now need to represent him in this preface, in which he desires, firstly, to return his thanks to his readers and critics in Spaceland, whose appreciation has, with unexpected celerity, required a second edition of his work; secondly, to apologize for certain errors and misprints (for which, however, he is not entirely responsible); and, thirdly, to explain one or two misconceptions. But he is not the Square he once was. Years of imprisonment, and the still heavier burden of general incredulity and mockery, have combined with the natural decay of old age to erase from his mind many of the thoughts and notions, and much also of the terminology, which he acquired during his short stay in Spaceland. He has, therefore, requested me to reply in his behalf to two special objections, one of an intellectual, the other of a moral nature.

The first objection is, that a Flatlander, seeing a Line, sees something that must be thick to the eye as well as long to the eye (otherwise it would not be visible, if it had not some thickness); and consequently he ought (it is argued) to acknowledge that his countrymen are not only long and broad, but also (though doubtless in a very slight degree) thick or high. This objection is plausible, and, to Spacelanders, almost irresistible, so that, I confess, when I first heard it, I knew not what to reply . But my poor old friend's answer appears to me completely to meet it.

"I admit," said he—when I mentioned to him this objection—"I admit the truth of your critic's facts, but I deny his conclusions . It is true that we have really in Flatland a Third unrecognized Dimension called 'height,' just as it is also true that you have really in Spaceland a Fourth unrecognized Dimension, called by no name at present, but which I will call 'extra-height'. But we can no more take cognizance of our 'height' than you can of your 'extra-height' . Even I—who have been in Spaceland, and have had the privilege of understanding for twenty-four hours the meaning of 'height'—even I cannot now comprehend it, nor realize it by the sense of sight or by any process of reason; I can but apprehend it by faith.

"The reason is obvious. Dimension implies direction, implies measurement, implies the more and the less. Now, all our lines are equally and infinitesimally thick (or high, whichever you like); consequently, there is nothing in them to lead our minds to the conception of that Dimension. No 'delicate micrometer'—as has been suggested by one too hasty Spaceland critic—would in the least avail us; for we should not know what to measure, nor in what direction. When we see a Line, we see something that is long and bright; brightness, as well as length, is necessary to the existence of a Line; if the brightness vanishes, the Line is extinguished. Hence, all my Flatland friends— when I talk to them about the unrecognized Dimension which is somehow visible in a Line—say, 'Ah, you mean brightness': and when I reply, 'No, I mean a real Dimension,they at once retort, 'Then measure it, or tell us in what direction it extends'; and this silences me, for I can do neither. Only yesterday, when the Chief Circle (in other words our High Priest) came to inspect the State Prison and paid me his seventh annual visit, and when for the seventh time he put me the question, 'Was I any better?' I tried to prove to him that he was 'high,' as well as long and broad, although he did not know it. But what was his reply? 'You say I am "high"; measure my "high-ness" and I will believe you.' What could I do? How could I meet his challenge? I was crushed; and he left the room triumphant.

"Does this still seem strange to you? Then put yourself in a similar position. Suppose a person of the Fourth Dimension, condescending to visit you, were to say, 'Whenever you open your eyes, you see a Plane (which is of Two Dimensions) and you infer a Solid (which is of Three); but in reality you also see (though you do not recognize) a Fourth Dimension, which is not colour nor brightness nor anything of the kind, but a true Dimension, although I cannot point out to you its direction, nor can you possibly measure it.' What would you say to such a visitor? Would not you have him locked up? Well, that is my fate: and it is as natural for us Flatlanders to lock up a Square for preaching the Third Dimension, as it is for you Spacelanders to lock up a Cube for preaching the Fourth. Alas, how strong a family likeness runs through blind and persecuting humanity in all Dimensions! Points, Lines, Squares, Cubes, Extra-Cubes—we are all liable to the same errors, all alike the Slaves of our respective Dimensional prejudices, as one of your Spaceland poets has said—

On this point the defence of the Square seems to me to be impregnable. I wish I could say that his answer to the second (or moral) objection was equally clear and cogent. It has been objected that he is a woman-hater; and as this objection has been vehemently urged by those whom Nature's decree has constituted the somewhat larger half of the Spaceland race, I should like to remove it, so far as I can honestly do so. But the Square is so unaccustomed to the use of the moral terminology of Spaceland that I should be doing him an injustice if I were literally to transcribe his defence against this charge. Acting, therefore, as his interpreter and summarizer, I gather that in the course of an imprisonment of seven years he has himself modified his own personal views, both as regards Women and as regards the Isosceles or Lower Classes. Personally, he now inclines to the opinion of the Sphere (see page 86) that the Straight Lines are in many important respects superior to the Circles. But, writing as a Historian, he has identified himself (perhaps too closely) with the views generally adopted by Flatland, and (as he has been informed) even by Spaceland, Historians; in whose pages (until very recent times) the destinies of Women and of the masses of mankind have seldom been deemed worthy of mention and never of careful consideration.

In a still more obscure passage he now desires to disavow the Circular or aristocratic tendencies with which some critics have naturally credited him. While doing justice to the intellectual power with which a few Circles have for many generations maintained their supremacy over immense multitudes of their countrymen, he believes that the facts of Flatland, speaking for themselves without comment on his part, declare that Revolutions cannot always be suppressed by slaughter, and that Nature, in sentencing the Circles to infecundity, has condemned them to ultimate failure—"and herein" he says, "I see a fulfilment of the great Law of all worlds, that while the wisdom of Man thinks it is working one thing, the wisdom of Nature constrains it to work another, and quite a different and far better thing" For the rest, he begs his readers not to suppose that every minute detail in the daily life of Flatland must needs correspond to some other detail in Spaceland; and yet he hopes that, taken as a whole, his work may prove suggestive as well as amusing, to those Spacelanders of moderate and modest minds who—speaking of that which is of the highest importance, but lies beyond experience—decline to say on the one hand, "This can never be," and on the other hand, "It must needs be precisely thus, and we know all about it"

  1. The Author desires me to add, that the misconception of some of his critics on this matter has induced him to insert (on pp. 74 and 92) in his dialogue with the Sphere, certain remarks which have a bearing on the point in question and which he had previously omitted as being tedious and unnecessary