In 1884, Edwin Abbott Abbott wrote a mathematical adventure set in a two-dimensional plane world, populated by a hierarchical society of regular geometrical figures-who think and speak and have all too human emotions. Since then Flatland has fascinated generations of readers, becoming a perennial science-fiction favorite. By imagining the contact of beings from different dimensions, the author fully exploited the power of the analogy between the limitations of humans and those of his two-dimensional characters.
A first-rate fictional guide to the concept of multiple dimensions of space, the book will also appeal to those who are interested in computer graphics. This field, which literally makes higher dimensions seeable, has aroused a new interest in visualization. We can now manipulate objects in four dimensions and observe their three-dimensional slices tumbling on the computer screen. But how do we interpret these images? In his introduction, Thomas Banchoff points out that there is no better way to begin exploring the problem of understanding higher-dimensional slicing phenomena than reading this classic novel of the Victorian era.
About the Author
Edwin Abbott Abbott (1838-1926), the author of more than fifty books on classics, theology, history, and Shakespeare, was headmaster of the City of London School and one of the leading educators of his time. Thomas Banchoff is professor emeritus of mathematics at Brown University and author of Beyond the Third Dimension.
Read an Excerpt
A Romance of Many Dimensions
By Edwin Abbott Abbott
PRINCETON UNIVERSITY PRESSCopyright © 1991 Princeton University Press
All rights reserved.
Part I This World
"Be patient, for the world is broad and wide."
§ 1: Of the Nature of Flatland
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 of 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 rep resents with us a Tradesman of the respectable class. Fig. 1 rep resents tire Tradesman as you would see him while you were bending over him from above; figs. 2 and 3 rep resent 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 le how we see him in Flatland)you would see nothing but a straight line.
When I was in Spaceland I heard that yon r sailors have very similar experiences while them 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 towards us in Flatland. As there is neither sun with its, 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 close to us we see his line becomes larger; if he leaves us it becomes smaller: bud still he looks like a straight lime; be he w Triangle, Square, Pentagon, Hexagon, Circle, what you will—a straight Line ha 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.
§ 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 oh our own. By a Law of Nature with use, 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 absentminded traveller suddenly running against them: and therefore, 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.
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.
§ 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 Class 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 condition. For, after a long series of military successes, or diligent and skilful labours, it is generally found that the more intelligent among the Artisan and Soldier classes manifest a slight increase of their third side or base, and a shrinkage of the two other sides. Intermarriages (arranged by the Priests) between the sons and daughters of these more intellectual members of the lower classes generally result in an offspring approximating still more to the type of the Equal-sided Triangle.
Rarely—in proportion to the vast numbers of Isosceles births—is a genuine and certifiable Equal-sided Triangle produced from Isosceles parents. Such a birth requires, as its antecedents, not only a series of carefully arranged intermarriages, but also a long-continued exercise of frugality and self-control on the part of the would-be ancestors of the coming Equilateral, and a patient, systematic, and continuous development of the Isosceles intellect through many generations.
The birth of a True Equilateral Triangle from Isosceles parents is the subject of rejoicing in our country for many furlongs round. After a strict examination conducted by the Sanitary and Social Board, the infant, if certified as Regular, is with solemn ceremonial admitted into the class of Equilaterals. He is then immediately taken from his proud yet sorrowing parents and adopted by some childless Equilateral, who is bound by oath never to permit the child henceforth to enter his former home or so much as to look upon his relations again, for fear lest the freshly developed organism may, by force of unconscious imitation, fall back again into his hereditary level.
The occasional emergence of an Isosceles from the ranks of his serf-born ancestors, is welcomed not only by the poor serfs themselves, as a gleam of light and hope shed upon the monotonous squalor of their existence, but also by the Aristocracy at large; for all the higher classes are well aware that these rare phenomena, while they do little or nothing to vulgarise their own privileges, serve as a most useful barrier against revolution from below.
Had the acute-angled rabble been all, without exception, absolutely destitute of hope and of ambition, they might have found leaders in some of their many seditious outbreaks, so able as to render dick superior numbers and strength too much for t lie wisdom even of the Circles. But a wise ordinance of Nature has decreed that in proportion as the working-classes increase in intelligence, knowledge, and all virtue, in that same proportion their acute angle (which makes them physically terrible) shall increase also and approximate to the harmless angle of the Equilateral Triangle. Thus, in the most brutal and formidable of the soldier class creatures almost on a level with women in then lack of intelligence—it is found that, as they wax in the mental ability necessary to employ their tremendous penetrating power to advantage, so do they wane in the power of penetration itself.
How admirable is the Law of Compensation! And how perfect a proof of the natural fitness and, I may almost say the divine origin of the aristocratic constitution of the States of Flatland! By a judicious use of this Law on Nature, the Polygons and Circles are almost always able to stifle sedition in its very cradle, taking advantage of the irrepressible and boundless hopefulness of human mind. Art also comes to the aid of Law and Order. It is generally found possible—by a little artificial compression or expansion on the part of the State physicians—to male some of the more intelligent leaders of a rebellion perfectly Regular, and to admit them at once into the privileged classes; a much larger number, who are still below the standard, allured by the prospect of being ultimately ennobled, are induced to enter the State Hospitals, where they are kept in honourable confinement for life; one or two alone of the more obstinate, foolish, and hopelessly irregular are led to execution.
Then the wretched rabble of the Isosceles, planless end leaderless, are either transfixed without resistance be the small body of their brethren whom the Chief Circle keeps in pay for emergencies of this kind; or else, more often, by means of jealousies and suspicions skilfully fomented among them by the Circular party, they are stirred to mutual warfare, and perish by one another's angles. No less than one hundred and twenty rebellions are recorded in our annals, besides minor outbreaks numbered at two hundred and thirty-five; and they have all ended thus.
Excerpted from Flatland by Edwin Abbott Abbott. Copyright © 1991 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
Table of Contents
Preface to the Second and Revised Edition ix
Part I This World
1 Of the Nature of Flatland 3
2 Of the Climate and Houses in Flatland 4
3 Concerning the Inhabitants of Flatland 6
4 Concerning the Women 8
5 Of our Methods of Recognizing one another 12
6 Of Recognition by Sight 16
7 Concerning Irregular Figures 20
8 Of the Ancient Practice of Painting 22
9 Of the Universal Colour Bill 24
10 Of the Suppression of the Chromatic Sedition 27
11 Concerning our Priests 30
12 Of the Doctrine of our Priests 32
Part II Other Worlds
13 How I had a Vision of Lineland 39
14 How in my Vision I endeavoured to explain the nature of Flatland, but could not 42
15 Concerning a Stranger from Spaceland 46
16 How the Stranger vainly endeavoured to reveal to me in words the mysteries of Spaceland 49
17 How the Sphere, having in vain tried words, resorted to deeds 55
18 How I came to Spaceland and what I saw there 57
19 How, though the Sphere showed me other mysteries of Spaceland, I still desired more; and what came of it 61
20 How the Sphere encouraged me in a Vision 66
21 How I tried to teach the Theory of Three Dimensions to my Grandson, and with what success 68
22 How I then tried to diff use the Theory of Three Dimensions by other means, and the result 70
What People are Saying About This
The best introduction one can find into the manner of perceiving dimensions. (From the Forward)
Most Helpful Customer Reviews
Great story...really gets one thinking about multidimensional possibilities. Ian Stewart's annotated version is nice. He explains mathemarical concepts as well as Victorian-era cultural influences. E.A. Abbott's perspective writing is insightful and ammazing!
If you've not read this introduction to dimensional geometry and satire of Victorian society, it's worth a read. It's only 80 pages, so it can read in a couple sittings. This free edition has almost no errors, which is quite rare in these scanned titles. If you're looking for a free edition, this one should suit.
Interesting novella, a sort of mixture of science fiction/social commentary and a Dummy's guide to dimensions and relativity.Very , very clever.
This book was given an overview in a silly book from the 1960s which my father once gave to me-- it was a book of math puzzles and the like. That book, however, did not hint to me that Flatland is really more of a Victorian social commentary than a book about math. I enjoy creative books about math, like 'The Math Devil'. The Math Devil is one fine book.Anyway, Flatland is interesting, yes, but-- well-- it's Victorian social commentary! Not something I enjoy reading for the sake of itself. Victorian social commentary is fine when there's an interesting plot to be had, but using MATH to make Victorian social commentary more interesting? Hmm. Not exactly the best decision. But it's still good, and it's very easy to see why this is a classic. Everyone should get around to reading it at least once-- and it's so short that this shouldn't be a problem for anyone, really.
You don¿t have to be a math geek, but it doesn¿t hurt. There isn¿t an extensive plot; it¿s not a novel in any real sense. And it is written in a style of the late 19th century, a style of writing that can leave readers working harder to decipher the language than the actual story. But there is a reason this exploration of dimensions has been around a really long time. It is fun to read, the language is actually pretty penetrable, and, while posing as an exploration of interesting things about the 2nd, 3rd, 4th, etc. dimensions (which it does well), it does a fairly decent job of pointing out the problems with society and people. While it¿s easy enough to think of this as showing the intolerance and narrow-minded thinking of the late 1800¿s, do not be fooled ¿ we have gotten no better. It is a relatively quick read (for me, a flight from Kansas City to Phoenix), but the kind that will keep coming back.
The book would not load. I guess you get what you pay for. :s I'll be trying another version.
It was mentioned in Infinite Jest, so I bought it. It's a little dry, but it contains some great concepts.
The introduction should be saved for last as suggested by the writer because it will bring perspective to the two dimensional world. As a physician, i encounter two dimensions everyday with reading of mri's, cat scans, etc. And never realized this. Overall, it was pure genius.
Is this a ROMANCE romance (guys and gals getting very close), or just a romance (kisses and hugs in full clothing)?
Flatland was a very interesting book. The world is completely different from the world we have today. For one in this book the people are shapes. In our world we have humans. Another huge difference is in the world of FLatland the miltary is nothing to them but in this world our miltary is everything to us. If we didn't have a miltary we wouldn't have half of the things we have today. The main difference to me being that I am a female is women in this book are treated like they are not important. They have separate doors away from the men's doors. In this world there are still some sexist people but it is still better than how the women in FLatland get treated. Although I disagree with some of the things in this book overall this book is good and I would recommend it to people and I would also like them to comment to see if they share the same feelings I have or if they have the complete opposite reaction from what I have.