ISBN-10:
0691123667
ISBN-13:
9780691123660
Pub. Date:
10/09/2005
Publisher:
Princeton University Press
Flatland: A Romance of Many Dimensions

Flatland: A Romance of Many Dimensions

by Edwin Abbott Abbott, Thomas BanchoffEdwin Abbott Abbott
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Overview

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.

Product Details

ISBN-13: 9780691123660
Publisher: Princeton University Press
Publication date: 10/09/2005
Series: Princeton Science Library , #81
Pages: 144
Product dimensions: 5.50(w) x 8.50(h) x (d)

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

Flatland

A Romance of Many Dimensions


By Edwin Abbott Abbott

PRINCETON UNIVERSITY PRESS

Copyright © 1991 Princeton University Press
All rights reserved.
ISBN: 978-0-691-16555-4



CHAPTER 1

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.


(Continues...)

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

Introduction xiii

Part I This World

Section

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

Customer Reviews

Flatland 3.8 out of 5 based on 0 ratings. 154 reviews.
Anonymous More than 1 year ago
Good read, don't buy it though. You can get it for free in public domain.
Anonymous More than 1 year ago
This is going to be really corny, but it's true. This book influenced my decision to pursue mathematics and science as a career. Parts of it are a little dry, but these are the social commentary sections. I credit the rest of this book with equipping me to visualize higher dimensions. Definitely worth a read.
Guest More than 1 year ago
I recommend this as required reading for any geometry student and/or anyone who has ever given the slightest thought to dimensions other than our lovely 3rd dimension.
Kim_Duppy More than 1 year ago
My friends in the literature department will tell you that this is a clever novel about Victorian England. If that's all it were, I couldn't recommend it to anyone. In point of fact, this book is a kind of bare bones look at culture itself (not merely Victorian Culture). By reducing everything to shapes, the author manages to show how cultures evolve—or perhaps better put: how nature influences the development of culture. Plus, if you don't know much about geometry (I don't), you may learn a little about that as well.
Guest More than 1 year ago
This must be the best book I have read in years! It helped me understand mathematically and logically understand other dimensions as well as our own. This book will give you a glimpse of what living in a two dimensional world might look like, and also an Idea of what the fourth dimension might have in store in a logical manner. It also has a fantastic story and description of a two-dimensional culture, government and relationships. I strongly recommend it for geometry or advanced algebra students or anybody who wants a better understanding of multiple dimensions!
Guest More than 1 year ago
This book is an excellent choice for future math teachers. I am a junior in college getting my BA in Middle Level Math Education. This is an excellent book that will help understand demensions beyond our own.
Guest More than 1 year ago
I've recommended this book to my students of Geometry, especially those who will be teachers. This is a delightful guide to the understanding dimensions beyond our own. Must be cautioned that it does seem sexist - maybe a reflection of the time it was written.
Anonymous More than 1 year ago
An absolute pleasure to experience and shocking to discover its origin of 1884 whence I, a humble resident of Spaceland, encounter it now for the first time in this year 2019, some 135 years later. Yet it’s truths still inspire and enlighten one to contemplate those further dimensions that we cannot name in direction but can conceptualize with the imagination.
Anonymous More than 1 year ago
Anonymous More than 1 year ago
Great way to conceptualize dimension increments. Iconic, classic work. As a story ... meh. This is a case where I recommend watching the video (faster) vs. reading the book.
dandelionroots on LibraryThing More than 1 year ago
I read the annotated version, which I think was a poor decision. Many of Stewart's side notes weren't terribly interesting or illuminating, and much of the explanation was unnecessary (although parts of it I did appreciate). Abbott's work however is brilliant. I love the satire. The journey through Pointland, Lineland, Flatland, Spaceland and beyond (nonsense!) is epic-ly amazing.
ieJasonW on LibraryThing More than 1 year ago
Many people, when discussing complicated issues like religion or god, do not understand what it means to observe an entity that exists in a space that has one dimension more than themselves. Conversely, they often do not consider what it means to understand how they might be seen by an entity that exists in a space with one dimension less than their own. While these points are not surprising on their own -- beings in other dimensions are not obvious things! -- what is surprising is the lack of use of this information by those who advocate the existence of such beings (ie. God). I think Flatland provides fodder for many deists but is, unfortunately, neglected by the same.
rretzler on LibraryThing More than 1 year ago
I find it very interesting that this book was written 130 years ago and has survived so long, but not many have heard of it. I found it an intriguing book - a reflection of society at the time, which in some ways still holds true today. It has a definite religious theme, as well as a philosophical one. While I cannot say that I liked it, I'm glad that I read it, because it was indeed thought-provoking, which I'm guessing was what the author intended. I would say that it is a must read - and should definitely be on that list of classics which everyone should read. It was an easy read as well, however, I believe it may help if you have a basic understanding of geometry.
Griff on LibraryThing More than 1 year ago
Flatland. A nice read. A nice suggestion from a friend. At one point early in the book, when the narrator describes the lot of women in Flatland (and the ¿obvious¿ reasons for that lot) I could not help but think back to time spent in Qatar and the points of reference historically, socially and religiously describing the view of women in Islamic and Arabic cultures.I thought the mathematic and geometric explanations were masterful. I was struck by the powerful description of the way in which the paradigm with which we view the world limits our ability to comprehend certain things, while for others with a different paradigm, it is a matter of course. The various passages related to this theme reminded me of two works which have affected me a great deal: George Engel¿s description of the his biopsychosocial model for medicine in ¿Where You Think You Stand Determines What You Think You See¿ ¿ and Thomas Kuhn¿s Structure of Scientific Revolutions.I smiled and laughed at one point during the description of Lineland, when it was pointed out ¿once a neighbor, always a neighbor.¿ I immediately thought Lineland would necessarily have to be rampant with incest and homosexuality (or both simultaneously) until the author (or Lineland Monarch) anticipated my thoughts and described the marriage and mating rituals and processes. I breathed a sigh of relief and read on.A brief, but enjoyable book.
BrynDahlquis on LibraryThing More than 1 year ago
A short, surreal trip that makes me very curious and almost suspicious about life. Never before have I enjoyed geometry so much, and I'll probably never look at it the same way again.
Liberuno on LibraryThing More than 1 year ago
I started reading this book thinking that I was just going to get a quick humorous read on geometry. I didn't expect a short story told from the point of view of a square in a plane to hold so many interesting questions ranging in subject: from metaphysics and religion to discrimination.This short book is definitely worth reading.
jorgecardoso on LibraryThing More than 1 year ago
Flatland: A Romance of Many Dimensions is a great little book by Edwin Abbott. Flatland is a mathematical adventure on geometry. It takes place on a two-dimensional world with a strict hierarchical society based on the shape of its individuals and it describes the consequences of the adventure of one of those individuals (a square) through the realms of three-dimensions. It's a great book that makes us think about more-than-three-dimensional spaces and objects through analogy with two- and one- and even zero-dimensional worlds. As I read this, I thought it would be interesting to see an animation version of this book, but it turns out there are already some movies on Flatland. There is even a recent one with Martin Sheen (voice).
LaurenGommert on LibraryThing More than 1 year ago
I love this book...even though most of it's way over my head! Still, an awesome book written with a concept that no one has come close to copying since its release. When the book was written readers weren't sure how to classify it, so it got lumped in with sci-fi...which it isn't really. It has more to do with geometrics and philosophy. I bet that's a pair you never thought you'd see together!
fieldri1 on LibraryThing More than 1 year ago
Imagine a world where things exist on a plane of two dimensions. There is no up and down at all. People this world with polygons whose social position is ruled by the number of sides they have (triangles are the plebs, circles are the priests) and the class structure is rigidly adhered to.Then imagine of young person in this world who is contacted by a three dimensional sphere and who offers to take her out of her plane world and show her how the universe really is! This is the concept behind Flatland.Part satire on the class structure of Victorian Britain, part teaching aid for teaching euclidian space, this is a classic book, aimed at children, but powerful and thought provoking enough for adults.Its a very slim volume, but the content and its ideas will sit with you for a very long time.
pratchettfan on LibraryThing More than 1 year ago
A fascinating allegory on Victorian society told with two and three dimensional geographical figures. Even though written in the late 19th century, aspects of the story about people's sometimes limited world views are still relevant today and the moralities of the book shouldn't be lightly dismissed. At only 80 pages it is a quick read which you shouldn't miss.
TheBooknerd on LibraryThing More than 1 year ago
A classic--all fans of science fiction should read this book.
csixty4 on LibraryThing More than 1 year ago
I had such high hopes for this book. I figured any speculative fiction that stood the test of time so well must be something really special. Instead, I got porn for math geeks. The whole first half of the book, a description of the inhabitants of Flatland, might have been more interesting if the details were revealed through narrative, but the explanations and diagrams would make a good cure for insomnia. The second half was more interesting, and indeed the last bits were exciting. But the cost to get there was too much.
m_dow on LibraryThing More than 1 year ago
Meh. Not great, but it's a really short read and somewhat entertaining. The "flatland" society is actually rather horrific, full of eugenics and chauvinism, but the story is kind of fun. I wouldn't discourage you from reading it, but I'm not going to run around shouting that this is the best book ever.
TheoClarke on LibraryThing More than 1 year ago
A two-dimensional being discovers the third dimension.
aethercowboy on LibraryThing More than 1 year ago
Set in a two-dimensional world that could be represented on a large sheet of paper, we meet A. Square, a free thinker in a land ruled by oppressive religious zealots who will hear nothing other than "the world is flat."One day, however, Square meets Sphere, and is bumped out of Flatland and sent on a multidimensional journey.If you've ever been interested in the mathematical concept of dimensions, and want any reason to believe that the fourth dimension is not time, per se, I suggest you read this book, as it will open your eyes to a whole new perspective by likening yourself to Square.