Flatland: A Romance of Many Dimensions

By Edwin A. Abbott

This masterpiece of science (and mathematical) fiction is a delightfully unique and highly entertaining satire that has charmed readers for more than 100 years. The work of English clergyman, educator and Shakespearean scholar Edwin A. Abbott (1838-1926), it describes the journeys of A. Square, a mathematician and resident of the two-dimensional Flatland, where women-thin, straight lines-are the lowliest of shapes, and where men may have any number of sides, depending on their social status.
Through strange occurrences that bring him into contact with a host of geometric forms, Square has adventures in Spaceland (three dimensions), Lineland (one dimension) and Pointland (no dimensions) and ultimately entertains thoughts of visiting a land of four dimensions—a revolutionary idea for which he is returned to his two-dimensional world. Charmingly illustrated by the author, Flatland is not only fascinating reading, it is still a first-rate fictional introduction to the concept of the multiple dimensions of space. "Instructive, entertaining, and stimulating to the imagination." — Mathematics Teacher.

• Language: English
• Publisher: Dover Publications
• Release date: Feb 29, 2012
• ISBN: 9780486110530

Book preview

e9780486110530_cover.jpge9780486110530_i0001.jpg

Note

FIRST PUBLISHED IN 1884 under the pseudonym A. Square, Flatland has maintained a unique place in imaginative scientific literature for over a century. The product of Edwin A. Abbott (1838—1926), an English clergyman and Shakespearean scholar whose avocation was mathematics, this charming narrative of a two-dimensional world has achieved renown both as an unequaled presentation of geometrical concepts and as a barbed satire of the hierarchical world of the Victorians.

DOVER THRIFT EDITIONS

General Editor: Stanley Appelbaum

Editor of This Volume: Philip Smith

Introduction © 1952, 1979 by Dover Publications, Inc. Copyright © 1992 by Dover Publications, Inc.

All rights reserved.

This Dover edition, first published in 1992, contains the unabridged, corrected text of Flatland (original publication: Seeley & Co., Ltd., London, 1884). The Introduction by Banesh Hoffmann first appeared in the 1952 Dover edition, and the Note has been newly created for the Dover Thrift Edition.

Library of Congress Cataloging-in-Publication Data

Abbott, Edwin Abbott, 1838—1926.

Flatland: a romance of many dimensions / Edwing A. Abbott; illustrated by the author; with an introduction by Banesh Hoffmann.

p. cm.—(Dover thrift editions)

Unabridged, corrected text of Flatland (original publication: Seeley & Co., Ltd., London, 1884)—T.P verso.

9780486110530

1. Fourth dimension. I. Title. II. Series.

QA699.A13 1992

530.1’1—dc20 92—18494

CIP

Manufactured in the United States by Courier Corporation

27263X18

www.doverpublications.com

Introduction

by Banesh Hoffmann

HERE IS A stirring adventure in pure mathematics, a fantasy of strange spaces peopled by geometrical figures; geometrical figures that think and speak and have all too human emotions. This is no trifling tale of science fiction. Its aim is to instruct, and it is written with subtle artistry. Start it and you will fall under its spell. If you are young in heart and the sense of wonder still stirs within you, you will read without pause till the end is regretfully reached. Yet you will not guess when the tale was written nor by what manner of man.

In these days space-time and the fourth dimension are household words. But Flatland, with its vivid picture of one and two and three and more dimensions, was not conceived in the era of relativity. It was written some seventy years ago, when Einstein was a mere child and the idea of space-time lay almost a quarter of a century in the future.

In those far off days, to be sure, the professional mathematicians were imagining spaces of any number of dimensions. The physicists too, in their theorizing, were working with hypothetical graph-spaces of arbitrary dimensionality. But these were matters of abstract theory. There was no public clamor for their elucidation; the public hardly knew that they existed.

One would think, therefore, that, in order to write Flatland, Edwin A. Abbott must have been a mathematician or physicist. But he was neither of these. True, he was a schoolmaster—a headmaster, no less, and a most distinguished one. But his field was classics, and his primary interests literature and theology, on which he wrote several books. Does this sound like the sort of man who would write an absorbing mathematical adventure? Perhaps Abbott himself thought it did not, for he published Flatland pseudonymously, as if afraid that it might besmirch the dignity of his more formal writings, of which he betrayed no reluctance to acknowledge his authorship.

Much has happened to our ideas of space and time since Flatland came into being. But despite all the talk of a fourth dimension, the fundamentals of dimensionality have not changed. Long before the advent of the theory of relativity, scientists thought of time as an extra dimension. In those days they regarded it as a solitary, isolated dimension that kept aloof from the three dimensions of space. In relativity time became inextricably intermingled with space to form a truly four-dimensional world; and this four-dimensional world turned out to be a curved one.

These modern developments have less significance than one might imagine for the story of Flatland. We do indeed have four dimensions. But even in relativity, they are not all of the same sort. Only three are spatial. The fourth is temporal; and we are unable to move freely in time. We cannot return to days gone by, nor avoid the coming of tomorrow. We can neither hasten nor retard our journey into the future. We are like hapless passengers on a crowded escalator, carried relentlessly forward till our particular floor arrives and we step off into a place where there is no time, while the material composing our bodies continues its journey on the inexorable escalator—perhaps forever.

Time, the tyrant, holds sway in Flatland as in our own world. Relativity or no relativity, we still have only one dimension more than the creatures of Abbott’s imagination; we still have only three spatial dimensions to their two. The inhabitants of Flatland are sentient beings, troubled by our troubles and moved by our emotions. Flat they may be physically, but their characters are well rounded. They are our kin, our own flesh and blood. We romp with them in Flatland. And romping, we suddenly find ourselves looking anew at our own humdrum world with the wide-eyed wonder of youth.

In Flatland we could escape from a two-dimensional prison by stepping momentarily into the third dimension and coming back on the other side of the prison wall. But that is because this third dimension is spatial. Our fourth dimension, time, true dimension though it be, does not permit us to escape from a three-dimensional prison. It does enable us to get out, for if we wait patiently for time to pass, our sentence will be served and we shall be set free. That is hardly an escape, however. To escape we must travel through time to some moment when the prison is wide open, or in ruins, or not yet built; and then, having stepped outside, we must reverse the direction of our time travel to return to the present. Neither we nor the inhabitants of Flatland can travel thus through time.

Though the crowded years go by, this nigh on seventy-year-old tale shows no sign of age. It remains as spry as ever, a timeless classic of perennial fascination that seems to have been written for today. Like all great art, it defies the tyrant Time.

e9780486110530_i0002.jpg

Fie, fie, how franticly I square my talk!

To

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

Preface to the Second and Revised Edition, 1884

by the Editor

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

Reviews for Flatland

[andycyca-1 · Rating: 3 out of 5 stars]

The life and times of a nobleman in 2d, a very interesting view of how dimensions work and how life could work out in a flat sheet

[drardavis · Rating: 5 out of 5 stars]

As one interested in mathematics in sci-fi, this book has been on my to read list for some time. I’m glad I finally got around to reading it. It is unique and a bit difficult to describe. Partly it is a satire of English culture in 1884. Partly it is anthropomorphizing geometric figures, with surprisingly convincing results. That’s the sci-fi, Abbott’s imagined universe. But basically it is a very clever mathematical proof of the existence of God. Having recently read Frankenstein, I was delighted at the main character’s, A. Square, description of himself as “a second Prometheus.”

[danielstj · Rating: 3 out of 5 stars]

Very strange, unique, and interesting.

[solitaryfossil · Rating: 3 out of 5 stars]

It was a fun re-read, I read it first about 100 years ago while in high school. Published in 1884, it usually gets categorized as Science Fiction, but it’s a social satire that skewers Victorian mores, especially how women were viewed by that society. And it examines dimensions. The main character is a 2 dimensional square who has a glimpse of the 3rd dimension, which sets him thinking in a new way.

It’s a pleasant little story.

[iejasonw · Rating: 4 out of 5 stars]

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.

[bratfarrar_1 · Rating: 4 out of 5 stars]

A bit dry, but an excellent way to get interested in geometry. Goes well with Euclid.

[nmarun-1 · Rating: 2 out of 5 stars]

I heard the audio book of Flatland and I found it very basic in terms of Geometry. I did like the author's creativity in creating the 1- and 2-dimensional world.

[adam.currey · Rating: 1 out of 5 stars]

I suspect it's an effect of this book having been written in the 1880's with language having changed quite a bit in the intervening time, but I found this book completely impenetrable - I only made it through 17 pages before giving up. Quite disappointing, as I was looking forward to it and was expecting it to be interesting.

[wendyrey · Rating: 3 out of 5 stars]

Interesting novella, a sort of mixture of science fiction/social commentary and a Dummy's guide to dimensions and relativity.Very , very clever.

[john257hopper · Rating: 4 out of 5 stars]

This is a short late 19th century concept novel, based on the imaginary thoughts of an inhabitant (a Square) of a land of two dimensions and the conversations he has with the Sphere, a being from Spaceland, a world of three dimensions. Lineland and Pointland also make appearances. I appreciated the author's intent in trying to show how the concept of dimension entirely governs one's notion of the universe; at the same time, I was amused by how the author's natural assumptions of nineteenth century middle class life governed even these other dimensional worlds; so, the Square's household (his wife is a line, his sons pentagons and his grandsons hexagons) has servants (triangles) and there is a criminal underclass of isosceles triangles. Even Lineland has a King. Pointland is (naturally) just a single being unaware of the existence of any other being. Simultaneously interesting and amusing little book.

[meriwetherr-1 · Rating: 4 out of 5 stars]

Impressed with how the author uses fiction to explain a complex concept and provoke thought.

[liberuno · Rating: 4 out of 5 stars]

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.

[pondlife_1 · Rating: 2 out of 5 stars]

The first half of this book appears to be an allegory of victorian class and gender discrimination, with some additional political things thrown in that went over my head but probably made sense to contemporary readers. For example, was the thing about colors and equality talking about the French revolution?The second half is a comparison of worlds of different dimensions from the viewpoint of someone living in a two-dimensional world. This part is easier to understand, but it's a bit overdone. I got the concept after a few pages, and after a while it felt repetitive.

[griff_2 · Rating: 3 out of 5 stars]

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.

[theboylatham_1 · Rating: 2 out of 5 stars]

Six out of ten.

Flatland is a two-dimensional world with a strict hierarchy of shapes and rules. The first half of the book is 'A Square' explaining in detail how the world of Flatland works. However, in the second half of the book his understanding of the world is ripped apart by trips to Lineland, Pointland and a possible world of three-dimension.Amazingly innovative. The book is meant as a parody of Victorian society that believed it had reached the edge of all human knowledge.

[michael5rimmer · Rating: 3 out of 5 stars]

I expected this book to be about the mathematics and physics of life in a two dimensional universe. Although there is some (well quite a lot, I guess) of this, it is mainly a satire on life in Victorian England. Despite my initial expectations being confounded, I did enjoy the book, which is very thought provoocing: I had a few "hand on chin" moments while reading it.

Enjoyable and recommended, but I still want to read a book more specifically about how life might evolve in two dimensions.

[kimmarie1 · Rating: 5 out of 5 stars]

Absolutely brilliant - a true masterpiece. The premise is so simple - just the basics of elementary school mathematics. Abbott makes characters out of basic shapes with such diversity and far reaching social commentaries that are as relevant today as it was in the time in which he wrote it.

[csixty4 · Rating: 3 out of 5 stars]

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.

[aratiel · Rating: 4 out of 5 stars]

Very interesting concept...the world of two-dimensional shapes explained by a square (named, appropriately enough, A. Square) who visits other planes and dimensions (Pointland, Lineland, and Spaceland). While it makes some valid points about humanity's false sense of superiority and true ignorance of higher consciousnesses, I could not overlook the blatant sexism. Yes, yes, I know it's supposed to be a parody of Victorian society, where women were shamelessly repressed and thought of as inferior to men, but I could not help bristling when I read "...among Women, we use language implying the utmost deference for their Sex...but behind their backs they are both regarded and spoken of - by all except the very young - as being little better than 'mindless organisms.' " The women in Flatland are ruled by emotions such as love and morality, which are thought to be silly ideas by the men who abide solely by logic. Parody or not, I cannot forgive the author for this. So, Mr. Abbott, if you were alive I would have this to say to you: "Your mother was a hamster, and your father smelt of elderberries." So there.

[ashishg_50 · Rating: 4 out of 5 stars]

It is difficult to imagine fourth spatial dimension, much less 11 that string theory suggests, but just because we cannot even conceive the idea doesn't mean it is not possible. After all, a being in flatlat finds our three dimensional world as unfathomabe as we find multi-dimensional world. Flatland is a short and simple read but opens mind to fantasy world of no dimension (pointland), one dimension (lineland) and two dimensions (flatland). Author's imaginations conjure up social hierarchy, ruling structure, culture and history of a fictional world forcing the reader to stretch his imaginations. An quick enjoyable stimulating read that will leave you smiling and thinking in the end.

[bryndahlquis · Rating: 5 out of 5 stars]

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.

[leticia.toraci · Rating: 4 out of 5 stars]

This is a very original book, where you will think in geometric figures at every line. It also sounds quite modern even if it was written in 1884. Its satirical tone makes it a priceless reading jewel and I would certainly recommend it.

[nx74defiant · Rating: 3 out of 5 stars]

A two-dimensional being records his journeys throughout various dimensions.Given his experience with the line I would have thought he would be more open to the possibility of a 3rd dimension.

[m_dow · Rating: 2 out of 5 stars]

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.

[hemlokgang · Rating: 2 out of 5 stars]

.I get it that this story of a two dimensional world is a metaphor for the superficiality and the divisiveness of the social structure of our three dimensional world. Did not like it at all.

[mrgan-2 · Rating: 3 out of 5 stars]

Deservedly a classic of popular mathematics, Flatland makes the problem of multiple dimensions into a comprehensible adventure of sorts. It's a great concept and something everyone should read about at some point.

However, there are much better books today from which you could learn the basics of Flatland. Being a product of the late 19th century, the book is saddled with stuffy, annoying prose; outdated and distracting satire; and only a marginally interesting "plot".

If you've heard about the book in the context of a work which covered the same ideas of how to think about dimensionality and other presumably unthinkable concepts, you can probably skip Flatland, as not much else about it is worthwhile.

[robble_1 · Rating: 4 out of 5 stars]

A wonderful philosophical novel which is a must read classic. This is basically the best book without real people that you are ever going to see. Abbott may be completely out of his mind, but he knows what he's talking about.

[tchesney_1 · Rating: 5 out of 5 stars]

In process:I read this book about 40 to 50 years ago.What a mind opener/expander!

[occassionalread · Rating: 4 out of 5 stars]

Flatland is a delightful book, the only that I know of that is narrarated by a two dimensional square. It provides a lesson in geometry and understanding dimensions. But it also operates, subtly, on another level, providing perhaps a tongue in cheek commentary on the British class system and social/sexual relations. It's politically incorrect at times, if taken seriously and not as a satire, but it's a lot of fun.

[lee7mervin · Rating: 4 out of 5 stars]

A wonderful mixture of science fiction and satire