Flatland: A Romance of Many Dimensions

By Edwin Abbott Abbott and Thomas Banchoff

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.

• Language: English
• Publisher: Princeton University Press
• Release date: Mar 22, 2015
• ISBN: 9781400866649

Edwin Abbott Abbott

Edwin Abbott (1838–1926) was an English educator and theologian best known for his 1884 novella Flatland: A Romance of Many Dimensions, a forerunner of modern science fiction. 

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PREFACE to the Second and Revised Edition 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 to 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 realise 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 infinitestimally 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 our Spaceland poets has said—

‘One touch of Nature makes all worlds akin.’ "

*

On this point the defense 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 defense 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 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.

---

* The Author desires me to add, that the misconception of some of his critics on this matter has induced him to insert 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.

FLATLAND: AN INTRODUCTION

Thomas F. Banchoff

Flatland first appeared over one hundred years ago and a dozen different editions have been published since then. Why now put out a new edition of this book in the Princeton Science Library? For each generation of readers, Edwin Abbott Abbott’s classic story of encounter between beings from different dimensions has had different significance. In 1884, the social satire of the limited perspective of Victorian England was as important as the comments on the use of analogy in treating higher dimensions, and both of these elements were clarified by the introduction to the second edition, purportedly written by an editor but actually written by Abbott himself. When the book was reissued in 1926, the main stimulus for considering higher dimensions was relativity theory, and a new introduction was written by William Garnett, a physicist student of Abbott’s. In 1952, when the first modern edition appeared, it was again a physicist, Banesh Hoffmann, who wrote the introduction, referring to the connetion between the dimensional analogy and the curvature of space. Other recent introductions have come from science-fiction and fantasy writers, a computer scientist, and a social historian.

In our own day, there are new reasons to reconsider this book and its fundamental ideas. Abbott challenged his readers to imagine trying to understand the nature of phenomena in higher dimensions if all they could see directly were lower-dimensional slices. That is precisely the situation that radiologists face today as they analyze the slices produced by CAT scans or magnetic resonance imaging, in attempts to reconstruct the forms of objects in space by studying their planar cross-sections. But Abbott did not want his readers to stop at two-dimensional representations of the third dimension: What would happen if we were to apply the same techniques one dimension higher, and attempt to conceive objects from a spatial fourth dimension by considering three-dimensional cross-sections? For mathematicians in Abbott’s day, that challenge was daunting. Their methods were adequate for visualizing surfaces in three-dimensional space, but it was only with great difficulty that these techniques could be applied to objects from higher dimensions. Frequently, researchers relied more and more on formal and abstract methods, leaving images behind.

Now, however there is a new interest in visualization, primarily because of the emergence of modern computer graphics, which can literally bring these higher dimensions into view. We can manipulate objects in four dimensions and see their three-dimensional slices tumbling on the computer screen. But how do we interpret these images? There is no better introduction to the problem of dealing with higher-dimensional slicing phenomena than this book, Flatland.

It is not necessary to read an introduction to appreciate Flatland, and at this point the reader might be encouraged to go ahead and read the book, returning to the introduction only later on to see if some of the questions that arise are answered here. In this short commentary, we shall address the satirical elements in Flatland, sketch Abbott’s background and interests, provide overviews of the style and historical context of the work, and investigate Flatland’s influence on contemporary approaches to new modes of visualization of higher dimensions, particularly computer graphics.

Social Satire in Flatland

When Edwin Abbott Abbott wrote his little masterpiece over one hundred years ago, he did it for several reasons. Some of these were obvious to his readers and remain obvious today. Others had to be explained by Abbott himself in the introduction to the second edition, which followed one month after the first, at the end of 1884. Still others were immediately clear to Victorian readers but need some clarification for readers in our

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.

[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.

[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.

[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.

[bratfarrar_1 · Rating: 4 out of 5 stars]

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

[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.

[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.

[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.

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

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

[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.

[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.

[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.

[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.

[amaraduende · Rating: 3 out of 5 stars]

This was so weird! Very different... and a quick read. It's about a man who lives in a very strictly ordered society in a world of 2 dimensions. He visits worlds of other numbers of dimensions... Lots of details included about what life might be like... even illustrations!

[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.

[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.

[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!

[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.

[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.

[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.