Flatland: A Romance of Many Dimensions (Dover Thrift Editions)

You've heard the classic criticism of a story is that it is "two dimensional". Well, Edwin Abbott's tale of an imaginary two-dimensional land adds a whole new twist to that phrase. Flatland, as he describes it, is about as rich as a two-dimensional story can be. And it is marvelously extended by its narrator's encounters with the unknown - the world of 3 dimensions. The challenges that narrator faces as he encounters the incomprehensible, quite closely mirror mine whenever I attempt to think about a 4th (or 5th or 7th) dimension. If you've faced the same struggle, you will be delighted by this book.

If you've ever wondered what a 4th dimension would look like, Flatland provides a lens through which you can imagine that extension of our 3D world. From here you can go on to read Rudy Rucker or Pickover or Hawkins - but this is the place to start your exploration of dimensions beyond experience. Abbot accomplishes this by describing the eye-opening extension of his narrator's 2D world when visited by a 3D apparition, a "sphere". His framing of the foundational issues through the experiences of what you'd expect to be the least interesting character in fiction are really quite engaging. The storyline, however sparse, is as interesting as the mathematics - albeit quite nineteenth-century'ish in tone.

Don't misconstrue Abbott's seemingly misogynist portrayal of women and of his class-stratified society. This element was intended to provide a third layer of sharp, Swiftian satire and critical commentary on the rigid social mores of his era. Abbot succeeds in this (but I, nevertheless, decided not to read it aloud to my 6th grade classes - worrying that they might not be attuned to this subtlety of tone.) Be forewarned.

The neat twist of this edition, the inclusion of Dionys Burger's 1983 "Sphereland", as an upside-down "second book", is quite a nice touch. Although I did not find Burger's stylings quite as engaging as those of Abbott, his extension of the mathematical ideas into non-Euclidian spaces is a nice introduction to that idea for non-mathematicians. Since it was intended as a standalone book, "Sphereland" commences with a detailed review of the "Flatland" story. This can be skipped without loss but is not a substitute for reading the original "Flatland", here, first.

Flatland is a timeless classic; a great book for the mathematician and non-mathematician alike.


A delightful sci-fi classic   January 12, 2003
Michael J. Mazza (Pittsburgh, PA USA)
34 out of 45 found this review helpful

"Flatland: A Romance of Many Dimensions," by Edwin A. Abbott, is a marvelous tale that I regard as a pioneering piece of science fiction. According to the introductory note in the Dover edition, Abbott was an English scholar and clergyman, and the book was first published under a pseudonym in 1884. The book is enhanced by the author's own delightful illustrations.

"Flatland" is told in the first person by an intelligent square who lives in a fantastic two-dimensional world. He describes in fascinating detail his own world of Flatland, going into such topics as architecture, war, genetics, medical arts, law, and family values. Particularly fascinating is his account of his society's rigid stratification by class and gender. The square tells of his visions of zero- and one-dimensional worlds, and of his life-changing contact with the three-dimensional world.

Abbott succeeds in a task attempted with varying success by generations of science fiction writers since him: he creates an alternate world which is utterly alien, yet disturbingly familiar--a world that is complete and consistently compelling. "Flatland" could certainly be read as a satire of Abbott's own world; parts of it are laugh-out-loud hilarious. Whimsical yet possessing a biting edge, this is a brilliantly conceived and wonderfully written book. For a companion text, try A.K. Dewdney's "The Planiverse" (also about contact between two- and three-dimensional worlds); also try Aldous Huxley's "Brave New World" (an equally intriguing view of a stratified sci-fi culture).


Unimaginable Dimensions   July 7, 2006
Jon Linden (Warren, N.J. United States)
18 out of 19 found this review helpful

Flatland is a unique and brilliant treatise on a trifurcated level. It is a sociological statement, a mathematical statement and a religious statement all rolled into an incredibly astute 82 pages. The book centers mostly on the differences between a two dimensional world and a three dimensional world; but comments on society, law, prejudice, religion, and proselytizing.

The book especially points out the difficulty in envisioning a greater reality and a greater vision than is commonly observed by any individual in any dimension or society. The author's premise relates to things existing in a "plane geometry" world as opposed to a "Euclidian Geometric" three dimensional figure universe. The book carefully illustrates to one denizen of Flatland how the three dimensional world of space works and/or exists. Upon finally understanding the "Gospel of Three Dimensions" our protagonist goes on to try and apply the same arithmetic logic and geometric analogs to a fourth dimensional universe. Shouldn't there exist a fourth dimensional universe that allows an entity to look down upon the three dimensional universe with as much transparency as one can from three dimensions to two?

Alas, things become different in dimensions other than the first, a world of lines, the second, a world of shapes and the third, a world of objects. In the zero dimension, all things are a point. Mathematically we know that any number raised to the "0" power equals 1 and therefore, all things in the zero dimension resolve into one single omnipotent point. This condition would also exist in the fourth dimension; as those of us in the third dimension have no model to compare it to. Envisioning a fourth dimension, even with time as the fourth dimension is truly difficult or impossible for us in the third dimension.

Interspersed with this witty and intellectual dialogue are comments on society and its structure. He specifically comments multiple times of the degradation of women in society to the lowest social status. Only men are educated in Flatland. Interestingly, he paints a picture of an authoritarian society in which people are judged by their shapes and angles. This reflecting the Victorian societal values around him at the time of his writing.

Flatland is recommended to all those who seek to enlighten their view of the universe and of potential universes. It is especially recommended to those seeking higher knowledge of any type. Flatland is truly a multi-dimensional experience and worth every minute.



It should be required reading in math and social studies   April 22, 2001
Charles Ashbacher (Marion, Iowa United States(cashbacher@yahoo.com))
15 out of 16 found this review helpful

This book should be required reading for students in both mathematics and social studies. The explanations by Abbott of how three dimensional beings would appear to a two dimensional creature have never been improved upon. I have seen them reproduced in nearly every book that I have read about space with dimensions more than three. Since we cannot visualize objects in four dimensions, the best that can be done is to describe them mathematically or by using an analogy to the dimensions that we can visualize.
However, the book is also a satire about the English social structure of the nineteenth century. Abbott was a champion of the rights of women, so his caricature of females must be read with that fact foremost in mind. His description of the society of Flatland is meant to be a critique of what he saw in an enormously class-conscious England with a distinct privileged class. Read with that foremost in mind, his caricatures of the inhabitants are quite amusing.
This is one of the very few books that I have read over three times. It has never failed to keep my interest, containing some of the best teaching analogies ever created. I even read it when it appears paraphrased in books written to describe the principles of space with more than three dimensions. I have no doubt that this is a book that will still be interesting to people thousands of years in the future.


A Multi-Demensional Story of Two Dimensions!   January 28, 2006
Kevin S Currie (Reisterstown, MD United States)
7 out of 7 found this review helpful

Edwin Abbott's "Flatland" is a story that can be read as many things: as an Orwellian satire on The State, as a contemplation of dimensions (even ones we do not know yet), as a philosophical variation of Plato's Cave, or simply as a delightful sci-fi story.

Flatland is broken up into two parts. The first sees our main character (a square who is a mathemetician - how cute!) telling us - those who live in "spaceland" - about the ins and outs of flatland. He teaches us both about how life is lived as a two dimensional object, explaining such things as how one can recognize by sight a squre from a hexagon without being able to see the angles. Also, the square explains how flatland's class and government system works: how women, for instance, are inferior to men and do not attend school, how polygons are superior to lowly squares and triangles, and how the circles make the laws.

It is this first section that is both a savy critique on government and the then prevailing social mores. Also, this seciton is fascinating from a mathematical point of view, answering questions that the astute reader is bound to have. (How, for instance, is sight possible in two dimensions?)

The second section of the book is where stuff gets even more intersting. This is the section where our square mathemetician both finds himself (in a dream) having to explain the concept of a second dimension to a "linelander", and is visited by a "spacelander" who must try to convince our "flatlander" that a third dimension is possible.

This section is most interesting from a philosophical perspective. How, for instance, is it possible to explain a third dimension to one who only knows two dimensions? To the flatlander, terms like "up," "down," "above," "below," "vertical," (etc.) have no meaning as they presuppose knowledge of a third dimension. That is, can a spacelander use "two diemsional language" (language presupposing no familiarity with any but two dimensions) to explain a third dimension? (Edwin Abbot thinks he can, but the astute reader will notice Abbott's linguistic fallacy in how he brought this about!)

At any rate, it is this part of the book that can also be seen as a variation on Plato's Cave. (Plato's Cave analogizes us to people looking at the back wall of a cave open at the other end, who see only shadows of objects outside the cave, but who are convinced that what they see is the real things.) That is, "Flatland" very beautifully articulates the idea of someone realizing that what they assumed was a complete view of reality was only a partial view of reality. And now, the flatlander bears the burden of trying to convince other flatlanders that the reality they take for granted as "all there is," is, in fact, not a complete view of reality at all.

Can he do it? Will he run into the same linguistic stumbling block already spoken of? More interestingly, if the "linelander" was wrong in assuming only one dimension, and the "flatlander" was wrong in assuming only two dimensions, might the "spacelander" be wrong in assuming only three dimensions, and so on? And how, if possible, could we find out?

These questions and more are artfully and skillfully broached in Edwin Abbott's mathematical classic "Flatland." Even if all you want is an intersting science fiction book, "Flatland" will do the trick. But for those who want a great "thinking" novel, "Flatland" is an outstanding choice!