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Flatland: A Romance of Many Dimensions

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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.

Although it isn't very long, Flatland does take a long time to read. This isn't because it is boring, or because it is hard to read, but because of the large amount of digestion one need's to fully comprehend (and to fully enjoy) this book. Even this book contains only 82 pages, it is by no means light reading. The book was originally released in 1884 under Abbott's pseudonym A Square. In the story we follow the journey of a square who lives in a land of two dimensions--a flat land. In it class, and ultimately intelligence, is determined by the amount of sides that a shape has. As the amount of sides a shape has decreases, we find that it also is more emotional and apt to cause destruction through their pointed corners. Women are depicted as straight lines, but one has to take into account the time that this book was published. One can also disregard the story as having any relations to anything in our society and enjoy it for what it is, a mind bending social criticism. In this tale we follow the aforementioned square through his everyday life. we learn what it is like to exist in only two dimensions. We learn of how rain falls form the north and disappears to the south and how gravity is a minute force that pulls to the south ever so slightly. We follow him through the government and through social classes, and the discrimination that comes with them. When his son talks of geometric impossibilities such as 23 (cubed) he has a dream of a lesser land than his, a land called line land. IN it there is not two but only one dimension of being. Through discussion with the kind of lineland, we are offered insight into why our hero the square cannot conceive of the third dimension. Later our hero is visited by a great being, a sphere that appears to him seemingly out of nowhere. This confuses the square very much, and even more when the sphere tries to explain how he passed into his dimension from the third. After heated debate, the sphere takes him and shows him the third dimension, turning our hero into an evolved form of him self, a cube. Form his higher vantage point the square is able to see the innards of those who reside in flatland. He receives tutoring from the sphere about this new dimension and all that it entails. He learns of how limited the field of vision is for those living in flatland, both literally and figuratively. With his previous limits of reality stripped and with his eye opened to the truth, the square quickly follows logic and asks to see the insides of the sphere, and wishes to ascend further into greater dimensions, fourth dimensions and fifth and onward and upward. The sphere is appalled by this heresy and send our hero back to the limited realm of flatland. Here he tries to convince others to be enlightened, but cannot find success. He has a second dream involving the dimension of pointland, no dimensions. The being inhabiting this land is of nothing and knows nothing but itself, which is nothing. There fore this being cannot be disappointed by anything, because it cannot conceive of anything other than itself. We can see the religious parallels to Hinduism and Buddhism here. The completely content creature is of nothingness, much like the state that Buddhists try to achieve, and the outward ranking by dimension not sides can be seen in Hinduism in the spiral path towards God that the Hindu believe they travel along passing from one point on the spiral to another with each passing life. In this land of math all of the lands are contained within each other, much like the rings of the spiral. Finally after this dream the square realizes the futility of trying to convince others through speech, and he feels he must do it through demonstration. Folks hear of his heresy and bring him to the court for the climax of the book. Whether or not the plot of the novel itself is very entertaining, the ability to get your head around concepts that can only be experienced through the mind is challenged thoroughly by this novel. It is a must read for anyone who thinks that they are well educated, as it will quickly tell you just where you stand, theologically, philosophically and mathematically.

Flatland is THE must-read for anyone interested in getting a feel for higher dimensions. The book is extraordinarily readable and succeeds even with people that are afraid of mathematics. Abbott's charm lies in his ability to write simply and clearly about a topic that has its share of very unreachable, esoteric books. You fall into the story (whose plot is by no means secondary to the mathematical ideas), and before you know it you find yourself in contemplation of things like the fourth and fifth dimensions. The visual image that this book provides is a necessary step to envisioning and then understanding the idea of higher dimensions, even for those already versed in the mathematics of it. You never know, after you read this, you might even be willing to try your hand at things like Einstein's relativity. A little on the social aspects of the book: keep in mind that it was written in the very late 1800's. Hidden within the philosophical and mathematical ideas is a satire of the social climate of the times: how women, the military, the upper echelons of society, and just about everyone else were viewed. Flatland makes you think, and think deeply, on many different and sometimes unexpected levels.

From the square character's world of two dimensions in Flatland to the Sphere's three dimensional Spaceland, one comes to recognize the role of dimensions in geometry and in thinking in Abbott's Flatland. Both a mathematical essay and a satire the book challenges readers to discover dimensions for themselves in an unusual story. Beyond the story of the square lawyer protagonist and his adventure with the Sphere is the satire on Abbott's English society. Women are depicted as lines with the power to destroy men with there sharp, pointed ends. They are forced to remain in a constant waving motion as a courtesy to men in order to remain visible. An interesting predicament surfaces when coloring becomes a popular practice in identification. Women from certain viewpoints appear the same color as priests, much to the priests' chagrin. In sum, the women appear to have an inferior role to the multi-sided men as women faced inequality in late 19th century society. Secondly, the shapes themselves present a hierarchy of society. From the irregular figures to the noble Circles, each shape has its own ranking and occupations. Moreover, each shape is subdivided into figures that have a higher status in the Flatland world. For example, the equilateral triangle is seen as superior to any of the other isosceles triangle with top angles of less than sixty. These shapes have little hope of progressing; hope lies in their offspring which may possess a more respected number of equal sides. This can be seen as an analogy to the lower classes struggle to achieve success in the society dominated by the wealthy or aristocratic. While the story of Flatland may be a mockery of Victorian England, its heart is its mathematical meaning. It serves as an interesting and understandable window into the subject of dimensions. From Lineland, which knows no left or right directions, to the abstract Fourth Dimension, where it is possible to look inside a solid object, readers are introduced to new ways of thinking not usually encountered in math class. Most importantly, the text of the book is not beyond the scope of someone with a casual interest in the topic. Anyone can appreciate the search for the meanings of dimension and truth in easy to comprehend analogies presented by the author. Another math topic addressed is the discovery of new ideas themselves. Abbott shows that math is a field where anyone with an interest has a chance to succeed just as the main character stumbles upon the meaning of dimensions from thoughts from his grandson. He pursues his hypothesis on the dimensions of Spaceland as well as develops the ideas for the Fourth Dimension on his own. Although he is imprisoned for his thoughts and attempts to teach others, the square keeps his theories, not letting the views of society interfere with his work. It is interesting that he faces this fate when trying to educate the public about the truth of their world and beyond. On the whole, Flatland is more than just a short book with intriguing mathematical ideas. It is an opening experience to the search from the truth behind the world through the subject of dimensions. While mocking the English , the book also introduces readers an odd world of shapes and figures. Lastly, math is encouraged even though it may go against the grain of society. Any book that introduces readers to a new way of thinking is worth reading.

Abbot was passionate about addressing a Christian concern that I share: differentiating mere superstition from the supernatural. Flatland was written with this in view: to make a parabolic case for the dynamic of the "higher order" of things and how it would thus relate to us. It is also hilarious social and spiritual satire. While often portrayed as a treatise on mathematics, the math was only the idiom. He was trying, by way of parable, to show the rationality of belief in the supernatural.

The first half of the book is a set-up for the second half where the action takes place. The first time I read it (age 17) I thought it was a bit tedious till I got to the middle. But the first half must be read for the MIND BLOWING second half to make sense. When older and on second read, the first part is now my favorite. For Abbot is not just describing Flatland, he is satirically lampooning Victorian culture, priggery and prejudice. To think how this book must have made people's blood boil back then, just as it does now!

As for style, Abbott has a way of constructing a phrase that is just plain delicious once you get into the flow of it. He waxes most eloquent (in the voice of the "square" narrator) when he is most absurd and wrong, and this--you must admit--is a little too close for comfort as well.

A friend of mine called me at 4am several years after I gave him Flatland, having finally read it straight through, and said "I think I now 'get' the Revelation!" I hope it has a similar encouraging effect on you--either by humorous, pithy satire or spiritual insight.

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.

Warning: this is a terrible reproduction of the wonderful classic Flatland. I bought copies for college students and was very disappointed in the presentation. Very small print, poor contrast, and no decent formatting. Try to find an older printing, used if need be.

Flatland is subtitled "A Romance of Many Dimensions." I wouldn't call this book is "Romantic" in either the original or the modern sense of the term, but it certainly does include many dimensions. . .

And much more. With this one little book at the centre, one can discuss a wide array of topics: literature (late 19th century British satire), history (relate the satire to Victorian society), mathematics (beyond what you get from your typical proof-laden geometry class), physics (String Theory fits nicely), art (the influence of such mathematics on artists such as Dali and Escher), theology (pythian theology: God as The Being of Infinite Dimensions), and even a little climatology (how does it rain in a two-dimensional world?).

The first part of Flatland is an extensive description of life in a two-dimensional society. This is where most satirical elements can be found, but you don't have to know about British Victorian-era society to benefit by learning to view

physical reality more perceptively. How often would you ordinarily stop to consider what the social interactions and the houses and the weather and the class structure and so forth of a two-dimensional world might be? In addition to gaining a new appreciation for planar geometry, you will learn how very fortunate you are to have the extra dimension--but what if someone other entity is thinking the same about. . .

In the second part of Flatland, things really get spicy from the mathematical/physical/philosophical perspective. If you thought the preceding material was mind-expanding, just wait until A. Square travels to Sphereland, Lineland, and (my favorite) Pointland. This engages your brain in a way that no ordinary, prosaic math book can. Everything is explained in a manner that is easily understandable--but at the same time impossible to comprehend. You'll know what I mean when you try to apply the transition from Pointland to Lineland to Flatland to Sphereland to a like voyage to a fourth spatial dimension, and it seems like you should be able to do so, but you can never quite visualize the next dimension. You simply CANNOT. But, oh, what a savory intellectual treat it is to try!

Flatland does not have a plot--at least not until the very end, which is the book's most enduring satirical moment. How sad that this is based in reality. (It is also much like something one might read in. . .dare I say. . .a dystopia.)

If you really want an intellectual trip, contemplate temporal dimensions in a like manner. Yummy. . .

. . . but Abbott poses an even greater enigma: why is it that Flatland residents with the most acute angles are the most mentally obtuse (and vice versa)??

~pythia~

I first read "Flatland" when I was studying mathematics as a university first-year. I have just been re-reading it.

In my naivity, I first thought that Abbot was exhibiting his own prejudice against women and lower classes. I realise now that it is his very clever way of satirising his own culture - and he chose an ingenious way of doing it, by choosing a setting that was as far removed from the real world as is conceivable - the world of abstract mathematics.

The culture and society of Flatland are wonderfully developed, and the very human personalities of the square, circular, linear and hexagonal characters are colourful and vivid - displaying all the passions, corruptions, and emotions of our own, three dimensional world.

In addition, this book is a remarkable lesson in how higher dimensions can be conceived by beings who inhabit worlds of lower dimensions. And Abbot's eloquent and humorous writing style is a great pleasure to read.

This classic short novel is centered about intelligent beings that live in only two dimensions. One of the inhabitants, Mr. Square, describes his world and visits a one-dimensional world. He later encounters beings from the third dimension. Mr. Square finally considers even higher dimensions. This tale by Edwin A. Abbott (1839-1926), an English clergyman and academic, has become quite popular with those physicists and mathematicians who study higher dimensions. It is also regarded as a classic in the development of speculative fiction. I really enjoyed reading it. I've had a copy on my shelf for over forty years and I have never met a Ph.D. in physics (or math for that matter) who has not read it. Any student of the development of science fiction should also have a copy.