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Flatland (une aventure a plusieurs dimensions) Poche – 24 janvier 1984

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A Romance of Many Dimensions










In the summer of 1973, I went on a camping trip in Sequoia National Park. I was a graduate student in physics at the time, and my two companions were also physicists. Carved out of granite by retreating glaciers, Sequoia National Park lies in the southern end of the great Sierra Nevada mountain range of California and is most famous for its giant sequoia trees, which attain heights of several hundred feet and ages of two thousand years. In Sequoia, one’s senses are overwhelmed. The land tilts and swerves from the ancient shifting of subterranean faults, snow-covered mountains jut into space, shady forests suddenly give way to bright meadows.

During this barrage of sensation, in which it seemed to me that every cubic inch of the world was filled to its maximum capacity, one of my fellow campers, John Schwarz, was at work formulating a new theory of nature—a theory that required seven additional dimensions beyond the usual three. Schwarz’s pioneering calculations, called “string theory” and later extended by other theoretical physicists, are now regarded as the best attempt to develop a quantum theory of gravity and to unify all the forces of nature. For technical reasons, such a theory demands more than length, width, and breadth. Fortunately, the extra dimensions are curled up in such tiny circles that they cannot be experienced by macroscopic creatures who are already strained by a mere three.

Almost a century before that excursion into the sequoias of California, in 1884, there quietly appeared in England a little book titled Flatland, which invited its readers to consider the outrageous possibility of four dimensions and more. Flatland slyly accomplished this suggestion by portraying the highly limited life of a world of only two dimensions, whose inhabitants cannot imagine, and do not want to imagine, a third dimension. In Flatland, all existence and experience is confined to a plane. Nothing has thickness. People come in the shape of triangles, squares, pentagons, and so on, the greater the number of sides, the higher the status. Since all geometrical shapes appear as straight lines when viewed edge on—and edge on is the only possibility in Flatland—inhabitants must feel each other’s angles for proper recognition. Interiors of closed figures are invisible. Rain slides across the world plane from the north; consequently each house is oriented so that its “roof” side faces that direction.

The author of this extended fable was the Reverend Edwin Abbott Abbott, born in 1838, educated at St. John’s in Cambridge, and ordained in 1862. Abbott was a classicist, Bible scholar, and, from 1865 to 1889, headmaster of the great City of London School, which he himself had attended before university. Abbott came from a line of headmasters, his father, Edwin Abbott, having been headmaster of the Philological School, Marylebone.

In the edition of the British Dictionary of National Biography for persons who died in the period of 1922 to 1930, no reference is made to Flatland, A Romance of Many Dimensions (1884). Abbott is indeed celebrated as a writer, with special notice of his school primer How to Write Clearly (1872), his literary criticism such as Shakespearean Grammar (1870), and his many theological writings such as Philochristus (1878), Onesimus: Memoirs of a Disciple of Paul (1882), and Johannine Grammar (1906). But his greatest achievement, from the vantage of a few years after his death, was as a teacher. Although frail and delicate in physique, he could keep discipline without effort, for he had the presence of a commander and “the mark of the spiritual leader in that he could impart to others something of the virtue that was in him. He was aflame with intellectual energy: without driving or overtaxing his pupils, he made intellectual effort a kind of religion for them.” A number of his students, including H. H. Asquith, went on to Cambridge and Oxford and became leading men of letters and government, of which he was most proud.

Today, Abbott is remembered principally as the author of the little book not even mentioned in the D.N.B. None of Abbott’s now forgotten other forty books hint at anything nearly so imaginative as Flatland. Although Flatland contains the same moralizing and didactic tone that runs through all of Abbott’s writing, the fable manages to break out of its period with the inspiration and eternal meaning of a classic.

In its mathematical underpinnings, fancifulness, and wit, one cannot resist comparing Abbott’s fable to Lewis Carroll’s Alice in Wonderland, published only twenty years earlier. Both Abbott and Carroll came out of the English school system, both excelled in the classics, both lent their particularly mathematical and logical minds to the invention of imaginary worlds for the amusement of their readers. Abbott’s book is far more pedantic, its geometrical and moral lessons more overt. Although a great deal has been read into Carroll’s two Alice books, any didacticism, if it exists at all, is well hidden. Perhaps a closer comparison to Flatland would be the Russian-born physicist George Gamow’s Mr. Tompkins books: Mr. Tompkins in Wonderland (1940) and Mr. Tompkins Explores the Atom (1945). In these books, Gamow creates fantastic worlds in which his hero Mr. C.G.H. Tompkins, “a little clerk of a big city bank,” travels at speeds close to the speed of light, or shrinks to the size of subatomic particles. Gamow’s explicit purpose is to popularize relativity and quantum theory; he succeeds also in demonstrating the extreme limitation of human sensory perceptions and all intuitive notions of the natural world based on those perceptions. All of these books, including my own Einstein’s Dreams, constitute a literary tradition in which a fantasy world created by some physical or mathematical conceit invites the reader to ponder philosophical questions in the actual world of human existence.

Flatland is divided into two halves. The first half bristles with the portrait of what today can only be described as a repulsively elitist and sexist society. Priests, who are circles, are the highest class and administrate the two-dimensional realm. Everyone aspires to having the highest number of sides, approaching the exalted circles, and the children of aristocrats are actually fractured to increase their number of corners. Tradesmen and soldiers, among the lowest classes, with a scant three or four sides to themselves, are barely human. Women are not even worthy of three sides. They are one-sided figures—straight lines, in other words—and they must be constantly avoided or handled gingerly so that their two extremely sharp ends will not puncture incautious males in the vicinity. Women talk too much and are so dumb that they aren’t even aware of their wretched status at the bottom of the social hierarchy. Occasionally, they go on a berserk rampage and massacre hundreds of males, a practice that is accepted as keeping the population of the lower classes in check (and might have been applauded by a two-dimensional Thomas Malthus).

This description of society is conveyed earnestly by the narrator of the fable, a two-dimensional inhabitant of Flatland. At first, one has the uneasy feeling that the narrator stands for Abbott and his own views of society, but the fictional system is so caricatured and ridiculous and witty that it does not seem possible that Abbott could have shared these views, even in his day. I read this first half of Flatland as social satire, in the tradition of Jonathan Swift.

Also raised in the beginning section are the venerable philosophical controversies over nature versus nurture, free will versus determinism. Among the lowest of the low in Flatland are the “Irregulars,” miserable creatures whose sides are of unequal lengths (as opposed to the equilateral triangles, the squares, the regular pentagons, and so on) and who behave accordingly. Debate rages over whether the Irregulars are natural monsters, born with defective moral constitution, or instead acquire their bitterness and perversion only after being outcast and mistreated by society. The narrator openly confesses that he favors operating on the Irregulars to make them Regulars, and killing them if that process doesn’t work. The Priests claim that personal conduct depends completely on immutable Configuration. If one is born bad (irregular), one remains bad. Will, training, and encouragement are useless. Then again, the Priests have a great deal at stake in protecting their exalted position. The practice of operating on Irregulars to make them Regular or of fracturing the children of aristocrats to give them more sides also echoes the controversial idea of eugenics, in which the “best” human beings are segregated and bred together to improve the species. This notion, discussed today in the context of genetic engineering, was put forth most forcefully by Francis Galton in his book Hereditary Genius (1869), a book that well could have been read by Abbott.

The second half of Abbott’s book takes a sharp turn when the narrator becomes aware of the limited scope of his world and begins conjecturing on three, four, and higher dimensional possibilities. And here begins to emerge the meaning of the book for science. Indeed, in 1920, an article titled “Euclid, Newton, and Einstein,” published in the prestigious scientific journal Nature, referred to Flatland at some length. Einstein’s recent theory of General Relativity (1917), based in part on his earlier Special Relativity (1905), treats time as a fourth dimension. Time is not absolute. This fourth dimension is not rigidly marked out like ticks on a ruler but instead can expand and contract depending on gravity and on the motion of the observer through the three spatial dimensions. The Nature article compares the motion of our three-dimensional space against a hypothetical fourth dimension to Abbott’s description of motion of his two-dimensional Flatland relative to a three-dimensional sphere: If the latter passes through Flatland, it will be witnessed by the two-dimensional creatures as a circle (the intersection of a sphere with a plane) whose diameter starts as a point when the sphere first touches the plane, grows larger and larger to a maximum size, then contracts down to a point and disappears. The analogy between Einstein’s relativity and Abbott’s Flatland description is striking although not quite exact, since Einstein’s fourth dimension of time behaves very differently from an additional dimension of space. Nevertheless, the writer, signing him or herself only as “W.G.,” recognized a point of modern scientific significance in Abbott’s little book.

For me, the importance of the second part of Flatland lies not in its literal geometrical and dimensional discussion, but in its more shrouded warning of too much complacency in the scientific enterprise—and, by extension, all of life. At the time Flatland was published, in 1884, much of science, and especially physics, hummed along in a state of self-satisfaction. Newton’s celebrated laws of mechanics and gravity were unchallenged. Dalton’s and Avogadro’s modern concept of the atom provided a good working basis for the understanding of matter and chemical composition. The nature of heat and the laws of thermodynamics had become well established earlier that century by Rumford, joule, Clausius, Kelvin, and others. In 1865, Maxwell brilliantly elucidated the complete theory of electricity and magnetism, including a fundamental understanding of the properties of light. Soon after, Mendel published his laws for biological heredity; Mendeleyev put into place the periodic table and was successfully predicting new chemical elements. The theory of natural selection, although not unanimously endorsed, offered a scientific explanation of the specialized diversity of species. Science was indeed content with itself.

In an ironical dream, the two-dimensional narrator of Flatland visits Lineland, where the pitifully ignorant Monarch “was persuaded that the Straight Line, which he called his Kingdom and in which he passed his existence, constituted the whole of the world, and indeed the whole of Space.”

“Behold me—I am a Line,” said the Monarch of Lineland, “the longest in Lineland, over six inches of Space.”

“Of Length,” responded the narrator.

“Fool,” said the Monarch. “Space is Length. Interrupt me again and I have done.”

The first interruption of nineteenth-century science came with Roentgen’s discoveries of X rays, barely a decade after the publication of Flatland. X for unknown. Nothing had ever been seen like the highly energetic and penetrating radiations that streamed from an electrified gas. Then, the next year, powerful radiations emerged from uranium and radium, which were apparently spitting out tiny pieces of themselves. To the astonishment of scientists, the immortal and indestructible atom could disintegrate of its own accord. In 1905, the twenty-six-year-old Einstein proposed his new theory of time, contradicting all our intuition and experience with the physical world. Two events that are simultaneous to a man on a bench are not simultaneous to a man in a passing train. To Einstein’s theory of relativity, W. F. Maggie, Professor of Physics at Princeton (Monarch of Lineland), responded in 1911:

I do not believe that there is any man now living who can assert with truth that he can conceive of time which is a function of velocity or is willing to go to the stake for the conviction that his “now” is another man’s “future” or still another man’s “past.”

The final insult to all common sense was delivered by Heisenberg and Schrödinger’s quantum theory, which decreed that the position and velocity of an individual particle cannot be completely specified, even in principle. As a result, one cannot predict with certainty the future position and velocity of a particle; such predictions can be done only in terms of probabilities, which apply only to the average behavior of a large number of particles. In short, the world hovers in a state of uncertainty. Einstein, once the pioneer of revolutionary scientific ideas, now resisted the new ones and opposed the indeterminancy inherent in quantum physics. In a letter to fellow physicist Max Born, Einstein wrote:

The idea that an electron exposed to a ray by its own free decision chooses the moment and the direction in which it wants to eject is intolerable to me. If that is so, I’d rather be a cobbler or a clerk in a gambling casino than a physicist.

Back in 1884, the narrator of Flatland, who has had his worldview shattered by a visit from a three-dimensional Sphere, launches on a personal mission to “arouse in the interiors of Plane and Solid Humanity a spirit of rebellion against the Conceit which would limit our Dimensions to Two or Three or any number short of Infinity.” The flatlander begs the Sphere to take him to the “blessed region of the Fourth Dimension.” And the Sphere, who has previously scoffed at the flatlander’s small worldview, replies: “There is no such land. The very idea of it is utterly inconceivable.”

Today, at the end of the twentieth century, scientists are not so complacent as a century ago. We are particularly aware of the limits of human sensory perception. Our instruments detect X rays and radio waves at frequencies that the eye cannot see, measure relativistic time dilation at speeds far greater than human travel, confirm that subatomic particles behave as if they were in two places at once. Biologists now have the ability to alter the cellular instructions of animals, or to clone others from single cells in a test tube. We freely acknowledge that the world is far stranger than it seems. Thus Schwarz’s string theory of ten spatial dimensions, Stephen Hawking’s calculations of the evaporation of black holes, and Alan Guth’s theory of an exponentially expanding universe are all taken seriously.

But Abbott, if we read him deeply, has challenged us to question more than our tenets of geometry and physics. If the very dimensionality of space is open to question, then what beliefs remain sacred? What else should we question? For example: Is there really a sharp division between animate and inanimate matter? Could human consciousness be some kind of collective phenomenon, even though each of us has the strong sensation of individual thoughts and minds? Does the earth behave as a single living organism, with all of its physical and biological systems purposefully connected (as proposed in the “Gaia Hypothesis”)? Do nonphysical dimensions exist? Does modern technology diminish, rather than enhance, the quality of life? I confess that I do not know how to ask these kinds of questions, or even what areas of thought they involve. I cannot conceive of a world with these possibilities. And that is the point. The inhabitants of Flatland could not conceive of a third dimension. By definition, it is extremely difficult to imagine worlds outside of our experience. For that, we are as likely to receive guidance from our artists and philosophers, as from our mathematicians and scientists.


Alice’s Adventures in Wonderland by Lewis Carroll, first published in 1865

Through the Looking Glass, and What Alice Found There by Lewis Carroll, first published in 1872

Mr. Tompkins in Wonderland by George Gamow, first published in 1940 (Cambridge: Cambridge University Press, 1965)

Mr. Tompkins Explores the Atom by George Gamow, first published in 1945 (Cambridge: Cambridge University Press, 1965)

“Euclid, Newton, and Einstein” by W.G. in Nature, no. 2624, vol. 104, p. 627 (February 12, 1920)

Einstein’s Dreams by Alan Lightman (New York: Pantheon, 1993)

Gulliver’s Travels by Jonathan Swift, first published in 1726 (New York: Bantam, 1981)

--Ce texte fait référence à l'édition Broché .

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Format: Broché Achat vérifié
Ce roman devrait réjouir tout les admirateurs de science-fiction et de physique. Ecrit à la fin du XIXè siècle, il raconte l'histoire d'un carré qui évolue dans un espace à deux dimensions et qui va être amené après avoir eu des visions, à reconsidérer l'espace à deux dimensions dans lequel il est piégé. Tous les protagonistes de ce roman sont des figures géométriques, ce qui en fait un livre complètement original et atypique. De plus pour tous ceux qui ont du mal avec la physique, la lecture de Flatland pourrait les aider à comprendre la différence entre un monde à 0, 1, 2 ou 3 dimensions. La possibilité d'une 4ème dimension est même évoquée.
Il est dommage que le roman soit presque totalement dépourvu d'intrigue mis à part les voyages de ce carré dans les autres dimensions. En effet, la majeure partie du livre est consacrée à décrire la hiérarchie de la société ou le moyen de reconnaissance des autres figures géométriques dans Flatland. De plus le livre est très court (moins de 100 pages).
Ce roman serait aussi une critique déguisée de la société Victorienne du XIXè siècle.
Donc 4 étoiles pour l'originalité du livre sur un sujet qui aurait pu être totalement ennuyeux mais avec Abbot on ne s'ennuie pas une seconde, on regrette même que le livre ait été si court.
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Format: Poche Achat vérifié
Un très bon livre que je recommande de lire que ce soit en anglais ou en français. Relativement court, il se lit très rapidement.
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Un produit valant son prix extrêmement raisonnable.
L'édition est de bonne qualité et e contraste sur le papier est optimal pour une lecture sans gêne.
Je vous conseille cette édition valant bien les autres.
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Amazon.com: HASH(0xa18a81a4) étoiles sur 5 540 commentaires
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HASH(0xa13e3f90) étoiles sur 5 Mind bender anyone? 31 mai 2001
Par Un client - Publié sur Amazon.com
Format: Broché
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.
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HASH(0xa170d3f0) étoiles sur 5 Unimaginable Dimensions 7 juillet 2006
Par Jon Linden - Publié sur Amazon.com
Format: Broché Achat vérifié
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.
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HASH(0xa170d30c) étoiles sur 5 Horrible Edition 3 septembre 2010
Par M. Gajdosik - Publié sur Amazon.com
Format: Broché Achat vérifié
This edition is essentially unreadable and not representative of traditional printings. It's printed directly from the digitized (and free) copy from Google Books and has clearly had NO editing work done. The book is filled with references to figures that were not included, mangled words, and seemingly random breaks and markings in some spots. This would be fine for a free digitized text online, but is entirely unacceptable for a paid-for product, especially a short book that would be similarly priced in a physical store.
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HASH(0xa170d360) étoiles sur 5 The book that started it all for higher dimensional analysis 13 décembre 1998
Par Un client - Publié sur Amazon.com
Format: Relié
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.
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HASH(0xa170d3d8) étoiles sur 5 Math at its Best 19 mai 2000
Par Un client - Publié sur Amazon.com
Format: Broché
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.
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