The Annotated Flatland: A Romance of Many Dimensions

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Flatland is a unique, delightful satire that has charmed readers for over a century. Published in 1884 by the English clergyman and headmaster Edwin A. Abbott, it is the fanciful tale of A. Square, a two-dimensional being who is whisked away by a mysterious visitor to The Land of Three Dimensions, an experience that forever alters his worldview.

Like the original, Ian Stewart’s commentary takes readers on a strange and wonderful journey. With clarity and wit, Stewart ...

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Flatland is a unique, delightful satire that has charmed readers for over a century. Published in 1884 by the English clergyman and headmaster Edwin A. Abbott, it is the fanciful tale of A. Square, a two-dimensional being who is whisked away by a mysterious visitor to The Land of Three Dimensions, an experience that forever alters his worldview.

Like the original, Ian Stewart’s commentary takes readers on a strange and wonderful journey. With clarity and wit, Stewart illuminates Abbott’s numerous Victorian references and touches on such diverse topics as ancient Babylon, Karl Marx, Mary Shelley’s Frankenstein, Mt. Everest, H.G. Wells, and phrenology. The Annotated Flatland makes fascinating connections between Flatland and Abbott’s era, resulting in a classic to rival Abbott’s own, and a book that will inspire and delight curious readers for generations to come.

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Editorial Reviews

From Barnes & Noble
The classic mathematical fantasy Flatland has been beloved by generations for both its wit as social satire and its instructiveness as a means of thinking about higher spacial dimensions. However, as with many classics, some of the social references need explaining as the years go by, not to mention the many mathematical concepts blended into the story. Here, Ian Stewart -- one of the best math popularizers and the author of the "updated" Flatterland -- provides an illuminating and entertaining annotation of this most famous two-dimensional world.
From The Critics
Rewarding...[Stewart] does an admirable job of expanding and updating the scientific ideas evoked by Flatland.
Enhances the original work for readers not acquainted with the life and times of the author and with Victorian England.
Boston Globe
Anyone with even a passing fancy for this discipline will find much that is instructive, enlightening, and engaging.
Science Fiction Chronicle
Chance not only to re-read one of the classics of this genre, but to read it in an entirely different way.
Publishers Weekly - Publisher's Weekly
In The Annotated Flatland: A Romance of Many Dimensions, Ian Stewart (Does God Play Dice?) introduces and explains Edwin A. Abbott's 1884 math-geek classic. Stewart, a mathematics professor at Britain's University of Warwick, discusses Abbott's milieu and friends (including George Eliot and H.G. Wells), Victorian literary conventions (e.g., his protagonist gains understanding of the three-dimensional world in a dream), the low social status of Flatlandian women, Abbott's class and political affiliations, and a host of other entertaining and enlightening tidbits. Photos and illus. ( Dec.) Copyright 2001 Cahners Business Information.
From The Critics
The scientific fantasy by English clergyman and headmaster Abbott (1838-1926) has reputably never been out of print since its appearance in 1884, introducing generations of undergraduates to concepts of the fourth dimension. Stewart (mathematics, U. of Warwick) finds that just below the plane where the story lies roils a biting satire of Victorian values and perhaps an allegory of a spiritual journey. His marginal paragraphs explore whatever realms he thinks readers may find interesting. He has also recently written Flatterland, a sequel set a century later. There is no index. Listed in Books for College Libraries, 3d. Ed.. Annotation c. Book News, Inc., Portland, OR
The Barnes & Noble Review

Only on the surface, so to speak, does Flatland belong to the arcane subgenre of mathematical fantasy. In fact, this harrowing memoir could be more properly grouped with such fictional representations of totalitarian mind control as Arthur Koestler's Darkness at Noon, George Orwell's Nineteen Eighty-Four, and Anthony Burgess's A Clockwork Orange. I'm only exaggerating a little.

In Flatland, the narrator, A. Square, describes a two- dimensional world, as smooth as a piece of paper, upon which dwells a civilization of geometrical shapes. Its citizens range from the triangular workers to the square-like middle class to the aristocratic polygons (who would like to think of themselves as perfect circles). All these are male. The female residents of Flatland, no matter what their class, are simply short straight lines.

In their planar realm, A. Square tells us, the Flatlanders can "move freely about, on or in the surface, but without the power of rising above or sinking below it, very much like shadows — only hard and with luminous edges." Note that word "hard." Any sort of sharp point, whether the apex of certain triangles or the end of a straight line, is a potentially deadly weapon, being able to pierce the sides of an innocent square or hexagon.

Even more dangerous, though, is the entire class of irregular figures, which lack proper angles and proportions, and are innately destined for moral delinquency and criminal behavior. As the exalted Pantocyclus affirmed, "Configuration makes the man; that if, for example, you are born an Isoceles with two uneven sides, you will assuredly go wrong unless you have them made even — for which purpose you must go to the Isoceles Hospital; similarly, if you are a Triangle, or Square, or even a Polygon, born with any Irregularity, you must be taken to one of the Regular Hospitals to have your disease cured, otherwise you will end your days in the State Prison or by the angle of the State Executioner." A. Square himself declares that he has "never known an Irregular who was not also what Nature evidently intended him to be — a hypocrite, a misanthropist, and, up to the limits of his power, a perpetrator of all manner of mischief."

While Flatland has long enjoyed a placid history, it has nonetheless weathered occasional crises. For instance, during the "Chromatic Sedition" an incendiary passion for color led to domestic violence and social chaos, culminating in a political movement demanding a more democratic government. Fortunately, such misguided idealists, these sponsors of a Universal Color Bill, were firmly dealt with by the high-ranking Circles and the country quickly returned to its monochromatic serenity, as each citizen was again taught to "attend to your Configuration" and not get too uppity.

Configuration, in fact, is destiny, though some people do speak of right and wrong "as if they believed that these names represented real existences, and that a human Figure is really capable of choosing between them." With women, that volatile and deadly sex — the needle-like creatures can inflict great injury and damage — there is, of course, a tendency among males "to speak of 'love,' 'duty,' 'right,' 'wrong,' 'pity,' 'hope,' and other irrational and emotional conceptions, which have no existence, and the fiction of which has no object except to control feminine exuberance; but among ourselves, and in our books, we have an entirely different vocabulary and I may almost say idiom. 'Love' then becomes 'the anticipation of benefit'; 'duty' becomes 'necessity' or 'fitness'; and other words are correspondingly transmuted."

Such double-speak, as Orwell would call it, reveals the misogyny and myriad inequities of a ruthlessly hidebound and hierarchical world, one oddly reminiscent of Victorian England. Still, it is the only world A. Square knows — until that fateful evening when he is visited by a being from another dimension.

"It was the last day of the 1999th year of our era. The pattering of the rain had long ago announced nightfall; and I was sitting in the company of my wife, musing on the events of the past and the prospects of the coming year, the coming century, the coming Millennium." While in this pensive mood, recalls A. Square, "Straightaway I became conscious of a Presence in the room, and a chilling breath thrilled through my very being." An intruder has simply appeared out of nowhere.

It is, in fact, a Sphere, perceived on Flatland's two-dimensional surface first as a point and then as a larger and larger circle, until it finally stops growing. (Were the Sphere to continue passing through Flatland's plane, the horizontal circle — which is all that A. Square can see — would gradually shrink until it disappeared.) Already bewildering, the situation grows frightening when this strange, probably insane creature begins to rave about an incomprehensible "Gospel of Three Dimensions." There is, in particular, some nonsense about "upward."

After several unsuccessful attempts to prove that there is more to the universe than just breadth and width, the Sphere furiously picks up our narrator and simply carries him aloft into Space. There, A. Square undergoes a transcendental out-of-body experience: "An unspeakable horror seized me. There was a darkness, then a dizzy, sickening sensation of sight that was not like seeing; I saw a Line that was no Line; Space that was not Space; I was myself, and not myself." From the vertiginous heights the unmoored Flatlander looks down into the rooms of his own two-dimensional pentagonal house. And slowly at first, but then with the enthusiasm of the newly converted, A. Square believes. He knows, he can actually see with his own eyes that there is a third dimension.

But, as in the way of disciples, A. Square also starts to grow a bit fanatical. Could there not be, in fact, a fourth dimension? he asks the Sphere. "As you yourself, superior to all Flatland forms, combine many Circles in One, so doubtless there is One above you who combines many Spheres in One Supreme Existence, surpassing even the Solids of Spaceland. And, even as we, who now in Space, look down on Flatland and see the insides of all things, so of a certainty there is yet above us some higher, purer region, 'some yet more spacious Space, some more dimensionable Dimensionality, from the vantage ground of which we shall look down together upon the revealed insides of Solid things."

The Sphere, as limited in his way as A. Square had once been, refuses to give credence to any dimensions beyond three. But, as our increasingly mystical narrator insists, "of a surety there is a Fourth Dimension," even if it is only one that can be perceived by "the inner eye of thought." At which point, the now thoroughly irate Sphere releases the Flatlander who falls back into his old world. Home again, A. Square attempts to explain his new understanding of three dimensions to a bright grandson, who thinks he must be joking. Meanwhile, the government — aware that the Millennium brings out dangerous quackery — has promulgated strict prohibitions against any theories or beliefs that might upset the complacency of the populace. So, one day without having done anything truly wrong, Josef K. — I mean A. Square — is arrested and sentenced to life in prison.

There, as the lonely years grind on, the once bright memory of a three-dimensional universe begins to fade. Hoping to preserve something of his revolutionary vision, A. Square sets down the memoirs we have just read, trusting that they "may find their way to the mind of humanity in Some Dimension, and may stir up a race of rebels who shall refuse to be confined to limited Dimensionality."

Since Flatland first appeared in 1884, it has gradually come to be seen as a minor masterpiece. Be assured: My summary describes just the major arc of the book and only hints at its ingenious detail, charm and, not least, sly satire. In these pages Edwin A. Abbott (1838–1926) carries on an unrelenting critique of class barriers, an implicit plea for women's rights modeled after Swift's ironic "Modest Proposal," and an impassioned defense of freedom of thought. Yet could Abbott also be hinting, as A. Square's conjectures seem to imply, that God and a spirit world exist in a "Fourth Dimension" beyond our limited senses?

Both a liberal clergyman and a distinguished teacher, Abbott — according to Ian Stewart in The Annotated Flatland — doesn't go that far. From all accounts, the author of Flatland was a very down-to-earth sort and typically viewed religion as a source of moral values in this world while shying away from any otherworldly speculations. Still, his book — written in the early days of nineteenth-century Spiritualism and rich with mystical language — does seem to point to a metaphysical, as well as mathematical, aspect to multiple dimensions.

That said, Flatland has always been regarded as an early attempt, a baby step, toward our current theorizing about the realm of hypercubes, space-time and superstrings. As A. Square exclaims: "In that blessed region of Four Dimensions, shall we linger at the threshold of the Fifth, and not enter therein? Ah, no! Let us rather resolve that our ambition shall soar with our corporal ascent. Then, yielding to our intellectual onset, the gates of the Sixth Dimension shall fly open; after that a Seventh and then an Eighth." And why stop there?

At least some of Abbott's ideas almost certainly derive from Charles H. Hinton (1853–1907), a fellow Brit who ended his life working at the Naval Observatory in Washington, D.C. Hinton's major works — including What Is the Fourth Dimension?, A Plane World, "Many Dimensions," and, late in his life, An Episode of Flatland — were first gathered together in two volumes titled Scientific Romances, a term that H. G. Wells and others would soon adopt. (The essays and stories may also be found in the Dover anthology Speculations on the Fourth Dimension: Selected Writings of Charles H. Hinton, edited by Rudolf V. B. Rucker.) Since Abbott and Hinton, writer-mathematicians have regularly revisited Flatland or imagined similarly constrained worlds. Outstanding examples include Dionys Burger's Sphereland, A. K. Dewdney's The Planiverse, Rudy Rucker's Spaceland, and Ian Stewart's Flatterland.

In The Time Machine, Wells hypothesized that duration was the fourth dimension, complementing height, width and breadth. I certainly grew up believing that. But today's theorists tend to imagine instead the never-ending implications of non- Euclidean space. In our digital age, we can sometimes even advance beyond what A. Square calls "the inner eye of thought" and actually model a tesseract rotating through four dimensions. It looks like a cube designed by M. C. Escher sliding around, in and through itself.

Fortunately, at least for those of us who never went beyond tenth- grade math, Edwin A. Abbott's Flatland is primarily a literary work, a critique of nineteenth-century mores coupled with a philosophical plea to think outside the box or, in this particular case, above the plane. As A. Square tells us, "To be self-contented is to be vile and ignorant, and . . . to aspire is better than to be blindly and impotently happy." Sapere aude!

Michael Dirda is a Pulitzer Prize–winning columnist for The Washington Post Book World. He is the author of the memoir An Open Book and several collections of essays, including Classics for Pleasure. His latest book, On Conan Doyle, has been published by Princeton University Press.

Reviewer: Michael Dirda

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Product Details

  • ISBN-13: 9780465011230
  • Publisher: Basic Books
  • Publication date: 7/7/2008
  • Edition description: Reprint
  • Pages: 272
  • Sales rank: 380,701
  • Product dimensions: 7.40 (w) x 9.10 (h) x 0.80 (d)

Meet the Author

Edwin A. Abbott was born in London on December 20, 1838. Educated in St. John’s College in Cambridge, he was ordained in 1862 and three years later was appointed headmaster of the City of London School, where he served until 1889. Abbott wrote over fifty books, most of them scholarly works. He died in Hampstead on October 12, 1926.

Ian Stewart is Professor of Mathematics at the University of Warwick and Director of its Mathematics Awareness Centre. His many books include Why Beauty Is Truth, Nature’s Numbers, Does God Play Dice?, and Letters to a Young Mathematician. He lives in Warwick, England.

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Read an Excerpt

What is Flatland, and why should it be annotated?

Flatland is a work of scientific fantasy written by the English clergyman and headmaster Edwin Abbott Abbott and published in 1884. It is a charming, slightly pedestrian tale of imaginary beings: polygons who live in the two-dimensional universe of the Euclidean plane. Just below the surface, though, it is a biting satire on Victorian values- especially as regards women and social status- and an accomplished and original piece of scientific popularization about the fourth dimension. And, perhaps, an allegory of a spiritual journey.

It deserves to be annotated because- just as Euclid's plane is embedded in the surrounding richness of three-dimensional space- so Flatland is embedded in rich veins of history and science. Investigating these surroundings have led me to such diverse items as The Good Grave Guide to Hampstead Cemetery, phrenology, ancient Babylon, Karl Marx, the suffragettes, the Indian Mutiny of 1857, the Gregorian calendar, Mount Everest, the mathematician George Boole and his five remarkable daughters, the Voynich manuscript, H.G.Wells's The Time Machine, the 'scientific romances' of Charles Hinton, spiritualism, and Mary Shelley's Frankenstein.

I first read Flatland in 1963 as an undergraduate newly arrived at the University of Cambridge (England) to study mathematics. I enjoyed it, added it to my science fiction collection- it fits a broad definition of the genre- and forgot about it. Years later, I re-read it, and the idea of a modern sequel began to form in my mind. I wasn't the first person to think of that, or to do it, but recent advances in science and mathematics made it easy for me to invent a new scenario. The result was Flatterland, whose genesis I have related in its own preface. While Flatterland was being readied for publication, my editor Amanda Cook at Perseus Books came up with the idea of a companion volume- a republication of the original Flatland, but with added annotations.

I started with the idea that I would focus mainly on the mathematical concepts that Flatland uses or alludes to, so the writing ought to be simple and straightforward. But when I started looking into the life and times of its author, his associates, and the scientific and cultural influences that led up to the writing of Abbott's unique book, I was hooked. My amateur-historian investigations led into ever more fascinating byways of Victorian England and America, and I began to rediscover many things that are no doubt well known to Abbott scholars, but are far from common currency.

At first, I was concerned that I might not be able to lay hands on the necessary material. But a glance at one of the more obvious and accessible sources- Abbott's entry in the Dictionary of National Biography- brought to light a curious coincidence. Abbott's professional life revolved around the City of London School- he is its most famous Headmaster, a post that he took up in 1865. Now, there exists in London an institution called Gresham College. It was founded in 1597 with a legacy from Sir Thomas Gresham (1518/19-1579), originator of Gresham's Law ('bad money drives out good') and founder of the Royal Exchange in 1566-68. Gresham was a philanthropist, and his will instructed the Mercer's Company ( one of the livery companies created by King Richard II) and the City of London to 'permit and suffer seven persons by them from time to time to be elected and appointed... sufficiently learned to read... seven lectures.' The College has no students- only the general public- and until recently it appointed seven professors, in Astronomy, Divinity, Geometry, Law, Music, Physic, and Rhetoric. To these have been added an eighth: Commerce.

Anyway, between 1994 and 1998 I was the Gresham Professor of Geometry. The first such was Henry Briggs (1561-1630, appointed in 1596), inventor of 'natural' logarithms; others include Isaac Barrow (1630-1677, appointed 1662), who recognized that differentiation and integration, the two basic operations of calculus, are mutually inverse; Robert Hooke (1635-1703, appointed 1664), who discovered the law of elasticity named after him, suggested that Jupiter rotates, and laid the early foundations of crystallography; and Karl Pearson (1857-1936, appointed 1890), one of the founders of statistics. The College is still funded by the Mercer's Company and the City of London. The City also has a long interest in the City of London School, and as a Gresham Professor I had lectured at its sister insitution, the City of London School for Girls (founded 1894). So I had an easy introduction to Abbott's professional home. The City of London School had been badly damaged in World War II and had moved to new premises; I wrote a letter asking whether, despite that, it still had any Abbott documents, pictures, or other information. In response, Head Porter Barry Darling sent me a history of the School (City of London School by A.E. Douglas-Smith) which contained extensive information about Abbott, and invited me to visit and look through the School's archives.

A week later, I was ushered into a small, rather disorganized, room, lined with shelves and crammed to the celing with old books, magazines, photographs, and bound volumes of letters. On the top shelf, tucked away in one corner, was an almost complete collection of Abbott's books, including a rare first edition of Flatland. (I knew that a second revised edition had followed hard on the heels of the first, because the preface to that second edition says so. What had he changed? Now I could find out.) I went away with a stack of photocopies, and three framed photographs of Abbott at various stages of his career, loaned to me for copying. I had his obituary in the School magazine, a review of Flatland in the same journal, samples from geometry texts used by the School in Abbott's day, extracts from his publications- even a copy of his letter of resignation.

Of course, the Abbott scholars had been there before me- but even so, I felt like Sherlock Holmes hot on the trail of Moriarty.

Other sources now came into their own. I could surf the net, because I had some idea of what to look for. Entering 'Flatland' into Yahoo turned up thousands of sites about off-road vehicles, but 'Edwin Abbott Abbott' was much more helpful. An article by Thomas Banchoff (the leading expert on Abbott, currently working on a biography) explained the crucial connection to Charles Howard Hinton, whose wild but ingenious speculations about the fourth dimension undoubtedly inspired Abbott's fable. A conversation with a colleague, Bruce Westbury, in the Warwick University Mathematics Common Room, put me on to the four-dimensional mathematics of Alicia Stott Boole. As a science fiction aficionado I already knew that H.G. Wells had used four-dimensional geometry in The Time Machine; now the web turned up a brilliant historical survey by the science fiction author Stephen Baxter, and another by James Beichler, linking Wells to Hinton. Rudy Rucker's The Fourth Dimension opened up dozens of further leads... and so it went.

What is the purpose of an annotated edition? Martin Gardener, in the classic among all such books, The Annotated Alice: the Definitive Edition, says: 'I see no reason why annotators should not use their notes for saying anything they please if they think it will be of interest, or at least amusing.' Which is exactly my feeling. So I have pursued trails wherever they led, and reported anything that seems to fit the overall story. The most extreme case is a series of associations that links Abbott to Mary and Percy Bysshe Shelley, Lord Byron, Augusta Ada Lovelace, Charles Babbage, Sir Edward ffrench Bromhead, George Boole, Mary Boole, Charles Howard Hinton, Alicia Stott Boole, and the Dutch mathematician Peiter Schoute- with a side branch to the science fiction writer H G. Wells.

Something important emerges from such chains of connections: Victorian England was a tightly-knit society. The intellectuals all knew each other socially, traded and stole each other's ideas, and married each other's sons and daughters. It was an exciting period of scientific and artistic discovery, for the staid and repressive attitudes of the Victorian era were sowing the seeds of their own destruction. Abbott knew many of these people- most of them more colourful than he was-and they influenced his thinking in profound ways. It's been fun ferreting out their stories. For example, along the way I discovered that I once held the same job as Abbott's mathematics teacher- but 146 years later.

As a strictly amateur historian, I know that I will have made some mistakes, misinterpreted some events, or left out some vital items of information that are well known to all the experts. This happens with any book: it is virtually impossible to track down all the relevant documentation, all the names, all the dates. (Oh, yes: I've been moderately obsessive about giving dates for almost everything and everybody- except for minor figures- because the timing is so crucial in this kind of investigation. When I don't know a date- and sometimes when nobody does- I've either put a question mark or omitted it.) So I invite anyone who has constructive criticisms, useful observations, wild theories, or new information, to e-mail them to me. I can't promise you a reply- though I'll do my best- but I do promise that I'll take note of anything I think is interesting. And when (I'm sure it will be 'when') it is time to prepare a new edition, I'll make the necessary changes.

However, I also promise that nearly everything I say is true- or, if it's an opinion, plausible. I've tried to do my historical and scientific homework. I hope you'll come to agree with me that there is so much more to Flatland than meets the eye, even if it is a world of only two dimensions.

Coventry, May 2001
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Table of Contents

Preface     ix
Introduction     xiii
The Annotated Flatland     1
Introduction   William Garnett     9
Preface to the Second and Revised Edition, 1884$dthe Editor     21
This World     31
Of the Nature of Flatland     33
Of the Climate and Houses in Flatland     38
Concerning the Inhabitants of Flatland     43
Concerning the Women     49
Of our Methods of Recognizing one another     57
Of Recognition by Sight     65
Concerning Irregular Figures     73
Of the Ancient Practice of Painting     79
Of the Universal Colour Bill     84
Of the Suppression of the Chromatic Sedition     90
Concerning our Priests     97
Of the Doctrine of our Priests     101
Other Worlds     111
How I had a Vision of Lineland     113
How I vainly tried to explain the nature of Flatland     121
Concerning a Stranger from Spaceland     129
How the Stranger vainly endeavoured to reveal to me in words the mysteries of Spaceland     136
How the Sphere, having in vain tried words, resorted to deeds     149
How I came to Spaceland, and what I sawthere     155
How, though the Sphere shewed me other mysteries of Spaceland, I still desired more; and what came of it     164
How the Sphere encouraged me in a Vision     181
How I tried to teach the Theory of Three Dimensions to my Grandson, and with what success     187
How I then tried to diffuse the Theory of Three Dimensions by other means, and of the result     191
The Fourth Dimension in Mathematics     203
Bibliography of Edwin Abbott Abbott     231
Bibliography of Charles Howard Hinton     235
Sources and References     237
Further Reading     239
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Customer Reviews

Average Rating 3.5
( 2 )
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  • Anonymous

    Posted March 27, 2013

    Annotations distract from a brilliant story

    I actually read this book years ago and decided to read it again, annotated. The book itself is a thoughtful engagement of philosophy and a reflection of Victorian ideologies and society, and I was hoping the annotations would provide more insight into that culture.

    The annotator, however, decides to spend more time on mathematics (this book is obviously geared towards a mathematical audience, do we really need further lecturing on geometry?) along with frequent pointless ramblings that quickly become disruptive, tiresome, and boring. He does make a few points on the relation to Victorian Culture, but they are few and far between. I most definitely didn't get what I had hoped for out of the annotations.

    Flatland itself is a great book, but skip the annotated version.

    0 out of 1 people found this review helpful.

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  • Posted June 3, 2011

    The Essential Guide

    I can not imagine reading Flatland without Stewart's commentary on the time of XIX Centuary England, the Life of Abbott, and the mathematics and symbolism of a seemingly simple story. Along the lines of Martin Gardner's commentary of Alice in Wonderland, Abbott was genius in working meaning into everything about flatland necessitating a ponderous commentary to flesh it all out.

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